Talk:List of pitch intervals

Too many redlinks

Generally, something should only be linked if it makes sense to write an article about it. What material could be used in an article on the septimal minor third? It's a good interval, even one of my favorites, but that doesn't mean it's worthy of its own article. —Keenan Pepper 05:06, 8 December 2005 (UTC)

Flip the ratios

As suggested, indeed we should flip all ratios. See for example in the article Interval (music). The ratios should have values over 1. For example a fifth is 3:2. It is more common to think in frequencies than in string or tube length. −Woodstone 21:34, 20 February 2007 (UTC)

Makes sense to me, especially for the sake of consistency with the other article. Rigadoun (talk) 16:16, 21 February 2007 (UTC)

Yes this is all backwards, if I am not mistaken. An octave being "twice" the pitch should be 2:1, and intervals between unison and octave should be between 1:1 and 2:1, no? Also, the interval 4:7 (apparently 7:4) is listed as the "seventh harmonic" -- but strictly speaking, isn't the seventh harmonic 7:1? I'm also wondering if 7:4 is also called the "harmonic seventh" (perhaps the words got reversed?), the "blues seventh", and/or the "barbershop seventh"? Pfly 06:51, 9 March 2007 (UTC)

Well, it wasn't exactly wrong, as getting the intervals backwards just makes them descending instead of ascending, or in terms of string lengths and so forth, but having them greater than one is more common. Anyway, I've flipped them all for the sake of famiiliarity and consistency with most description. Yes, strictly speaking I suppose the seventh harmonic is 7:1, but it's more often thought of as being in the same octave as the nearby minor seventh (dividing by four just moves the pitch down two octaves). I'm not really sure abut the nomenclature, though. Rigadoun (talk) 18:07, 9 March 2007 (UTC)

True, it wasn't wrong, although the cents would be negative :) ...thanks for flipping. As for 7:4, the Scala's list of pitches (which seems to be a widely used standard) calls it "harmonic seventh",[1], as does Mathieu's book "Harmonic Experience", as I read last night. So I think I'll flip the words. It appears that Harmonic seventh was once a page here, or at least there are some redlinks to it. A quick google seems to imply that there is some mild controversy about 7:4, but the sources I am most familiar with seem comfortable enough calling it "blue seventh" as well as "harmonic seventh". But perhaps "blue notes" are inherently not well defined. Thanks for the response! Pfly 18:50, 9 March 2007 (UTC)

The reciprocal ratios are essentially the same in concert practice. If you hit two notes that are a fifth apart, then one of them is ten, and the other is fifteen. Making the ratio higher than one is nothing but a tradition. That is why colons are used instead of slashes, because the division is not done. So, do not edit war over this one. 216.234.170.74 (talk) 19:36, 3 September 2009 (UTC)

Distance in semitones

I think it might be much more usefull if at least the musical intervals in the equal temperament would also include their width in semitones. —The preceding unsigned comment was added by 86.121.142.63 (talk) 13:29, 20 March 2007 (UTC).

It's already there. The first column states the interval in cents. One hundred cent is a semitone. So where it says 500 cents, you can read 5 semitones. −Woodstone 15:02, 20 March 2007 (UTC)

Factorization column

In the table of intervals may be column named Factorization to indicate intervals of Just Intonations and its limit for them. If somebody have another opinion, please discuss. Commator (talk) 18:31, 13 December 2007 (UTC)

When writing tunes into synthesizers written for this stuff, factoring is very useful. If that is your work, then thanks. 216.234.170.74 (talk) —Preceding undated comment added 19:39, 3 September 2009 (UTC).

Unreferenced material

Intervals such as "ragisma" and "breedsma" need to referenced (dictionaries, texts, peer-reviewed journals). Frank Zamjatin (talk) 12:40, 15 April 2008 (UTC)

Midi microtuning

The "audio files" being added are actually ".mid" files, or "midi" files. In that case, in order to have any intervals other than equal temperament these files need to employ microtuning. How certain are you that most computer's midi player support this not too common feature? −Woodstone (talk) 22:05, 8 August 2008 (UTC)

Wikipedia:Media help. Hyacinth (talk) 00:48, 9 August 2008 (UTC)

That really didn't answer the question. Not all PC MIDI synths support the MIDI microtonal tuning standard. The TiMIDIty softsynth does, but consumer-grade PC soundcards are anyone's guess. — Gwalla | Talk 01:04, 9 August 2008 (UTC)

Test your midi pitchbend range. Hyacinth (talk) 01:32, 9 August 2008 (UTC)

So everybody who wants to use your MIDI samples needs to test their software first? I'm not sure I get your point. A response with your own words in actual sentences might help, because right now it looks like you're trying to dodge. Plus, are you using pitchbend or the microtuning standard? They're different things! — Gwalla | Talk 05:36, 9 August 2008 (UTC)

I went ahead and did the test and it worked. I have one of the most generic Realtek integrated soundboards. SharkD (talk) 01:09, 17 August 2008 (UTC)

Okay, Woodstone: What would you prefer we label the midi files, if not audio? How certain are you that most computers do not use this supposedly "not too common" feature?

Gwalla: Apparently everyone who uses audio samples on Wikipedia may need to test their software first, as you would see if you checked Wikipedia's media help, as all media directs one to do if one has trouble. I assume that this is a feature of Wikipedia's media help because it is a standard feature of technical assistance, and not unreasonable. Hyacinth (talk) 06:31, 9 August 2008 (UTC)

Lastly, I'm using pitchbend controls. Why? Hyacinth (talk) 06:31, 9 August 2008 (UTC)

It's not a question of what to label the MIDI files. It's a question of whether MIDI files are appropriate for this use, or if Ogg should be used instead. But if you're using pitch bend messages, those should be supported by pretty much everything, so there shouldn't be a problem, I think. — Gwalla | Talk 03:19, 10 August 2008 (UTC)

Audio examples

I think MIDI is better because it's a "mathematical" format, just like vector images are better than Raster images. SharkD (talk) 01:04, 17 August 2008 (UTC)

On the linux box here, I find .ogg more convenient to play, followed by .mp3. I don't even bother trying to play MIDI files. Only use I have for them is converting to standard score notation.

Vector images have advantages over raster images for many uses, but they are not "better" in all cases. Using MIDI for showing intervals as played on real instruments (whose timbre comes from partials that may be more or less harmonically related to the fundamental note, and so will affect the sound of the interval when both notes are played together) makes about as much sense as using a vocoder to show what any individual person's voice sounds like. Add uncertainty about accuracy of decoding at the receiving end, and most usefulness disappears. __Just plain Bill (talk) 02:44, 19 August 2008 (UTC)

Must instruments be used at all? Why not play back pure tones? SharkD (talk) 04:54, 19 August 2008 (UTC)

Unfortunately Wikipedia:Manual of Style (music) gives us no guidance as to general priorities in audio examples, though they are often requested on talk pages with {{Audio requested}}, and neither does Wikipedia:WikiProject Music/MUSTARD/Sounds and multimedia.

Just plain Bill has implied some general priorities for audio examples for this article. I would suggest using the same sound for all intervals, so that one is not distracted by novel timbres. We could also try using two different instruments/sounds in each example, to distinguish each note of the dyad. An instrument/sound with sustain seems like a must. I would also suggest having them all based on the same note, though this would not keep them in the average register it would keep the bottom note always in the same. Hyacinth (talk) 06:48, 19 August 2008 (UTC)

Pure tones will be a good start for simple examples. Can bump into some oddities without going to novel timbres: the partials of a piano string are slightly inharmonic, leading to the need for stretch tuning. Played on different instruments, perhaps in different registers, the same interval may have a slightly different flavor. Going a bit further, I believe dense multi-voice harmonies will be poorly represented by pure tones. __Just plain Bill (talk) 11:33, 19 August 2008 (UTC)

The audio example for Just Perfect Fourth is incorrect. It sounds like some sort of major third.--Mark Pfannschmidt (talk) 15:06, 17 September 2009 (UTC)

Color coding

I suggest color-coding the table items depending on whether they are even tempered, just, pythagorean, etc. SharkD (talk) 22:57, 9 August 2008 (UTC)

Would these provide more information, such as revealing a pattern? Hyacinth (talk) 10:48, 10 August 2008 (UTC)

It would at least allow readers to sort the table items into groups in external software. No, I don't expect readers to see patterns at a simple glance. I would personally like to know which items belong to the same tuning systems (it's not obvious to me, a novice), and adding this sort of information here would eliminate the need to dredge through all the connected articles. Adding another column to the table is a possible alternative to color-coding. There is a "sortable" class of tables which would allow readers to sort the table based on these factors. The disadvantage of adding another column, of course, is that it takes up more space. SharkD (talk) 00:42, 11 August 2008 (UTC)

Wouldn't this be redundant? For example, fourteen of the intervals are already labeled as just. Seven are already labeled Pythagorean. Hyacinth (talk) 03:24, 11 August 2008 (UTC)

One problem is that some pythagorean intervals are also (5-limit) just intervals. The unison and octave are in every tuning. So they would have to have multiple colors. Also in just tuning, for some notes there are multiple choices, so picking out a matching set would not be easily achieved. We could experiment somewhat to see how it turns out. −Woodstone (talk) 07:11, 11 August 2008 (UTC)

I have been thinking about this and have an idea. How about adding a column with the factorisations of all the just intervals. So for example 10:9 = 2*5/3*3. Then the color could indicate the highest prime factor occurring. To complete this, there would be colors as follows:

• color1: 12-tone equal tempered (including unison and octave)
• color2: other non-fractional tones
• color3: pythagorean (3-limit, but not unison, octave)
• color4: 5-limit intervals (but not 3-limit)
• color5: 7-limit intervals (but not 5-limit)
• etc

I can work on it when I have time. What do you think? −Woodstone (talk) 09:26, 11 August 2008 (UTC)

I have completed the color coding as proposed above. If I may say so: it looks good. It actually brings out more pattern than I expected. −Woodstone (talk) 18:44, 11 August 2008 (UTC)

A couple of comments: the color-coding could be limited to the "Name(s)" column. I think that would be less distracting. The "Ratio" and "Ratio factors" columns could possibly be merged, or the factors could be turned into <ref>s and displayed in a "Notes" section at the bottom of the article. Turning the table into a sortable one as I described earlier, might be a good idea, but I'm not 100% sure.

Also, as I understand the table, the 13-limit just intonation includes the 11, 7, 5 and 3 limit intonations as well but aren't necessarily marked as such? I have to admit I'm a bit of a novice, so I'm not quite sure. SharkD (talk) 22:20, 11 August 2008 (UTC)

The labels for the equal tempered and Pythagorean (pink and yellow) tunings are clear, but I suspect the labels for the remaining tunings are a bit terse. Do these tunings have full, long names? SharkD (talk) 02:43, 12 August 2008 (UTC)

The factors play a role in judging the nature of the interval and are as such informational to the reader. The total fraction is less important, but easier to recognise for many people. Converting either to footnotes would create a very long list and would make it difficult to relate numbers to the tones. So I think both columns have a place in the table.

Choice of color and columns to be colored is a matter of taste. I experimented and to me this one looks best.

Yes, all intervals in "limit" tuning of course include the ones with a lower limit. To express that in the table would require 5 (or 6 for 2-limit or 7 including equal temperament) columns with different coloring. I'm afraid that would make it too messy.

Perhaps we could bring up the notes at the bottom, because they explain the terminology.

−Woodstone (talk) 07:29, 12 August 2008 (UTC)

I've been thinking about it further, and placing the factorized ratios in parentheses next to the simplified versions might be an option. For instance, 2401 : 2400 (74 : 25·3·52). SharkD (talk) 00:53, 13 August 2008 (UTC)

I notice that there are several "lines of symmetry" in the table surrounding the equal-tempered intervals. What I mean is that an equal-tempered interval may be surrounded by two 5-limit intervals, which are surrounded by two 3-limit intervals, which are surrounded by two 7-limit intervals, and so on. The undecimal neutral third is a good example of this, as it lies in the middle of several other major and minor thirds. Do you think we can draw attention to this pattern in any way? Maybe, instead of different hues we can have gradations of the same hue, from light to dark. On the other hand, it might be better to leave the table as it is and highlight this phenomenon in the diagram image instead. SharkD (talk) 21:47, 12 August 2008 (UTC)

I merged two of the columns into two others in order to reduce the width of the table and leave room for the possible future addition of more columns. SharkD (talk) 01:09, 16 August 2008 (UTC)

Merging the two ratio columns into one is not an improvement. It makes it difficult to see either of them in relation to their companions in the column. Also I kind of liked the narrowness of the "limit" columns, because there are so many of them. −Woodstone (talk) 11:58, 16 August 2008 (UTC)

I agree merging the column makes it harder to compare values. However, in essence both columns are the same (but on another scale), and I therefore felt the information to be not different enough to warrant being separated and taking up additional space. As for the narrowness of the columns, this will reoccur once more columns/tunings are added. SharkD (talk) 00:17, 17 August 2008 (UTC)

Intervals

I noticed that one of the Pythagorean intervals is missing from this list. Specifically, the interval(s) near 600 cents from the table found in Pythagorean tuning. However, there are two values listed there: 588.27 cents and 611.73 cents. Which of the two should we use? SharkD (talk) 21:34, 12 August 2008 (UTC) Also, half of the values for the Pythagorean tuning correspond to frequencies that have been tuned down from unison as opposed to tuned up, resulting in discrepancies equal to the Pythagorean comma. Is this OK, or should we only tune up or only tune down? SharkD (talk) 21:39, 12 August 2008 (UTC)

I've been reading Just Intonation Explained, and it says, "The smaller the numbers in an interval's ratio, the more consonant (sweet-sounding) it is, and the more useful it is for purposes of musical intelligibility." Does this imply that sticking unison smack-dab in the middle is the optimal solution? SharkD (talk) 00:53, 13 August 2008 (UTC)

I looked through the article Minor second and couldn't find any references to the ratio, 27:25 or "Maximus". Is this maybe an error in the table? SharkD (talk) 23:30, 12 August 2008 (UTC)

24-equal temperment

Should the equal tempered quarter tone, neutral second, neutral third, neutral sixth and neutral seventh be re-labeled as belonging to 24-TET? We could label the column "Q", for "quarter tone". SharkD (talk) 22:13, 14 August 2008 (UTC)

Sure, don't see why not. Hyacinth (talk) 08:19, 16 August 2008 (UTC)

Done. SharkD (talk) 10:18, 16 August 2008 (UTC)

Frequency ratio

Is there an article or article section that specifically defines the term, "frequency/pitch ratio"? I couldn't find it in Frequency or Pitch (music). I would like to link to it in the column header like "Cents" is. SharkD (talk) 22:13, 14 August 2008 (UTC)

That term doesn't currently appear in the article. A "frequency ratio" is a ratio ("a comparison of two numbers by division") where the two numbers are the frequencies of musical sounds or pitches. Hyacinth (talk) 08:18, 16 August 2008 (UTC)

Yeah, I know what it is. I just thought it weird that it's not defined in an article. SharkD (talk) 08:21, 16 August 2008 (UTC)

"Frequency" and "ratio" are both defined in articles. Hyacinth (talk) 08:36, 16 August 2008 (UTC)

Image

I was hoping to update the graph of intervals to include a few more limits, but am encountering issues with the software used to create the image. Namely, the text labels overlap in their current orientation and can't be rotated to be vertical. I can rotate the graph itself in order to accomadate the labels, but I don't find that to be a desirable solution. SharkD (talk) 00:32, 15 August 2008 (UTC)

As some other editors have pointed out earlier, the current table is rather non-informational. In essence both axes are the same (but on another scale). A better graph could be obtained by setting out the 12 tones horizontally and the relative frequency on a log scale vertically. Tones in the various tuning will stack up vertically for each note. The line connecting the equal temperament tones will be straight. The lines connecting pythagorean and 5-limit just tunings will snake around it. A few points on the curves wil coincide. −Woodstone (talk) 12:09, 16 August 2008 (UTC)

I can see how a logarithmic scale would be beneficial, as one would be able to fit many more intervals (especially beyond the first octave). However, I find the logarithmic scale a bit harder for general readers to comprehend (and therefore chartjunk), and intervals beyond the octave are rare. SharkD (talk) 00:13, 17 August 2008 (UTC)

• Chartjunk means unnecessary or distracting visual elements of a chart. Inappropriate choice of axis scale may be misleading or deceptive, or it may make the chart not very useful as a graphic presentation of information, but is not, strictly speaking, chartjunk.
• I don't see how a log scale would be of any use here. Cents for both, but let the horizontal axis run from zero to 1200, say, Let the vertical axis run from -50 cents to 50 cents or some such scale, with zero at the center. The horizontal axis could simply be "pitch numbers" from zero to 12, with the vertical stacks showing differences between similarly named pitches, with equal tempered frequencies at zero, and the other tunings snaking around it. Big job figuring it all out so it's clean and informative, certainly not a task I want to undertake.
• Intervals beyond an octave are commonly referred to in describing (or spelling) chords. __Just plain Bill (talk) 09:55, 22 August 2008 (UTC)

As discussed in Pitch (music), using a logarithmic scale is inappropriate for general readers. How are readers not going to be distracted by something that is not appropriate for their level? SharkD (talk) 11:06, 22 August 2008 (UTC)

Adding more equal temperaments

I've been looking at other equal temperaments, such as 19-TET. Should they also be listed here, or should they be restricted to List of meantone intervals? I don't want to duplicate content, but I also want the list to be comprehensive. SharkD (talk) 00:28, 17 August 2008 (UTC)

Stupid question

Is "List of musical intervals" really the proper title for this page? I mean, is there something particularly "musical" about the intervals (do they carry a tune?) Wouldn't it be better to rename the article to "List of music intervals" or "List of intervals in music"? SharkD (talk) 07:56, 18 August 2008 (UTC)

"Used in music" is a perfectly valid meaning of the term "musical". And since a tune is made out of them, I suppose you could say that they can carry a tune. — Gwalla | Talk 18:14, 18 August 2008 (UTC)

It still sounds odd to me. I think "List of intervals in music" would be better. SharkD (talk) 17:08, 20 August 2008 (UTC)

By any chance are you thinking of "musical intervals" in the sense of "musically pleasing intervals" or consonant intervals? I'd have a hard time calling a musical instrument an "instrument used in music," for example. By the way, carrying a tune is just one part of music; Aside from melody, there's also articulation, ornamentation, rhythm, harmony, sheets of sound, and who knows what else. __Just plain Bill (talk) 17:47, 20 August 2008 (UTC)

Yes, that's the ambiguity I was hoping to clear up. Note that I can recall periodic renamings of categories for similar reasons, for example Wikipedia:Categories for discussion/Log/2008 July 31#Category:Bridges and tunnels that are Registered Historic Places, Wikipedia:Categories for discussion/Log/2008 February 17#Category:Cycle races in... and Wikipedia:Categories for discussion/Log/2008 February 17#Category:US Maoist or Maoist sympathizing parties. SharkD (talk) 01:00, 22 August 2008 (UTC)

Intervals are basic musical items, pretty close to the foundation of music.

The OED's topmost definition of "musical" is "of or belonging to music" which is how it's used here. If someone doesn't get that right away, looking at the list itself may clear things up for them. I see no reason to use more awkward language to cater to some subset of readers who just might be unfamiliar with standard English usage; exposure to standard usage is an effective way for them to get familiar with it.

Who is going to say which intervals are more euphonious or pleasing or beautiful than others? For just one example, a minor second may give a sense of completion, when used as a step from leading tone to tonic. The same interval may not be so pleasant when a kitten steps on two adjacent piano keys, but it's still a musical interval, not a spacing or timing interval. __Just plain Bill (talk) 03:52, 22 August 2008 (UTC)

Yet, there are multiple definitions[2], and changing the adjective to a noun is meant to clear up this confusion, just as changing "Spanish" to "in Spain"[3] is. As for standard usage, maybe usage is incorrect in this case. Also, you can take Pitch (music) as another example of this. It isn't named Pitch (musical), it is named Pitch (music). SharkD (talk) 10:58, 22 August 2008 (UTC)

Usage is usage, it's how folks say what they mean to say so others will understand. Saying things like "maybe usage is incorrect" tells me that you might be interested in inventing your own brand of English. I don't have a problem with that at all, it happens all the time, but you need to know that it is ineffective unless others go along with it, understand it and adopt it. I don't see that happening here. Sorry if that seems harsh, but its the way I've got to call this one. __Just plain Bill (talk) 11:22, 22 August 2008 (UTC)

I assume this is not an aesthetic argument but a question regarding grammar.

This page consists of a list of intervals in isolation, none of which are actually being used in music in the immediate context of the list, they are simply being listed. Thus one may assume that "List of intervals potentially used in music and music theory" may be the most clear.

"Music" is a noun and thus "music intervals" would be inappropriate. "Musical" is an adjective which means "of or relating to music" [4], making it clear and accurate and making "in music" unnecessary. Hyacinth (talk) 05:04, 22 August 2008 (UTC)

Actually, English lets nouns modify other nouns, e.g. "cheese sandwich." Nobody says "cheesy sandwich" (unless they're talking about different degrees of cheesiness, I suppose.) Still, "musical intervals" strikes me as what cognizant English speakers would call the items on this list. __Just plain Bill (talk) 09:36, 22 August 2008 (UTC)

Please what? End of original comment by User:Hyacinth Thank you. Comment added by Just plain Bill

Yes, we could have "a list of cheese intervals". Hyacinth (talk) 02:25, 23 August 2008 (UTC)

I'll accept you arguing over nothing, but you may not add comments to my own. Hyacinth (talk) 02:54, 23 August 2008 (UTC)

I was asking for an end to pointless argument there, couldn't you tell? Too late to forbid, you saw your answer. You did ask... __Just plain Bill (talk) 03:42, 23 August 2008 (UTC)

I could not tell as I still haven't seen my answer, but I'll agree that I overreacted. Why can't we have a "list of musicy intervals"? Hyacinth (talk) 15:14, 23 August 2008 (UTC)

Umm, because it would conflict with the "list of musically intervals?" The original "oh please" had to to with a different line of this discussion, and was not pointed at you. The answer is still, "thank you!" __Just plain Bill (talk) 15:34, 23 August 2008 (UTC)

Adding audio files?

I'm trying to add links to audio files on list of meantone intervals just like this page. Am I doing it correctly? I need help to add the others.--206.248.172.247 (talk) 22:20, 10 November 2008 (UTC)

Uh, no. What you did doesn't really make sense. You added links to some audio files of intervals in equal temperament, but the whole point of that article is that these interval names can refer to different exact intervals in different temperaments. For example, the phrase "minor second" can refer to a 100-cent interval in 12-EDO, or a 126-cent interval in 19-EDO. So should the audio example be a 100-cent interval or a 126-cent interval? It's not really clear which one you should use. —Keenan Pepper 02:24, 11 November 2008 (UTC)

Prime vs odd limit

You use prime limit and ignore odd limit, yet you say "The lower the number, the more consonant the interval is considered to be". This is not correct for prime limit. See http://organicdesign.org/peterson/tuning/consonance.html for example. 243:128 has a prime limit of 3, but is not consonant at all. (243:128 = 3^5:2^7) It has an odd limit of 243, though, which indicates its dissonance. —Preceding unsigned comment added by 96.224.65.203 (talk) 22:09, 25 November 2008 (UTC)

I agree with the rebuttal, and in a different vein. For example, 3:2 is sometimes called the dominant harmony. I liked it in my own voice the first time I heard it, for entire songs. Other people did not like it that much. So, I will delete the statement in favour of what people might glean, and this is that perfect ratios *tend* to be more consonant than major ratios, which are usually more consonant than minor ratios. For that matter, I do not like using "consonant" in this context, because the first book I read on the subject favoured "assonant". Too much dominant harmony makes you sound assonant. 216.234.170.74 (talk) 19:58, 3 September 2009 (UTC)

Secor

I found the ratio for the Secor, (18/5)^(1/19), at http://tonalsoft.com/enc/s/secor.aspx. --Glenn L (talk) 00:41, 21 October 2009 (UTC)

Mean tone

Some information on meantone temperament was added. I have some doubts about how that fits in this table. The values in the column "ratio" so far have all been exact values. The ones now added for meantone are approximations. The definition of the equal and "limit" tunings are very clear and so can easily be marked in their own trailing columns, but there are many meantone systems and the correponding marking column gives a very hazy and incomplete view on that collection. −Woodstone (talk) 08:56, 21 March 2010 (UTC)

The values for meantone intervals in the ratio column may simply be removed if they bother you, but I'm not sure why they do. They definitions may be clear, though that may depend on one's level of knowledge. I assume you mean that since you find the definitions clear you find just and 12-TET intervals easy to mark (rather than definitions). Did you notice that there are many systems of equal temperament, besides 12, and that we include those in the table? We simple don't mark them. Hyacinth (talk) 11:30, 21 March 2010 (UTC)

I did notice the other equal temperaments and agree that they should not be marked in a separate column, since they are not prevalent enough to merit that. I understand that each of the meantone systems has well determined intervals, but it goes above my knowledge to calculate these exact values. If you are able to work out the exact value of the meantone intervals that would be great. Having approximate values sort of negates the underlying exact nature of the definitions. −Woodstone (talk) 11:48, 21 March 2010 (UTC)

Definition and "exact" values added. Hyacinth (talk) 13:38, 21 March 2010 (UTC)

I think where you "add" fractions like (81:80)^(1/4), you actually mean multiplying the ratio by that amount. I checked some with the given cents and it matches. −Woodstone (talk) 14:53, 21 March 2010 (UTC)

SVG file wrong

Personally, I believe what the svg file shows is wrong. For example, perfect fourth in Pythagorean tuning is about 498 cents but 500 in 12-TET. However, on the svg file, it shows the perfect fourth in 12-TET is shorter than Pythagorean tuning.

Mscdancer (talk) 15:34, 4 May 2013 (UTC)

Most often people create a chromatic scale in Pythagorean tuning by creating fifths both above and below the tonic or first note. The chromatic scale used in this picture only uses fifths above the starting point. Thus there is no perfect fourth, but rather an augmented third (enharmonic). While the Pythagorean perfect fourth is below 500 cents, the Pythagorean augmented third is above 500 cents.

Fifth	Letter	Interval	Just cents	ET cents	Relation to ET
0	C	Unison	0	0	Equal
1	G	Pythagorean perfect fifth	701.96	700	Above
2	D	Pythagorean major second	203.92	200	Above
3	A	Pythagorean major sixth	905.87	900	Above
4	E+	Pythagorean major third	407.82	400	Above
5	B+	Pythagorean major seventh	1109.78	1100	Above
6	F#++	Pythagorean augmented fourth	611.73	600	Above
7	C#++	Pythagorean augmented unison	113.69	100	Above
8	G#++	Pythagorean augmented fifth	815.64	800	Above
9	D#++	Pythagorean augmented second	317.60	300	Above
10	A#+++	Pythagorean augmented sixth	1019.55	1000	Above
11	E#+++	Pythagorean augmented third	521.51	500	Above
12	B#+++	Pythagorean augmented seventh	1223.46	1200	Above

Hyacinth (talk) 00:19, 17 August 2013 (UTC)

Sorry, I made the above image. What can I do to fix it? SharkD  Talk  00:26, 13 November 2016 (UTC)

Color codes

I propose the color coding below. Right now the colors are unobtrusive, but indistinguishable. Hyacinth (talk) 08:56, 13 May 2014 (UTC)

New Code	Compliant?	Legend (simplified)	Old Code
E	sort of	12TET.	E
Q	sort of	24TET.	Q
2	NO	2-limit.	2
3	NO	3-limit/Pythagorean.	3
5	sort of	5-limit/just.	5
7	YES	7-limit/ septimal.	7
11	sort of	11-limit.	11
13	NO	13-limit.	13
17	NO	17-limit.	17
19	NO	19-limit.	19
23	NO	23-limit.
29	NO	29-limit.
M	sort of	Meantone temperament.	M
U	YES	Measure/higher ET.	U
S	NA	Superparticular (no code).	S
H	YES	Higher harmonics.	H

As one climbs the limits one progresses through the rainbow.

12TET is made the same color (reddish) as 2-limit, since 2-limit only includes octaves.

24-TET is made a pinkish red.

Meantone is made an orange to match the color Pythagorean happens to be according to the rainbow pattern.

Units of measurement or equal temperament is made pink (12 is red, 24 is red/pink, > is pink).

Higher harmonics are gray.

Hyacinth (talk) 09:43, 13 May 2014 (UTC)

The highly saturated colors will make it difficult to read the text in the cells. Much fainter colours are required for legibility.

Why not use white or grey for equal temperament, which is the least colourful tuning system.

I do not understand why the legend has entries like " 2 3 5 7 11 13 17 19 = 19-limit (not 17-limit)". Would it not be simpler to have "7 = 7-limit". There are many rows that do not occur in the current legend list, such as "7 11 13 17 19". Any n-limit entry is also m-limit for m>n.

−Woodstone (talk) 12:02, 13 May 2014 (UTC)

There is a policy/guideline on legibility, specifically with colored backgrounds: Wikipedia:Manual of Style/Accessibility#Color points towards WCAG and provides a link to a test of contrast.

Hyacinth (talk) 01:45, 14 May 2014 (UTC)

D3D3D3 and 9ACD32 (yellow and light green) pass the contrast test as backgrounds for black text. Perhaps we should add more bright colors. Hyacinth (talk) 02:04, 14 May 2014 (UTC)

Inversion

How should this list treat inversions? Should the inversion of any interval present be included, or should only integer classes be included? In the second case there would be no intervals above 600 cents, in the latter case there may be many unnamed intervals. For example, should both 3/2 and 4/3 be included, and would that require that both 531441/524288 and 1048576/531441? Should we have only 4/3 and 531441/524288? Hyacinth (talk) 10:10, 13 May 2014 (UTC)

Musically, inversions are just as important. There are not really non-inverted ratios. Intervals come in pairs. In my opinion they should all be included. Limiting to one octave would be fine. −Woodstone (talk) 07:50, 14 May 2014 (UTC)

Title: pitch interval

See also: Talk:List of pitch intervals § Stupid question

Why is this list at "List of pitch intervals"? According to the edit summary when it was moved: "In ordinary terminology (e.g., in Grove), these are pitch intervals; 'musical intervals' are 'Major third' and the like". Major third is included in this list. According to its article, a pitch interval is, "the number of semitones that separates one pitch from another, upward or downward". This doesn't seem to describe this list. Hyacinth (talk) 00:52, 14 May 2014 (UTC)

Pitch interval is not the right name, since pitch is a physio-psychological measure. The entries in the table are frequency ratios. What is intended to be described here are intervals relevant in music theory. Musical intervals would be a suitable name. −Woodstone (talk) 07:55, 14 May 2014 (UTC)

I don't agree; the vast majority of these intervals have no relevance at all in music theory, except insofar as theory of tuning is considered part of theory of music. The functional musical intervals that are fundamental to music theory are listed at List of musical intervals. Justlettersandnumbers (talk) 09:50, 14 May 2014 (UTC)

That still means the article title should not contain "pitch intervals". Hyacinth (talk) 00:48, 17 May 2014 (UTC)

All these intervals, the harmonics excepted, are only relevant in musical context. The terminology is never used in any other field. Therefore I still propose to include the word music in the title. Perhaps "Intervals in music" is better than the old "Musical intervals" to which objections exist. −Woodstone (talk) 04:59, 17 May 2014 (UTC)

Redesign

The reason that there were separate columns for each of the "limit" tunings was the ability to sort by them. Sorting on the 5 column would bring all 5-limit tones together in the list, still sorted by frequency. With the current single column this is no longer the case. I think we should go back on this. −Woodstone (talk) 07:46, 14 May 2014 (UTC)

For the original version: 21:29, 13 May 2014‎

For the version with TET merged: 23:35, 13 May 2014‎

For the version with TET & Limits merged: 03:19, 14 May 2014‎

Hyacinth (talk) 20:00, 14 May 2014 (UTC)

Non-rational intervals

For the non-rational intervals, it does not make much sense to express them as an explicit ratio. I propose to remove all the superfluous ": 1" tails in expressions like "2^7/12 : 1". The ratio is just an irrational number. I have already removed the copies in the column "factors" as that is a meaningless breakdown for irrationals. −Woodstone (talk) 15:25, 22 July 2014 (UTC)

[copied from personal talk page]

You have removed prime factorization of non-rational numbers, but also rational powers of rational numbers can be factorized in an unique way as product of rational powers of different prime numbers. I have asked on Mathematics Stack Exchange. I don't have source, but two people have agreed that it is obvious. Can I put the factorizations back? And if you find most of my calculations too trivial, I think that quarter-comma meantone simplifications are useful, even if you do not want to call them factorization. BartekChom (talk) 16:53, 27 July 2014 (UTC)

In most of the cases the removed data in the "factors" column was exactly the same as in the "frequency ratio" column. So I think the table is in better shape without the redundancy. Compare several possible arrangements:

cents	ratio	factors	comment
386.31	5 : 4	5 : 22	current rational
400.00	24/12 : 1	21/3 : 1	before the contested edits
400.00	24/12 : 1		current irrational
400.00	24/12		(my preference)
400.00	21/3	(21/12)4	possible compromise
400.00	24/12	21/3	another compromise
400.00	21/3	24/12	still other compromise

But personally, I think it is better to reserve the factors column for harmonic relationships only. And as said, I propose to remove the meaningless ":1" after irrationals.

−Woodstone (talk) 08:50, 28 July 2014 (UTC)

The one before last does not make much sense for me as (2^1/12)⁴ is not unique - not very good as factorization. And I think that they are called ratios for some reason, so 2^4/12 : 1 is not worse than 2 : 1. But maybe (3 : 2)^1/9 without ":1" would be logical - I really think that Alpha scale and family are made of "parts" (roots) of ratios (if they are really so defined - {{Citation needed}} suggest that this interpretation is dubious). (2 : 1)^4/12 is analogical, but makes notation longer by "()".

1. Most important for me is that quarter-comma meantone simplifications are included somewhere in corresponding rows.
2. Besides I find my version ("before the contested edits") most useful, but current notation looks probably more serious (does no contain trivial calculations like 2^1/3=(2^1/12)⁴) and contains less original research, so I'm not going to quarrel.
3. I'm for removing ":1" in Alpha scale and family but mildly against removing it from edo intervals and for example savart. BartekChom (talk) 11:21, 28 July 2014 (UTC)

They are called ratio because they are a ratio of frequencies: the highest divided by the lowest of the two notes forming the interval. This ratio is a plain real number, most of them between 1 and 2. The uniqueness for non-rationals is questionable. The expression (2^1/12)⁴ shows the underlying structure of the interval: four identical factors. The structure of the quarter-comma mean tone intervals can also be expressed this way, using as few as possible different non-rational factors. −Woodstone (talk) 14:22, 28 July 2014 (UTC)

If you think so, I'm not going to quarrel, but why 2^4/12 : 1 is worse than 2 : 1? (2^1/12)⁴ shows that this interval equals four semitones, and this is its construction in practice, but it is also (2^1/6)² - makes sense in hexatonic scale, (2^1/9)³, (2^1/12)⁴, (2^1/15)⁵ and for example (2^1/306)¹⁰² - makes sense in theoretically good 306edo. So I prefer as exponent an irreducible fraction 1/3, but these are trivial calculations anyway. In case of quarter-comma mean tone intervals there are probably many interpretation that make sense (including currently usedone ) and I only think it is necessary to write one that is more easily transformable to the simplest.

I've written a lot, but final answer is shorter: I can agree on removal ":1" from irrational numbers. If you are still willing to agree on what you called "another compromise", we can use it and it will be the end of our discussion. BartekChom (talk) 14:01, 29 July 2014 (UTC)

I agree the ":1" is just as unnecessary in 2:1, but let's not touch the rational intervals. I removed the tail from the irrationals. For the "factors" column the question is how to distinguish it from the ratio column. I added another row to the table above. What reason is there to choose which of 2^(4/12) and 2^(1/3) is in which column? −Woodstone (talk) 07:25, 31 July 2014 (UTC)

"Ratio" is something like definition in case of non-natural intervals. So I think that prime factorization should be simplest notation. Besides factorization should be unique. That's why I prefer "another compromise" over "still other compromise". BartekChom (talk) 12:23, 31 July 2014 (UTC)

Requested move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

No consensus to move, despite extended discussion. I would encourage further discussion to reach a conclusion as to what is the best possible title for this page, and what should be done with respect to various other related titles, but no conclusion seems to be forthcoming in the time frame for this proposal. bd2412 T 13:39, 21 August 2014 (UTC)

List of pitch intervals → List of musical intervals – Pitch refers to absolute frequencies, intervals are frequency ratios. — Relisted. Jenks24 (talk) 14:17, 7 August 2014 (UTC) Omegatron (talk) 23:50, 30 July 2014 (UTC)

• Support as described above. Also fine with "List of intervals in music". — Omegatron (talk) 00:24, 8 August 2014 (UTC)
• Support. Pitch interval is not the right name, since pitch is a physio-psychological measure. The entries in the table are frequency ratios. What is intended to be described here are intervals relevant in music theory. Musical intervals would be a suitable name. −Woodstone (talk) 06:49, 31 July 2014 (UTC)

• Oppose. There's already a list of functional musical intervals at List of musical intervals, though someone had misguidedly redirected it here a few days ago. The intervals in this list have, contrary to what Woodstone writes, no relevance whatsoever in music theory: the function of a musical interval is not dependent on the way it is tuned - a major ninth, say, has the the same musical functions and requirements whether the voices or instruments playing it are using a Renaissance mean-tone tuning, a Baroque temperament such as Werckmeister IV or a modern 3-cents wide equal temperament. This is a list of intervals encountered in tuning or intonation, not in music or music theory. I'd have no objection to a move to List of tuning intervals. Justlettersandnumbers (talk) 07:32, 31 July 2014 (UTC)
  • That they aren't used in your particular music theory tradition doesn't make them non-musical, or mean that the people who use them aren't engaged in making music. — Omegatron (talk) 00:24, 8 August 2014 (UTC)

• Oppose. That "Pitch interval" requires renaming is obvious, as this name is meaningless. (Musical) intervals cannot be reduced to frequency ratios. In addition, many of the intervals described here are totally irrelevant to music and exist only in unbridled fantasies. Many of the sytems implied by many of the 'intervals' in this incredible list have never had any musical use – at least as intervals. Among the many inconsistencies of this article, let me quote a few:

• To describe "Pythagorean" (Pythagorean what, by the way: philosophy? intonation? tuning? consonance? interval?) as "3-limit just intonation" is a derision: "just intonation", which is a historical term dating from the 18th century, never meant Pythagorean intonation, it denoted systems producing 'just' triads. And the definition in the article Just intonation again is pure fantasy with respect to the historical meaning of the term.
• To state that "the prime limit is a number measuring the harmony of an interval" merely evidences a total misunderstanding of what "harmony" means ("the harmony of an interval" is properly meaningless).
• And to say that "The lower the number [of the prime limit], the more consonant the interval is considered to be" not only doesn't say by whom this is considered so, but is unbelievable in any conception of consonance: we would be made to admit that 9:10 (the minor tone) is as consonant as 4:5 (the major third), merely because both can be factorized down to 5.
• A cent, or a savart (or an eptameride which, by the way, is not identical with a savart) are not 'musical intervals', they are measure units without any direct function in music.
• Etc., etc.

The article not only is in need of a change in its name, but also of a thorough revision in its content.

The List of musical intervals, on the other hand, needs to be restored as soon as possible, because it is a useful list, doing exactly what its title says.

Hucbald.SaintAmand (talk) 19:47, 31 July 2014 (UTC)

Prime limit description corrected. Measure units should be included at least for comparison. You can add explanation, if you think that now it is misleading. If you want to correct definitions of just etc., add explanations from sources, but probably current definitions are also used by somebody. BartekChom (talk) 11:51, 3 August 2014 (UTC)

I don't intend to change anything to this article, because if I began, I'd change everything. And I won't "add explanations from sources", while the article as it is doesn't provide a single source to justify its "explanations" (the ones I question below). I don't see the purpose of such an article. My problems remain:

• I don't understand the meaning of the expression "pitch interval".
• While your corrected definition of the prime limit indeed is somewhat clearer, it fails to indicate the nature of the "frequency ratio" concerned. (You seem to believe that an interval always can be expressed as a frequency ratio, but that merely is not true: a tempered fifth in ET, for instance, 700 cents, cannot be expressed as a ratio, and therefore cannot be factorized).
• The idea that "Intervals with lower prime limit are used more often" is wishful thinking. More often than what? In what kind of music? Piano music?
• The claims that "Pythagorean means 3-limit just intonation" and that "Just means 5-limit just intonation" appear to contradict each other, unless the meaning of "just intonation" is not "an intonation that is said to be just".
• I don't understand the statement "every tone in a 3-limit unit can also be part of a 5-limit tuning". Don't you mean "every interval..."? As to "every tone", it can of course be one of the two tones of any interval.
• The article states "In general, a meantone is constructed the same way as Pythagorean tuning"... A meantone tuning, strictly speaking, is one in which the tone is the mean between the major and the minor tones; such tunings are also called "regular", because they use only one size of fifth. The Pythagorean tuning, or ET, are meantone tunings. ET is a 1/12 comma meantone, but the comma is here the Pythagorean one, not the syntonic one.
• The definition of "Equal-tempered" as ET "with intervals corresponding to X multiples per octave" is totally obscure to me. Would it not be simpler to say "with the octave divided in X equal intervals", if that is what is meant?

As to the list itself, most of these intervals do not seem "musical" to me, despite the article's initial claim. I really wonder what kind of music would make use of a ragisma, a breedsma, a septimal kleisma, an Orwell comma or the like.

Hucbald.SaintAmand (talk) 16:51, 5 August 2014 (UTC)

Yes, this is a beautiful modern-day example of the age-old division between the teorici such as Gioseffo Zarlino, who spent their lives calculating the values of theoretical musical intervals and attempting to demonstrate their connection with celestial mathematics, the nature of god and the like, and the prattici such as Lodovico Zacconi, who talk about how people actually play music. This page is a sort of musical numerology, verging on WP:FRINGE. Justlettersandnumbers (talk) 19:24, 5 August 2014 (UTC)

I'd be more severe: this article, as I read it, really is an example of what WP defines as a Fringe, a domain that justifies itself merely by autoreference. I think that neither Zarlino nor Zacconi should get involved: after all, they both were theorici and prattici, even if they had a different sensibility about that. Even Huyghens, often cited, probably wouldn't have endorsed the claims made here. I am by nature indulgent, and I don't want to intefere with what others may be trying here. WP is a free project, where everybody is free to do as (s)he feels, as long as (s)he leaves the same freedom to others. I find it quite aggressive that the article List of musical intervals should be reduced to a redirection to this article, the purpose of which obviously is different, or even more that this one should be renamed to the name of the other: I won't do anything against the existence of this "pitch intervals" article; its authors similarly should not prevent the "musical intervals" article to exist. I presume that, if this were to become a matter of real conflict, WP authorities have procedures to resolve it. – Hucbald.SaintAmand (talk) 20:34, 5 August 2014 (UTC)

Generally I know the topic only from internet, so my answers are only guessing. So:

Explanations are more or less sourced in other articles.

• I understand pitch interval as interval between pitches, i.e. ratio of frequencies, as opposed to a view in which equal tempered major third and equal tempered diminished fourth are different intervals.
• I understand that ratio can be any number, not necessary ratio of two integer numbers. And some irrational numbers can be factorized (rational powers of rational numbers). Anyway, prime limit is only applicable to just intervals (rational ratios). Ok. I understand that it sounds strange, but in Polish "rational ratios" is wymierne stosunki, so I find it a coincidence, although rational numbers are ratios of integer numbers.
• If you do not agree, this can be removed. But I think that this is true: taking into account only 7-limit intervals are less standard, 11-limit ones (and so one) even more so. Of course in practice 12edo is most common.
• I understand that just intonation denotes ratios of (usually small) integer numbers, but in practice most often are used 5-limit ratios, so this is another meaning.
• Every interval. Unless we assume that we have common base tone.
• "They use only one size of fifth", so are "constructed the same way as Pythagorean tuning" - by adding fifths. Two perfect fifths (minus octave) give major second, three - major sixth and so one.
• It probably really would be more correct. I understood "multiples" as "steps" and didn't notice this problem.

For example ragisma can be encountered in 7-limit tuning if we make analogue of comma pump like just major sixths and lesser just major second up, and septimal major third and two just minor third down:

${\frac {5}{3}}\cdot {\frac {10}{9}}\cdot {\frac {7}{9}}\cdot \left({\frac {5}{6}}\right)^{2}\cdot {5}{9}={\frac {5^{4}\cdot 7}{2\cdot 3^{7}}}$

Hucbald.SaintAmand, now do you like organisation of this topic on Wikipedia? BartekChom (talk) 20:20, 6 August 2014 (UTC)

BartekChom, I won't comment on this anymore, I am merely not interested and I think to have said all I had to say on Talk:List_of_musical_intervals. I have first-hand knowledge of Benedetti's 16th-century comments on pitch drift in just intonation. I also studied schismatic exchange in 13th-century Arabic theory – and in 14th-century Western tunings. My own interest is in understanding these ancient theories – and these early musics – in their own terms, not through the lenses of modern fantasies. I don't think WP talk pages are the right places to further discuss such matters. – Hucbald.SaintAmand (talk) 21:02, 6 August 2014 (UTC)

• Oppose – the article is more about pitch intervals (pitch ratios), as anyone can see and as the discussion above clarifies. Dicklyon (talk) 05:12, 10 August 2014 (UTC)

• Oppose Comment - I think, let me see if I have this straight. The proposal is to move this to List of musical intervals? Someone above wrote "There's already a list of functional musical intervals at List of musical intervals, though someone had misguidedly redirected it here a few days ago." But currently List of musical intervals is a not a redirect or a list page, but a disambig page, linking here and to Interval (music)#Main intervals. So that's all a bit confusing.

That aside, the main complaint here seems to be that "pitch interval" isn't the right term, although I'm not clear on why. One person wrote that the term "pitch interval" is "meaningless", yet I have no trouble understanding it. Perhaps it is technically wrong and should be "pitch frequency ratios" or something, but that seems more likely to be confusing, even if pedantically more accurate. So I think I agree with Dicklyon, who seems to be saying that "pitch interval" means "pitch ratio", "as anyone can see". Still, a change to an alternate term that isn't too unwieldly would be okay with me.

Also in this thread some seem to argue that some of the intervals on this page are not "musical", or are even "totally irrelevant to music and exist only in unbridled fantasies". I disagree there. "What kind of music would make use of a ragisma, a breedsma, a septimal kleisma, an Orwell comma or the like?" A music that moves between tuning systems, for one. I myself have made music that shifts around by very small commas and the like. Not a lot of people make music like that, but that doesn't make the intervals "unmusical". Very small intervals are not melodic, mostly, but they are certainly important harmonically when making music like this. Very very small intervals like a ragisma or breedsma are not likely to be distinguished by ear, but their effect can be very noticeable in transforming harmonic spaces. As for the septimal kleisma, it is what is tempered out in Septimal meantone temperament, similar to how the Pythagorean comma is tempered out in 12 tone equal temperament. Is the Pythagorean comma "musical"? Not in a melodic sense, usually. Is it musically important? Without a doubt. And the septimal kleisma is used and important in other ways too.

Also, I would agree that a lot of this tiny interval stuff would be fringe theory with little or no real application if we were still in the Renaissance, Baroque, or 13th-century Arabic theory, or even in most of the 20th century. But these days people can and do make music with computers, which have made many impossible things possible, theoretical things practical. Pfly (talk) 06:20, 11 August 2014 (UTC)

@Pfly. The several discussions that led to replacing List of musical intervals, first merely by a link to this page, then by the disambiguation page that it now became, are spread on the talk pages of both articles. In the history of List of musical intervals, you'll see that the existing list has been removed three times in less than a month (and restored twice). The whole could have happened in a less aggressive way.

I think that vulgarization is not about replacing difficult concepts by approximations, but about trying to explain them as simply as possible. "Pitch interval" is not a good expression because a pitch is a frequency (e.g. 440 Hz) associated with a note name (e.g. A4). The intervals discussed in this article are not between pitches properly speaking, if only because they are independent from any particular frequency. The expression can be understood, which is why it has not (yet) been changed, but I persist thinking that it is essentially wrong.

A "musical interval", to me, is an interval used in music, and not in the discourse about music – otherwise it should be renamed as something like "music theoretic interval", or "intervals used in tunings and temperaments". But neither of these are really satisfying. A cent, an eptameride, a savard are not intervals, they are merely units for measuring intervals. The 12-TET intervals (and most others listed in the column headed TET) cannot be expressed as fractions and cannot be factorized (they are irrational). These are but a few of the approximations of the article.

I agree that computers make everything possible (or so it seems). Yet, many of the intervals in this list result from excessive "digitalisation", i.e. express in numbers what in real music can neither be perceived, nor even conceived in this manner. You speak yourself of a music that "shifts between tuning systems". Many historical tunings similarly have been shifting between segments of simpler tunings. 'Just intonation' (of Zarlino's type) can be described as segments of Pythagorean tuning shifted by a syntonic comma; this does not mean that the syntonic comma was used as a musical interval. 18th-century irregular temperaments make use of all sorts of shifts, many of which cannot be expressed as ragisma or breedsma or anything of the kind, merely because they cannot be expressed as fractions.

All this considered, it appears to me that what the List of pitch intervals article does is digitalizing intervals, i.e. approximating them in whole numbers. But the article does not say so, is not even aware of doing so. These intervals may or may not have been used in music. Let's admit that they all could be used in music – it is not exactly the same... The article rests on implicit notions of what a note is, of what a pitch is, of what an interval is, and supposes that these notions are universal. This to me is both vulgarization and eurocentrism at their worst.

Hucbald.SaintAmand (talk) 08:54, 11 August 2014 (UTC)

To be clear, my 'oppose' was mainly to do with the fact that the proposed move is to a currently existing disambig page. I was not and am not up on whatever other discussions and moves are going on. I just saw the move proposal, then saw the disambig page and thought, given that, the proposed move wasn't a good idea. I'm not against the page being renamed. I agree that "pitch intervals" is non-ideal. I almost posted a "comment" rather than an "oppose". In fact I think I'll change it above.

The debating about whether some of the intervals listed here are musical, theoretical, or whatever, seems mostly tangential to the proposed move and associated issues of redirects, disambigs, and so on. I see now there is a larger issue involving several pages and various recent changes, and that some stuff needs to be worked out. What concerned me when I read this page move thread was the the comment "The article not only is in need of a change in its name, but also of a thorough revision in its content." That worried me because I personally find this page very useful. But reading more carefully I see you later wrote "I won't do anything against the existence of this "pitch intervals" article; its authors similarly should not prevent the "musical intervals" article to exist." Sounds good to me, and makes sense. I hope you all can work it out! I'm not active enough to dive too deep just now.

PS, just to continue the debate a little, I'm not sure I understand your comments about "digitalisation", fractions, irrational numbers, and so on. I understand that the intervals of 12-TET (and many other intervals) involve irrational numbers. But they are still "ratios" between two frequencies. A 12-TET semitone is often described as 2^1/12:1. Our semitone page says the 12-TET semitone "is a ratio of 2^1/12." I didn't realize we had a page on the Twelfth root of two! This page, "pitch intervals", in the "frequency ratio" column, says a 12-TET semitone is "2^1/12," just like the semitone page. My inner pedant complains that "2^1/12" is not a ratio—it should say 2^1/12:1! Would we say the frequency ratio of an octave is "2"? That only makes sense when ratios are confused with fractions, I think. People often write 2/1, and we know that fractions in the form x/1 can just be written as x. But intervals are ratios, not fractions! (okay, my inner pedant is done for now)

Anyway, maybe your main concern is with this page's lead text rather than the list itself so much. If that's the case I agree—the lead text could be much improved. It does seem to suggest that all intervals involve whole number "rational" ratios. Pfly (talk) 06:45, 12 August 2014 (UTC)

The question of the requested move appears settled. My concern indeed is mainly with the lead, which doesn't at all contextualize the contents of the article. What is exposed here is a modern, Western view of intervals of all kinds, many of which are there only as numerical (digital) descriptions of what more often is described as movable degrees, in Western culture in former times and even today, and certainly in non-Western cultures. I find it most revealing that Ján Haluška, who seems to be one of the main sources of this article, at one point in his book claims that Turkish music counts 53 degrees in the octave: this typically is the point of view of a mathematician believing he knows something of music theory (and possibly a confusion with Holder's comma of the 18th century, which in turn is an approximation of the Pythagorean comma). I strongly doubt that any Turkish musician would endorse such a claim. Even the 20th-century idea that Arabic music is based on 48 quarter tones in the octave tends to be rejected today. And I'd very much like to know the historical origin of the fancy names of many of these intervals. But enough about that, which does not deserve so lengthy comments. – Hucbald.SaintAmand (talk) 11:28, 12 August 2014 (UTC)

Even a mathematician without an ounce of musical training might tend to reject the idea of cramming 48 quarter tones into an octave.—Jerome Kohl (talk) 15:46, 12 August 2014 (UTC)

• Support List of musical intervals as per the current titling within the article. No opposition to variations on this, but the current sounds like Association_football_pitch#Other_markings. --SmokeyJoe (talk) 06:38, 15 August 2014 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

List or errors in the article

I must apologize for what follows, but I HATE this article, which I consider one of the most carelessly written I ever saw on WP (or elsewhere). Having devoted much of my life to the study of historical tunings and temperaments, I cannot easily admit that this history be treated so lightly. I do understand that computers have made it possible to tune anything and to produce any type of temperament. Some may even think that it is possible to make music with that: it is up to them to prove so (Nicola Vicentino had tried some 500 years ago, without much success). Anyway, I decided to produce a list of errors or inadequate formulations in the article. This will take me some time, but here follows a first instalment, concerning the introductory text of the article, its "lead" so to say. I will soon add a list of inadequacies in the list itself. This list of errors may be considered a list of suggestions for improvement of the article, but I don't intend myself to realize any of these improvements.

• The introductory text does not give a single reference to published evidence.
• "Below is a list of musical intervals [why not "pitch intervals", as in the title?] encountered in tuning or temperament." Such wording appears to suggest that one may encounter these intervals while tuning an instrument. It might be more correct to write "in tunings or temperaments" (plural), referring to the results of the tuning. But one may wonder what type of instrument could be tuned to many of these intervals. Does one tune a computer? It might be more accurate to say that these intervals are encountered in (modern) theories of tuning or temperament, or something of the kind.
• "Intervals with lower prime limit are used more often." A properly unjustifiable statement. How does one determine, say, that 4ths (3-limit) are used more often than pure major 3ds (5-limit); are 'natural 7ths' (7-limit, about 969 cents) used more often than minor 7ths (about 1000 cents)?
• "Any n-limit entry is also m-limit for m>n." What this odd statement apparently means is that any n-limit interval can also be encountered in systems of higher limit, but not of lower limit. This could be better formulated by saying that the limit represents the system of lowest order in which the interval can be encountered: it is a low limit.
• "Similarly, septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit just intonation." The list itself also contains entries for "19-limit just intonation" and one can assume that any higher prime number could also form a limit (one has to stop somewhere in practice, but in theory there is no ... limit). But what is the point in calling all of them "just"? What does "just" mean if anything is just? Historically, "just intonation" has meant 5-limit, and there is no reason to change this historical meaning.
• "By definition every tone in a 3-limit unit can also be part of a 5-limit tuning and so on." This appears fully redundant with "any n-limit is also m-limit for m>n", see above. Any interval of a given limit can also be part of a system (or tuning) of higher limit.
• "In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in a meantone, each fifth is narrowed by the same small amount". A complex way to convey a simple idea. A meantone is a "regular" tuning, i.e. using only one size of fifth (but different from doubly augmented 4ths or diminished 6ths). In Pythagorean tuning, the fifths are perfect; in quarter-comma meantone the fifths are tempered (narrowed) by 1/4 (syntonic) comma; in 12-T equal temperament, the fifths are tempered by 1/12 (Pythagorean) comma; etc.
• "Equal-tempered refers to X-tone equal temperament with intervals corresponding to X multiples per octave." Apparently, "multiple" here means "division"… 12-T equal temperament corresponds to 12 divisions per octave.

2d instalment – Intervals from 0 to 100 cents (I refer to the intervals by their value in Cents, as given in the first column of the table).

• 0.00 This is a list of intervals. It makes no sense to call an interval the "tonic" (even if the fault exists in the reference cited, Kyle Gann's website) or a "fundamental". There probably is a confusion with another possible list, that of harmonic partials, in which partial 1 indeed would be the fundamental (and, by a common aberration, the "tonic"). Partch's definition of the interval as monophony also is odd. That the interval of unison in a list of intervals from C should be a C does not seem to need a reference to John Fonville's book.

• Other intervals are defined as "n^th harmonic", which strictly isn't an interval either; one may understand that what is meant is the interval between the harmonic and the highest power of 2 below it, but it would be better to say so.

• 0.01536 This interval is missing in the table. It is the Atom of Kirnberger, for a definition of which see Schisma.
• 3.99 An eptaméride is not the same thing as a savart and neither exactly corresponds to 10^1/1000. The eptaméride divides the octave by the decimal log of 2 counted with three decimals and multiplied by 1000, i.e. 301 parts of 3.9867 cents each; the savart does the same but using log(2) to the fifth decimal, i.e. 301,03 parts of 3.9863 cents each. They are therefore but approximations of 10^1/1000 (which counts an infinity of decimals), rounded to three or five decimals.
• 7.71 Marvel comma should refer to [Marvel temperaments]
• 13.79 Starling comma should refer to [Starling family]
• 21.82 This interval is missing in the table. It is the unit interval in 55-T ET, the pendant to 53-T ET, with diatonic semitones of 5 commas instead of 4. In this reasonably well documented temperament, all intervals corresponing to the chromatic scale are between 12-T ET and 1/4-comma meantone intervals.
• 27.91 This interval is missing in the table. It is Sauveur's méride, the 1/43 part of the octave. The heptaméride is the 1/7 part of the méride. Sauveur considered that the mérides were sufficient to describe "the tempered intervals of the diatonic system" (Principes d'acoustique, 1701, p. 12).
• 38.71 31-T ET deserves a reference. It is the tuning system advocated by Nicola Vicentino in the 16th century and discussed among others by Constantin Huygens. It provides an excellent approximation of 1/4-comma meantone.
• 63.16, missing in the table, should be mentioned. It is 19-T ET, with a chromatic semitone of 1/3 tone, a diatonic one of 2/3 tone. It provides an excellent approximation of 1/3-comma meantone.
• 90.22, reference [11]: considering that Helmholtz' treatise in Ellis' translation counts only 576 pages, the reference to page 644 must be a mistake. The expression "Low semitone" does not seem used in Ellis' translation, who speaks (p. 329) of the "small semitone or limma 256:243"; but Ellis describes a variety of intervals as "small semitones".
• 98.95 One wonders where the Huygens-Fokker Foundation found this "Arabic lute index finger". Al-Farabi (10^th century) describes 18:17 as one of the several positions that he names "neighbor to the index" (the index itself being at 9:8, 204 cents). 18:17 is exactly at the middle of the distance to 9:8 along the string (dividing a string distance in equal parts is a most interesting way to establish intervals). Other "neighbors to the index" include 256:243 (the Pythagorean limma), 162:149 and 54:49 (both missing in the table, also obtained by equidistance, but that will be for the next instalment).[To be continued]

There will be no third instalment, because I think I identified the general problems of this article, for which I will rather open a new section: see below, "Why this article does not achieve what it claims". – Hucbald.SaintAmand (talk) 19:01, 21 August 2014 (UTC)

While fully sympathizing with you on this, Hucbald, one small thing needs to be pointed out. Your complaint that "the introductory text does not give a single reference to published evidence" should not be a problem, supposing (as you do) that we are talking about the article's "lead" (or "lede"). The lead should never have references in it, for the simple reason that it should only summarize what is presented more amply in the main body of the article. However, only the first two paragraphs here are actually the article's lead. Beginning with "Some terminology used in the list:" we have got what should be a section with a separate header, and indeed you are correct that these terms all need references. If on the other hand this really is meant as part of the lead, then sources for the terms should be added at each appearance in the list. This would be cumbersome, however, so I suggest a "Terminology" header replace that phrase I just quoted, and references be demanded for any terms that are not in the "sky is blue" category.—Jerome Kohl (talk) 20:31, 21 August 2014 (UTC)

Critics should bear in mind that this is a "list" article, not a treatise on tuning and temperament. It aims to collect all possibly relevant intervals in a systematic categorised way, without theorising about them. Many of the remarks above are valid, so why don't you improve the header part accordingly? −Woodstone (talk) 05:59, 22 August 2014 (UTC)

You're "errors" are, at most, slight problems with the wording of the introduction. Hyacinth (talk) 09:58, 22 August 2014 (UTC)

If you want. But so many slight problems make a big problem. – Hucbald.SaintAmand (talk) 10:18, 22 August 2014 (UTC)

So, why don't you fix the errors you find? That's kind of the point of a wiki. The central premise, even. Be bold. — Gwalla | Talk 18:14, 22 August 2014 (UTC)

Because, as Hyacinth rigthly stressed, these are not "errors", merely "slight problems". I don't want in any way to endorse the general attitude of this article with respect to tunings and temperaments, because this attitude, even if may seem to lead only to "slight problems", denies all what I believe history of theory may be about.

I mentioned above the case of the "Arabic lute index finger" (98.95 cents). Anyone having ever played a lute (Arabic or other), or any other stringed instrument (a violin, for instance) knows that the normal position of the index finger is about a tone higher than the empty string. But some guy, probably having misread Arabic theory, declares otherwise, without leaving any possibility to check what he states. And somebody else, believing, as seems to be the case on WP, that anything published is true, reproduces it here. I am not myself an enthusiastic Wikipedian, in such conditions.

Al-Farabi, as I shortly explained, construed some of his intervals as equidistant divisions of the string. But this principle, equidistant division, even although it was conceived a thousand years ago (!), seems uninteresting to the self-appointed mathematicians building the kind of interval list we are dealing with here. I didn't check (I probably will), but I trust that many of the intervals so construed merely are absent of the list, their importance in Arabic theory notwithstanding. Max Weber, the early 20th-century sociologist, wrote in his book about musical sociology (I am merely quoting a vague reminiscense) that equidistant divisions have been an important technique in early times of music: he, at least, had some respect for these ancient theories.

I'd be willing to participate to the redaction of this article if I understood what it is all about. It implicitly claims to concern universal tunings, but in fact deals only with modern Western tunings.

Enough about this. In view of the above, I won't fix anything in this article. I repeat that anyone should feel free to make use of my comments (even if some, I'm ready to admit, are mere expressions of my irritation), and I might even answer questions, if any. But my comments ain't systematic, and it is not my intention to make any effort to make them more complete: I have other things to do.

Hucbald.SaintAmand (talk) 21:29, 22 August 2014 (UTC)

Dear Hucbald: While the point is well-taken, I must correct an error which you have now stated twice: As far as Wikipedia is concerned, if something is published it is verifiable—"truth" does not come into it. The famous formulation is, "The threshold for inclusion on Wikipedia is verifiability, not truth". This can be infuriating at times, especially so when the necessary refuting source cannot easily be found, but in the long run I think it is better to mention a well-documented error and then debunk it by citing the contrary evidence from an even more reputable source than it is either to disregard or attempt to conceal wrong-headed but widely disseminated notions that some bright child is only going to unearth later and try to insert as refutation of the stronger position we have worked so hard to establish in the first instance.—Jerome Kohl (talk) 21:52, 22 August 2014 (UTC)

Of course, Jerome, and we both know that truth may not exist. Change "true" in "verifiable": what I wrote becomes "believing ... that anything published is verifiable". I don't think that Wikipedia is that naive, but some Wikipedians are. If something is unverifiable in the source, it remains unverifiable once quoted. Woodstone wrote (above) that this article "is not a treatise on tuning and temperament" and does not want to theorize. Agreed, but it should at least base itself on verifiable theory... Hucbald.SaintAmand (talk) 17:32, 23 August 2014 (UTC)

Ah, but by the Wikipedia definition here, appearing in a reliable, published source is the way claims are verified (theories are another matter entirely). This is, amongst other things, a form of insurance against libel action. The tricky part of course is in demonstrating the reliability of a source. Even a source that is universally regarded as being reliable generally may contain the occasional error. In (Wikipedia) practice, any published book or article in a journal or magazine is more likely to be regarded as reliable than, for example, a self-published website or, worse, a blog. There are of course self-published books and magazines, too, so at a certain point the reputation of the publisher becomes involved. This in turn can lead to disagreement amongst editors, which is usually resolved through (sometimes tedious) discussion. The "open editing" of Wikipedia makes this inevitable, I think. As I said before, this can take some getting used to, and I have heard of many cases of superbly qualified editors giving up on Wikipedia in frustration. I do not want you to become one of them.—Jerome Kohl (talk) 19:54, 23 August 2014 (UTC)

Jerome, contrarily to those editors who gave up Wikipedia, I am not self-confident enough to give up. But I'll always feel free to express my opinion, even if it enters in conflict with WP's policies. After all, what I am doing here is just that, expressing my opinion, and this too is one reason why I don't want to work on the article itself. I note however that the Huygens-Fokker website cites none of its authors (I think to know some of them, but that changes nothing), and I remain totally puzzled at the idea of an "Arabic lute index finger": what to they mean by that? On the one hand, the index on the lute is moving as needed; on the other hand, most of the theorists who tried to define the position of the index on the 'Oud placed it a tone, not a semitone, higher than the open string. When I quote al-Farabi, I give an information that is verifiable (i.e. falsifiable – it could be proved to be false, if that were the case). The one given by the Huygens-Fokker foundation is neither, because they don't even bother to explain what it is supposed to mean. – Hucbald.SaintAmand (talk) 21:06, 23 August 2014 (UTC)

I am relieved to learn you will not be giving up easily! Opinions of course may be expressed freely on talk pages—in fact, one of their main purposes is to discuss what we think ought to be changed about an article, and this often involves exchanging opinions with other editors. As to the "Arabic lute index finger", I really have no idea what they can be talking about, but unsigned articles on websites are always of suspect reliability. This reminds me of a "reliable source" that was called to my attention thirty or more years ago now. This was when the mandolin method was first published in the Mel Bay series, and the person who indignantly told me about it was the owner of a music shop who had just received a copy as part of a consignment of new editions. Whoever had written this book was obviously a guitarist who may have picked up the mandolin occasionally, but plainly had never learned to play it properly. He instructed exactly what you describe as the "Arabic lute index finger", placing the index finger on the first fret, but then compounded the problem by adding fingers chromatically above it. My informant was in fact a professional mandolinist and violinist, and of course the mandolin is fingered very much like the violin: diatonically, starting with the index finger a whole step from the nut. He wailed that, because of the huge coverage in the market commanded by the Mel Bay name, whole generations of beginners were going to be brought up trying to play the mandolin the wrong way and would give up in frustration upon discovering they could play practically nothing in the mandolin repertoire using this method. I imagine this book is still in circulation, and I am afraid to look at what the Wikipedia article on the mandolin might have in it.—Jerome Kohl (talk) 22:37, 23 August 2014 (UTC)

Why this article does not achieve what it claims

It took me some time to identify in general terms the shortcomings of this article: my apologies for this. The article claims to be

a list of musical intervals encountered in tuning or temperament.

and one would suppose to find among others intervals found in historical temperaments. But one is soon disappointed.

The article mentions meantone, "the most common of which is quarter-comma meantone", and it refers to the article Meantone temperament where one would search in vain for some indication of the history of such tunings... Whatever it be, one may expect that "List of pitch intervals" would at least give the intervals of 1/4-comma meantone on a keyboard of 12 notes in the octave; but it gives (or identifies) only two, the minor third (310,36 cents) and the perfect fifth (696,58 cents). The just major third is there too, of course, and the list gives a "just augmented fifth" that turns out to also belong to 1/4 comma meantone, but neither is identified as being a 1/4 comma meantone interval. ALL other intervals of 1/4 comma meantone are missing (that is, more than half of its intervals). No mention is made of so many other varieties of meantone, 2/7 comma, 1/3 comma, 1/5 comma,, 2/9 comma, 3/10 comma, etc. etc., all meantone tunings described in any good history of tuning and temperament. The article does refer to the List of meantone intervals, it is true, but none of these is found there either.

The same is true of all kinds of tuning, the numerous historical varieties of just intonation (i.e. 5-limit!), or the irregular systems described by Grammateus, Ganassi, Artusi, Colonna, Mersenne, Rameau, Kirnberger, Werckmeister, Marpurg, etc. Nothing is said of the tuning of fretted lutes, the only mention of Arabic tuning is meaningless (or unverifiable), etc. etc.

In short, this article is about some intervals, but I fail to understand how the choice has been made; certainly, it is not about intervals "encountered in tuning or temperament".

Hucbald.SaintAmand (talk) 08:32, 24 August 2014 (UTC)

That's called being incomplete, isn't it? Hyacinth (talk) 12:47, 25 August 2014 (UTC)

Explicitly, what is your definition of "tuning" and what is your definition of "temperament"? Hyacinth (talk) 13:56, 25 August 2014 (UTC)

A tuning is a way to tune an instrument. Here, more specifically, a way to tune an instrument of fixed pitches (i.e., excluding the members of the violin family, which also are tuned, but usually with one single interval, or wind instruments with finger holes, the tuning of which involves other problems). Traditional instruments of fixed pitches include the keyboard instruments, the harp, the fretted instruments, etc., to which were added some electric and electronic instruments in the 20th century. As J. Murray Barbour wrote, "The tuning of musical instruments is as ancient as the musical scale. In fact, it is much older than the scale as we ordinarily understand it." (Tuning and Temperament, East Lansing, 1951, p. 1.) Mark Lindley, in the New Grove Online, gives three definitions of Tuning: (1) "The adjustment, generally made before a musical performance, of the intervals or the overall pitch level of an instrument"; (2) "the set of notes to which an instrument is tuned"; (3) "the 'tuning system' employed, referring to a model of the scale corresponding to some mathematical division of the octave." It is this third use that retains us here.

A temperament is a special type of tuning in which one or more interval(s) is (are) tuned false in order to resolve problems linked with the limited number of pitches available and, in many cases, the necessity to repeat them at the octave. Mark Lindley defines temperaments as "Tunings of the scale in which some or all of the concords are made slightly impure in order that few or none will be left distastefully so".

Murray Barbour gives other, more concise and in a way less explicit definitions, in which "tuning" and "temperament" are mutually exclusive, contrarily to the situation in Lindley's (or my) definitions:

• Temperament – A system, some or all of whose intervals cannot be be expressed in rational numbers.
• Tuning – A system all of whose intervals can be expressed in rational numbers.

(Murray Barbour, op. cit., p. xii)

The Pythagorean system is a tuning: it involves no tempered interval; Meantone temperament tempers the 5th, in order to tune the major or the minor 3rd closer to pure; Just intonation again is a tuning, involving no tempered interval.

Hyacinth, I am very sorry to be engaged in such a quarrel about this article. I merely think that if the authors want to change common and generally accepted definitions of what they claim to be speaking of, they should at least say so. Let me remind you that this discussion began when somebody suggested that this article should replace another one devoted to "musical intervals". I would never have bothered looking at "List of pitch intervals" without that. Of course, the article 'merely' is incomplete; I fear, however, that it could not be made complete, because there is something wrong in its conception itself.

The 12-degree meantone temperaments discussed by Murray Barbour, which range from 1/3 to 1/10 comma meantone (i.e. the fifth tempered by any amount between about 8 and 2 cents), yield the following values rounded to the nearest cent (that is that any figure here may stand for several, if more decimals were accepted).

C: 0

C#/Db: 64 72 70 76 79 83 85 89 92 95 97 99

D: 190 191 192 193 194 195 197 198 199 200

D#/Eb: 269 301 302 303 305 307 308 309 310 312 313 316

E: 379 383 384 386 389 390 394 395 396 397 398 399

F: 500 501 502 503 504 505

F#: 569 574 576 579 582 585 586 587 590 593 596 598 599

G: 695 696 697 698 699 700

G#/Ab: 758 768 773 777 781 787 791 794 797 798 808 812 814 817

A: 884 887 888 890 892 893 895 896 897 898 899

Bb: 1000 1001 1002 1003 1005 1006 1007 1008 1009 1010

B: 1074 1078 1080 1083 1085 1088 1090 1092 1095 1097 1098 1099

C: 1200

None of these intervals (except 0 and 1200) can be expressed as a ratio - i.e. none of them belongs to any n-limit! The article makes so much case of these n-limits, without realizing that temperaments, by definition, cannot fit within any of them!!!

Hucbald.SaintAmand (talk) 14:34, 25 August 2014 (UTC)

I need to take responsibility here, I believe, as (though I haven't actually gone through the history to be sure) I think that it was I who added that stuff about "encountered in tuning and temperaments". And I think I did so more or less unthinkingly because some of those intervals (such as major tone, Pythagorean comma) may be encountered when discussing tuning or temperament. I know for sure that most of them are not used in music-making in the ordinary way. That is why I made the now-suppressed List of musical intervals, so that ordinary people could come to Wikipedia and find out about the intervals that are commonly used or discussed in Western-style music discourse. And yes, I'm aware that there is an element of systematic bias in such a statement.

It was also I who gave this horrible mess its current title, for much the same reasons. If it's wrong, please go ahead and move it, to some appropriate title - List of intonational intervals? List of intervals you will never encounter as a working musician? List of totally fantastic made-up intervals with no real-world application whatsoever?

What I'd like to know now is this: if these intervals and this terminology (all this stuff about prime limits and the like) are not used in music-making (as they surely are not) and are not used in tuning (as Hucbald says more eloquently and cogently than I could - I can't fault him on one word), then where exactly are they used? Or is this all just the musical equivalent of "the length of the side of the great pyramid of Cheops measured in East Anglian barleycorns is exactly the same as the height of Rouen cathedral measured in lignes of the pied du Roi prior to the reform of 1622"? Justlettersandnumbers (talk) 18:50, 25 August 2014 (UTC)

I do think indeed that it would be important to know what this article is about. I belive that its list of references gives some cue. They are not so numerous:

– An article by John Fonville, "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", Perspectives of New Music, 1991.

– A "List of intervals" anonymously published on the web by the Huygens-Fokker Foundation, which I take to be a Dutch gang assembling (a) experts in historical tuning and temperaments (some of whom I know); (b) fanatics of microtonal music. Even their own [website] does not give any name to make them less anonymous.

– A web site by Kyle Gann (I have no idea who he is), with somewhat odd and uncontrolled ideas about the question.

– A book by Ján Haluška, who I take to be one of those mathematicians fascinated by the odd mathematics of musical systems; I don't know him, but I know some of his kins quoted in his book.

– The website of Xenharmony, apparently a project linked to the Huygens-Foffer Foundation, or of similar aims.

– A few citations from older books (Ellis, Helmholtz), probably intended to somehow legitimate the whole.

In short, the article seems to stem form enthusiasts of modern microtonal music. Some of them do make music with that. I believe therefore that the article would be perfectly legitimate, if it only described the context in which it is proposed. But when it claimed to list "musical intervals" at large, I could not but react. The intervals listed here perhaps belong to some (modern) type of microtonal music. I suggest therefore that the article be renamed "List of intervals used in some modern microtonal music" – or anything better, provided the authors explain what...

Hucbald.SaintAmand (talk) 20:27, 25 August 2014 (UTC)

I think I can help in a very small way by directing you to the Wikipedia article on Kyle Gann, who is a respected American music critic long associated with the Village Voice, and also a composer and professor of music. He has also been known to drop in as an editor at Wikipedia from time to time.—Jerome Kohl (talk) 20:42, 25 August 2014 (UTC)

Thanks, Jerome. Following your message, I had a closer look at Kyle Gann's [Anatomy of an Octave]. I had found it odd that he named the interval of unison the "tonic". I see now however (1) that his list is not one of intervals, but one of pitches, which makes much more sense (but does not justify "tonic"), (2) that the origin of this "tonic" is in Daniélou's Tableau comparatif des intervalles musicaux (1958) to which Gann gives a link (more about this below). Gann clearly declares what his list is, and implicitly what it is not:

• Once again, it is a list of pitches, not of intervals; it gives "more than 700 pitches within an octave".
• Gann does not claim that these pitches are "musical"; at best, one is left to imagine that they may be used in music.
• He gives precise and systematic criteria for inclusion in the list:

– all pitches result from ratios, which means that none stems from a temperament

– the list includes (1) all harmonics up to 128 (reduced inside one octave, even if he doesn't say so, but it is obvious); (2) all ratios between whole numbers up to 32; (3) 5-limit ratios up to 1024; (4) 11-limit ratios up to 128; (5) 31-limit ratios up to 64; (6) a choice of historically important ratios.

No justification is given for these criteria, but no claim is made, either. The reader at least gets a clear idea of what (s)he is reading, and is left to decide for her(him)self whether it is interesting.

I didn't yet read Daniélou's Tableau comparatif, which counts iv+143 pages (and I am not sure I will do so). Daniélou is a controversed Indologist and the introduction to his Tableau includes somewhat puzzling statements, for instance (my translations):

• "All musicians have noted that, in the infinite scale of frequencies, if one chose an arbitrary point as tonic, or fundamental sound, some other sounds appeared to harmonise with this basic sound, appeared to form precise, definite, ineluctable intervals, and any deviation from these particular points appeared offensive to the ear and was immediately described as either too high or too low sound". (p. i)
• "It seemed to the ear that when the voice meandered on the scale of frequencies, one found, with respect to the tonic, a series of precise points corresponding to perfectly expressive intervals, kinds of melodic words." (p. i)
• "These elementary observations on the intervals led to the division of the octave in twenty-two of twenty-four regions within each of which was found a perfect interval of precise meaning. [...] This scale, known to the Greek, was called enharmonic (Enarmonikos) and was considered in the Hellenic world and its heirs as the basis of all musical scales. The Arabs took over the Greek tradition which they discussed and interpreted in a manner at times differing from that of the Europeans." (pp. i-ii)

Daniélou also states that his table does not include ratios higher than 7-limit, unless they are superparticular or they are "mentioned in the diverse writings about music theory". It must be stressed that his table is given as a list of intervals, while it should be read as a list of pitches (Daniélou gives frequencies with respect to the "scale of the physicists", i.e A4=426.6 Hz, because he believes that C should be a power of 2).

Reading this table reminded me that there exists another interesting list of pitches (or intervals), based on different premises, in Hugo Riemann's Musiklexicon, at least in the 3d edition of its French translation, that I must have somewhere in my library.

All in all, I think that Wikipedia might offer various lists of [relative] pitches (rather than intervals), e.g. pitches in this or that (family of) historical temperament(s), or in this or that prime limit, or whatever, each clearly stating what it concerns, possibly with a disambiguation article refering to these various lists. I might let me be convinced to participate in parts of such a project, which seems to me more reasonable than trying to make sense of a "List of pitch intervals".

– Hucbald.SaintAmand (talk) 09:45, 26 August 2014 (UTC)

I do not think it is helpful to refer to one of the sources as a "gang of fanatics". I also don't think it's all that important that Gann is a bit fuzzy with the term "pitch": his list is by relative distance (in ratios and cents), not by absolute frequencies. — Gwalla | Talk 18:52, 26 August 2014 (UTC)

Keep in mind that English is not my native language. My Webster defines a "gang" as "a group of persons having informal and usually close social relations", and it is in this sense that I understood it. And I maintain that some of them are fanatics ("marked by excessive enthusiasm and often intense uncritical devotion", Webster again) of microtonal music. As to Gann, did I ever said that he is "fuzzy" with the term "pitch"? I merely said that his choice of proposing a list of pitches "makes more sense" than a list of intervals. I am perfectly aware that pitches may denote relative distances. On the other hand, he is "fuzzy" with the term "tonic". – Hucbald.SaintAmand (talk) 19:58, 26 August 2014 (UTC)

"Gang" typically implies criminal activity, though it is also used for informal but close-knit groups of young friends. Similarly, "fanatic" has very negative connotations, of insanity or obsession. And you didn't say he was being fuzzy with the term "pitch", I did: it looked like you were saying that it is significant that he calls the members of his list "pitches", while I don't think it is. — Gwalla | Talk 22:13, 27 August 2014 (UTC)

You will be hard pressed to prove that an interval isn't used. Hyacinth (talk) 07:42, 5 September 2014 (UTC)

Given the definition of tuning as "a way to tune an instrument", how many ways are there to tune instruments? Hyacinth (talk) 07:48, 5 September 2014 (UTC)

Let’s first agree that what we are speaking of here is not the tuning as performed by instrument makers or professional tuners, but the building of systems, in a sense approaching that of the Greek systema teleion – the overall scalar arrangement underlying music. The tuning of instruments is involved mainly because some of them provide a concrete illustration of the system: particularly keyboards (many theory textbooks begin with an image of a keyboard, meant to represent the system), also harps, guitars, etc.; more generally, instruments 'of fixed pitches'.

Not all musical cultures know an underlying scalar system of this type; I won't discuss that aspect now. "Tuning", as discussed here, is mainly (but not exclusively) a matter of Western music. There are, I think, three main ways of 'tuning', of building musical systems: (1) tuning consonances, (2) temperaments, and (3) equidistant division, which I’ll consider in turn.

1. Tuning consonances. Consonances have been associated either with numeric ratios of whole numbers, or with harmonic partials; these two boil down to the same, because the harmonic series is the series of whole numbers. Three such systems are documented in Western history: Pythagorean tuning, based on tuning perfect fifths (3:2); "just" tuning (also called Zarlino’s system), tuning perfect fifths and major thirds (3:2 and 5:4); and unsuccessful 18th-century attempts (mainly by Euler) to add to these consonances the 'natural seventh' (7:4) – unsuccessful because the natural seventh does not easily fit in the diatonic/chromatic system of common practice Western music. Harry Partch rekindled the reflection on 'consonances' with his concept of prime limit, extending the number of harmonics considered … without limit.

2. Temperaments have been conceived, from the late 15th or early 16th century onwards, mainly for instruments of fixed pitches with 12 pitches in the octave. Temperaments consist in 'tempering' a consonant interval (usually the fifth), i.e. making it very slightly dissonant, in order to increase the compatibility of the degrees between themselves. There are mainly two types of temperaments:

– regular temperaments (also called meantone temperaments), tempering all fifths by the same amount;

– irregular temperaments, often meant to make some keys closer to just intonation than others.

Regular (especially equal) temperaments can of course be extended to produce as many subdivisions in the octave as one wants; several such extended systems are historically documented because they provide satisfactory approximations of systems calculated otherwise.

3. Equidistant divisions should not be confused with equal temperaments. They consist in dividing a geometrical space, e.g. between frets on a fingerboard or between fingerholes on a wind instrument, in equal distances. (Equal temperaments divide in equal intervals, resulting in distances in an exponential series.) I have seen a folk 'spinet' (a monochord) where the maker placed the frets using a stapler and a centimeter, putting a staple at each centimeter. More often, intervals obtained by 'consonance' (perfect fourths, in most cases, 4:3) are divided in several (e.g. three) 'equidistant' intervals: this is the case in several medieval descriptions of Arabic musical systems. Equidistant divisions coincide with consonant divisions, but are conceived on a different principle. The string of an Arabic lute, for instance, could be divided in 40 equidistant spaces, of which only the last 10 are used: 30 spaces produce the perfect fourth to the open string (30/40=3/4), 32 spaces produce the just major third (32/40=4/5); the other divisions produce less usual intervals. This corresponds to dividing the space of the 4th in 10 equidistant spaces. Although these 10 intervals (39/40, 38/40=19/20, 37/40, 36/40=9/10, 35/40=7/8, 34/40=17/20, 33/40, 32/40=4/5, 31/40 and 30/40=3/4) are relatively well documented in medieval Arabic theory, only 4 of them are listed here, which once again makes me wonder what this article is about.

Does this answer your question? – Hucbald.SaintAmand (talk) 18:15, 5 September 2014 (UTC)

If there are an infinite number of ways to tune instruments, and there are an infinite number of pitches, you may never be satisfied with this article's completeness. Hyacinth (talk) 23:34, 5 September 2014 (UTC)

Exactly, Hyacinth. This is why I wrote above: "All in all, I think that Wikipedia might offer various lists of [relative] pitches (rather than intervals), e.g. pitches in this or that (family of) historical temperament(s), or in this or that prime limit, or whatever, each clearly stating what it concerns, possibly with a disambiguation article refering to these various lists." Even a list of "prime-limit intervals" (or "relative pitches") could not be complete, unless it stated from the start which limits were retained. A list of Pythagorean intervals or relative pitches could be made reasonably complete. A list of meantone intervals or pitches could at least say which meantone temperaments are retained, and to how many degrees they are extended. A list of intervals or pitches in medieval Arabic theory could be more or less complete, and could easily say which theorists are considered. Etc.

My only complain about the present article is that it is unable to clearly state which intervals were retained, and why. I tried to clarify things by modifying the lead, but it remains that this article merely cannot in any way be 'complete'.

Hucbald.SaintAmand (talk) 07:46, 6 September 2014 (UTC)

See: Wikipedia:Wikipedia is a work in progress. Hyacinth (talk) 20:00, 8 September 2014 (UTC)

Meantone

I corrected the definition of meantone temperaments in the "Terminology" section. I don't think that a question asked (without answer) on a forum can count as a verifiable source for 1/2-comma meantone. Pending the answer, I don't think that any meantone with fifths narrowed more than a 1/3-comma ever were in use. But I have no time to verify sources, so that my correction also needs a citation. Murray Barbour's book on Tuning and Temperament certainly is one, but I've no time to give page numbers just now. – Hucbald.SaintAmand (talk) 21:10, 27 August 2014 (UTC)

OK. So is usage in a program enough? [5] BartekChom (talk) 21:46, 27 August 2014 (UTC)

I suppose so; it would be even better if you could provide a reference, but I presume that it may not be easy to find. The [Logic Pro website] does not mention meantone at all, or I were unable to find it. But a sofware like Logic Pro can in fact produce any tuning or temperament, known or unknown: it allows inventing them...

What is certain on the other hand is that Fogliano (about whom the question was asked on the forum that you had mentioned) never described 1/2-comma meantone (Lodovico Fogliano, [Musica theorica], Venice, 1529, sectio tertia). – Hucbald.SaintAmand (talk) 10:55, 28 August 2014 (UTC)

The new picture in the lead is an excellent improvement on the doubtful former one. Actually it now could easily be enhanced by adding the most common 5-limit just tuning. That would give some information on the 4 most fundamental systems: ET, pythagorean, quarter-comma, just. If the picture becomes too cluttered one could leave out one or more of the other ones. −Woodstone (talk) 16:36, 28 August 2014 (UTC)

Glad you like it. I once drew it with just intonation added, but that does indeed make it somewhat cluttered (removing one of the meantone temperaments wouldn't really help).

Just intonation is made of segments of Pythagorean intonation separated by a syntonic comma. It could be represented in a Tonnetz, as this:

		Eb	Bb
F	C	G	D
A	E	B	F#
C#	G#

(This is a somewhat unusual presentation because the major thirds, in each column, must be read downwards; the advantage is that reading from left to right and from top to bottom, you get the full cycle of fifths). Each of the rows forms a segment of Pythagorean tuning (perfect fifths), parallel to the blue line in my figure. To pass from one segment to the next, you must go down in the figure by the distance of a syntonic comma, and get there a new segment of line parallel to the blue one. The line representing just intonation therefore would form a zigzag crossing the whole. It could be added, I'll think of it. But I trust that anyone with an idea of what just intonation really is easily would figure it out; and the others may not be much helped by an improved figure.

Note that I speak here of "just intonation", not of 5-limit. Neither ET nor the two meantone temperaments represented can be expressed in terms of a prime limit; Pythagorean tuning is 3-limit, and therefore also 5-limit. I don't see how one could justify all this without reaching in a mess... Comparing historical systems under their historical names, Pythagorean tuning, ET, meantone temperament and just intonation, is so much clearer. [This is not to say that I don't understand the logic behind prime limits; I merely think that they should be used with more caution.] – Hucbald.SaintAmand (talk) 18:22, 28 August 2014 (UTC)

Most of the intervals listed with a "M" for "Meantone" in the "M" column of the table actually do not belong to meantone temperament. Correcting that will be quite a work... – Hucbald.SaintAmand (talk) 17:54, 31 August 2014 (UTC)

Newer image

Why not make the image in the introduction (File:Meantone.jpg) bigger? Hyacinth (talk) 20:01, 8 September 2014 (UTC)

It seems unclear what the syntonic comma is between. Maybe the line should go from symbol to symbol. Hyacinth (talk) 21:21, 8 September 2014 (UTC)

Instead of creating too much clutter, adding a 5-limit just chromatic scale (8/5 6/5 9/5 4/3 1/1 3/2 9/8 5/3 5/4 15/8) would help clarify. See: File:Meantone comparison.png. Hyacinth (talk) 21:25, 8 September 2014 (UTC)

[The answer below was written before your suggestions for changes in the image itself and concern only your suggestion to changing its size. It may however in a way answer tot the whole.]

Well, I don't know, Hyacinth, maybe merely because as its author I didn't want to push too far... Also, I thought that it nicely occupied the space left at the right of the table of contents. You have more experience in Wikipedian matters and I'll gladly give in to your opinion.

On a more general level, I think, Hyacinth, that we should stop arguing about this article, and that we should better seriously consider how to improve it or to replace it by something better. I think that the core of the problem has to do with this matter of prime limit, which is not treated as rigorously as it should, neither here nor in the other articles that I have read.

You induced me to reread Ellis' Additions to Helmholtz' treatise, and this made me realize that the concept of prime limit may somehow stem from there. Indeed, Ellis does speak of "septimal" chords, based on harmonic 7, and claims for instance that the "septimal minor triad G ⁷Bb D is by far superior to the Pythagorean minor triad D F A". The notation "⁷Bb" is striking. The passage raises several questions, for instance why he describes the Pythagorean triad as D F A, not G Bb D, etc. He also speaks of septendecimal chords, using harmonic 17 ("¹⁷Db").

These considerations do not belong to this particular article on "pitch intervals", but rather on the one on prime limit. But there exists an array of articles on such matters, also including those on the various types of temperaments, which do deserve some of our energy. My figure about which you raise the question above also does not really belong in this article, but more probably in one on meantone tuning and temperaments.

We won't solve any of this, and we may keep quarreling without true reason, unless we attack the problem from the start. This is a problem similar to that about diatonic/chromatic/enharmonic, as recently discussed on the Talk:Music_theory page, if I remember well, without that we reached any concrete conclusion.

Such a project begins with an inventory of the articles that need revision and coordination. But it requires many collaborations. I think that here too, an appeal should be made to the participants to the Music Theory Project. I would more easily participate to that sort of thing than to hopelessly try to improve the List of pitch intervals article as it is.

I am not an iconoclast, Hyacinth, even if at times I do not show enough patience. I do hope we can get out of the quarrel and begin something worthwhile.

Hucbald.SaintAmand (talk) 21:53, 8 September 2014 (UTC)

Let me add a few comments to your proposed image including just intonation. I shall first stress that what an image in the lead of this article really should show is 3-limit, 5-limit, etc. There is no other reason for the image to show meantone temperaments (which are not really present in the list itself) than that it is mentioned in the terminology section. But a representation of 3-limit and 5-limit intonations is an interesting challenge, worth a discussion. Needless to say, such an image would better belong to the Prime limit article. But imagining it here raises interesting questions.

Let's begin with 3-limit, i.e. extended Pythagorean tuning. The figure as it stands is limited to 12 degrees (plus one for the enharmony) because of the chromatic scale and the usual keyboard of 12 degrees in the octave. If one wants to show Pythagorean tuning extended to more than these, one merely has to extend the line representing it (blue in my figure, red in yours) on both sides, in order to reach as many degrees as desired. Or else, one may conceive the drawing as if drawn around a cylinder, with G# at the right on the same vertical axis as Ab on the left, and with a second turn after G#, continuing with D#, A#, E#, etc., and a third turn 'before' Ab, descending Db, Gb, Cb, etc.: the extended Pythagorean would then be represented as a spiral (this is a well-known representation). The unfolded cylinder would show several lines parallel to the main one, and the number of parallel lines would depend on the extension given to the Pythagorean system. This would form a convenient illustration of 3-limit, and a complete one if the limit of the extension were stated. [I didn't check the extension of the 3-limit in the list; I can only suspect (or hope) that it may be complete to some unstated extension.] I will not draw the figure, but I may represent it in tabular form as follows:

816

G#

D#

A#

E#

B#

Fx

Cx

Gx

Dx

Ax

Ex

Bx

F###

840

792

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

816

768

Bbbb

Fbb

Cbb

Gbb

Dbb

Abb

Ebb

Bbb

Fb

Cb

Gb

Db

Ab

792

i.e. three lines, each with the same slope as the line labelled "Pythagorean", and each a Pythagorean comma higher than the preceeding one.

In your representation of just intonation, your brown zigzag line labelled "just", presented in the form of a Tonnetz (fifths as rows, major thirds as columns), would be like this:

		Ab	Eb	Bb
	F	C	G	D
	A	E	B
F#	C#	G#

As you can see, your presentation is somewhat irregular, with F# being the major third of no other note: F# should better stand at the right of B, i.e. on the same line as A E B in your drawing. [Note also that the four lines of this Tonnetz are distant from each other by a syntonic comma, as is usual in such representations.] It is inherent to just intonation itself that it can be (and has been) presented in many different ways: any representation that you would chose would represent only one version of just intonation. As a matter of fact, one would need each degree both as the fifth of another and as the major third of another still. This soon results in way more than 12 degrees, e.g. in instruments like Vicentino's arcicembalo with 36 keys in the octave, and in the Tonnetz itself being a torus: see File:TonnetzTorus.gif.

Never mind. Our problem now is to represent extended 5-limit intonation. That can be achieved by extending each of the fragmentary brown lines in your image at both sides, to reach both Ab and G# in pure fifths. One would again have parallel sloping lines as in the extended 3-limit described above, but now the lines would be a syntonic comma apart instead of a Pythagorean one! Combining 3-limit and 5-limit in a single figure would result in something like this:

816

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

840

814

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

838

792

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

816

770

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

794

768

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

792

with the parallel lines 816-840 and 814-838, or 770-794 and 768=792 being a schisma (~2 cents) apart, the difference between a Pythagorean and a syntonic comma. Such a figure would indeed represent extended 3- and 5-limits, with a number of intervals to be measured from C in the center. In addition, the lines a syntonic comma apart could also be extended to spiral aroung the cylinder, but the whole would soon become utterly unreadable, reducing to a set of parallel lines 2 cents apart.

To sum up, if my figure were to be changed, (1) the two lines representing meantone temperaments should be removed, as they have no place here; (b) both the "Pythagorean" line and your zigzag line representing just intonation should be extended to represent 3- and 5-limits. I cannot figure how to represent 7-limit, unless by adding a third dimension – our computer screens cannot yet do that, but it will come.

Enough for today. – Hucbald.SaintAmand (talk) 08:16, 9 September 2014 (UTC)

Sorry, I believe I meant the display size, and not the image itself. But I was being a sarcastic jerk, for which I apologize. In the meantime it appears that I have created an image: File:Meantone comparison.png, as well as File:Meantone comparison just.png and File:Meantone comparison Pythagorean.png. Hyacinth (talk) 20:51, 27 May 2015 (UTC)

These indeed are interesting images, Hyacinth. Yet, I have a problem with the representation of just intonation. Such images work well with "regular" temperaments (i.e. temperaments with one size of 5th), as they are meant to show the size of the 5ths in comparison with some other case (ET, or Pythagorean, etc.). They hardly show the size of 3ds, which are considered resultant from three or four 5ths. They also show the size of the 5ths in the case of just intonation, particularly the fact that some 5ths are not the same size as others; but they hardly show why. It is only when one observes that the line representing just intonation sort of follows that representing 1/4 comma meantone that one might begin to suspect the reason. An additional problem is that just intonation can be built in many ways. (This already was discussed above, 9 September 2014, when I noted that in your description F# is the 3d of no other note.) In your comparison meantone-just, the increasing divergence between the lines at the right of the figure is caused by your particular construction of just intonation. If you had built a series of three 5ths A-E-B-F# (as is more usual), the figure would be less divergent. It would be tricky to explain this to the reader (I don't even know whether you can follow what I just tried to explain!). — Hucbald.SaintAmand (talk) 14:04, 28 May 2015 (UTC)

Hyacinth, although you removed your last changes before I saw them, they made me realize that I was wrong on my last point: your comparison meantone-just did include an A-B-E-F# span, which I overlooked. Sorry for that. (Or else you changed your image, but I don't think so.)

On the other hand, I have some doubts about the inclusion of the "schismatic temperament", which obviously belongs to another level of reflexion. Meantone temperaments, ET and Pythagorean tunings are all historical tunings, documented since centuries, while the schismatic temperament (1) is a recent construction (2007, I think); (2) has never been and couldn't be utilized to tune real instruments; (3) is only a mathematical construct aiming at explaining micro intervals. Not that this would be a wrong purpose, but that it distorts the image given, mixing historical facts with recent speculations. I may be too traditional a musicologist too easily accept this, but my opinion on this point might be worth recording here, if not in the article itself... Hucbald.SaintAmand (talk) 20:19, 1 June 2015 (UTC)

The image is also "distorted" (I assume you mean "distorting") in that it contains five other unlabeled syntonic commas. Hyacinth (talk) 04:28, 2 June 2015 (UTC)

I'm sure Helmholtz died before 2007, yet his On the Sensations of Tone mentions "skhismic temperament" on page 435. Hyacinth (talk) 05:02, 2 June 2015 (UTC)

;-)) You should never forget that English is not my native language. It is not your image that is distorted, it is the view that it gives of the situation (the image given by your schema, if you want). As to Helmholtz, several points:

1. The mention of the "skhismic temperament" on p. 435 is among the "Additions by the translator", i.e. by Ellis. What Helmholtz himself has to say about this can be found on p. 280 ff. of the translation, about the "Arabic and Persian musical system"; see especially Ellis' footnote * on p. 281.

2.1. I fail to see why Ellis calls this a "temperament". He says that the 5ths have to be "perfect", 701,955 cents [note that a temperament, as Lindley evidenced, always tempers the 5ths], i.e. Pythagorean fifths, and the Skhisma disregarded, which results in his "K" (the comma) amounting to 23.460 cents (i.e. the Pythagorean comma) and "T" (the major third) 384,360 cents, i.e. a Pythagorean 3d diminished by a Pythagorean (instead of a syntonic) comma.

2.2. This is not a temperament because no interval is tempered. The fifths are perfect and the only question is how one comes to such thirds. Ellis gives the answer when he says (p. 435) that c:fb is "such a major third", then quotes a:db:e and e:ab:b as "quite smooth" triads (see also p. 280). This is what Liberty Manik (Das arabische Tonsystem im Mittelalter, 1969) described as Safi al Din's Schismatische Verwechslung, and what was used by Arnaut de Zwolle and others in the 15th century as an early just intonation (as commented by Mark Lindley in several papers — see Schismatic temperament#History of schismatic temperaments).

3. It certainly is not what your figure describes as "schismatic temperament", in which (if I read correctly) the 5ths are tempered by 1/12 of a schisma and measure 701,8 cents — that is that the difference between Ab at the left and G# at the right of your line for the schismatic temperament is a syntonic comma of 21,506 cents instead of a Pythagorean one of 23,460 cents. This is a 1/12-schisma temperament and it belongs to the schismatic temperaments described by Milne, Sethares and Plamondon in Computer Music Journal 31/4 (2007) — but I agree that this is not the earliest description of such things.

Note that the Schismatic temperament article is a mess, probably another product of the gang of microtonalists [I know that my use "gang" here is improper English but, as said above, I am not a native speaker]. I fail to see what "tempering the schisma to a unison" might mean; the article at times speaks of "tempering out", at other times of "tempering to (some interval)", neither of which makes sense to me [and this is not because I don't understand English]; the name "Helmholtz temperament" is undocumented and probably comes from [6]; Mark Lindley never said (nor believed) that schismatic tuning as described in the article ever was in use during the late Middle Ages: there is a confusion here between Ellis' (or Safi al Din's) skhismic temperament (see above) and the "schismatic temperaments" to which the article is devoted. Helmholtz did describe something ressembling a schismatic temperament (p. 512 of the German version, 316 of the translation), but did not describe it in terms of fractions of a schisma: he used 1/8 of the amount by which a 5th is tempered in 12-T ET — it boils down to almost the same, but not exactly.

Hucbald.SaintAmand (talk) 09:29, 2 June 2015 (UTC)

You got the year 2007 from Syntonic temperament. Hyacinth (talk) 17:14, 2 June 2015 (UTC)

2007 is the date of the paper by Milne e.a. in Computer Music Journal. I just had a look to Syntonic temperament, which I had never seen before. It seems almost identical with Schismatic temperament. Both might better be reassembled under, say, "Regular temperament", which I don't think is a term coined by Erv Wilson; it is used, if my memory doesn't fail, by Murray Barbour and several others. Regular temperaments are also meantone ones... Wikipedia obviously does not provide (and does not have) a very clear idea of all this. — Hucbald.SaintAmand (talk) 19:23, 2 June 2015 (UTC)

See: Schismatic temperament#Construction. Hyacinth (talk) 19:04, 3 June 2015 (UTC)

Lets look at this together, Hyacinth.

In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents). The major third above C, E, is considered five fifths above C.

We will easily agree that E really is four fifths above C, not five. Never mind.

This causes the Pythagorean major third, E+ (407.82 cents), to differ from the just major third, E♮ (386.31 cents): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma).

Helmholtz's "skhismic temperament" (Helmholtz/Ellis 1885 p. 435) instead uses the note eight fifths below C, F♭ (384.36 cents), the Pythagorean diminished fourth or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E♮, and thus more in tune than the Pythagorean major third. As Helmholtz puts it, "the Fifths should be perfect and the Skhisma should be disregarded.

I think that it is Ellis who says that at this point, not Helmholtz, but never mind. This, if I may repeat myself, is Safi al Din's Schismatische Verwechslung as discussed by L. Manik, or Arnaut de Zwolle's (and others') tuning system as discussed by M. Lindley. It is NOT a temperament: the 5ths remain pure, as clearly stated later in the same article (Schismatic temperament#History of schismatic temperaments: "Mark Lindley and Ronald Turner-Smith (1993) argue that schismatic tuning was briefly in use during the late medieval period. This was not temperament but merely 12-tone Pythagorean tuning, though typically tuned from G♭ to B in ascending just fifths and descending just fourths, instead of the prevalent A♭ to C♯ or E♭ to G♯ schemes."

In his 1/8th-schisma "Helmholtzian temperament" (Helmholtz/Ellis 1885 p. 435) the note eight fifths below C is used as the major third above C, but it, rather than the perfect fifth, is assumed to be perfectly in tune.

No, by no means! In Helmholtz/Ellis/Safi al Din/Arnaut de Zwolle's tuning, the 5ths remain perfectly in tune, nothing is tempered by 1/8th-schisma, this is a confusion between two different systems! The 3d is a schisma flat, but this is assumed negligeable. The Wikipedia article, I repeat, confuses here two systems, both described by Helmholtz, one that Ellis names "skhismic temperament" (improperly so because it is not a temperament), and the other Helmholtz' tuning of his harmonium as described on p. 316 of Ellis' translation (and which truly is a temperament). Even this temperament, however, is not strictly speaking an 1/8th-schisma temperament, because Helmholtz clearly says that the 5ths are tempered by 1/8th of the difference between a perfect 5th and a 5th in equal temperament. This difference is almost equal to a schisma, but not exactly so. And, as a result, the major third in this temperament of Helmholtz is exactly in tune, not a schisma flat. Because the harmonium has 34 keys in the octave, it cannot really be said that the major third is C-F♭, even although in essence that is what it is.

As Helmholtz puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded.

This is an inaccurate repeat of a quotation made two lines above and once again, it is not Helmholtz, but Ellis who puts this. What Ellis really writes (p. 435) is "The conditions is that the Fifths should be perfect and the Skhisma should be disregarded". That is to say, the fifths should remain perfect (as in Pythagorean tuning), and the resulting difference of a schisma in the thirds should be neglected. There is no mention whatsoever — never, neither in Helmholtz nor in Ellis — of a temperament by 1/8th-schisma. With all due indulgence for Wikipedia, this article is obviouly confused and mistaken. It renders unduly complex things that in essence are much simpler. — Hucbald.SaintAmand (talk) 21:35, 3 June 2015 (UTC)

Moved to: Talk:Schismatic temperament#Image
Hyacinth (talk) 15:18, 4 June 2015 (UTC)

Color codes II

I propose the color coding below. Right now the colors are less obtrusive, but more indistinguishable. Hyacinth (talk) 12:04, 16 July 2015 (UTC)

New Code	Compliant?	Legend (simplified)	Old Code	Compliant?
E	YES	12TET.	E	YES
Q	YES	24TET.	Q	YES
2	YES	2-limit.	2	YES
3	YES	3-limit/Pythagorean.	3	YES
5	YES	5-limit/just.	5	YES
7	YES	7-limit/ septimal.	7	YES
11	YES	11-limit.	11	YES
13	YES	13-limit.	13	YES
17	YES	17-limit.	17	YES
19	YES	19-limit.	19	YES
23	YES	23-limit.
29	YES	29-limit.
M	YES	Meantone temperament.	M	YES
U	YES	Measure/higher ET.	U	YES
S	NA	Superparticular (no code).	S
H	YES	Higher harmonics.	H	YES

As one climbs the limits one progresses through the rainbow.

12TET is made the same color (reddish) as 2-limit, since 2-limit only includes octaves.

24-TET is made a pinkish red.

Meantone is made a brown to match the color (orange) Pythagorean happens to be according to the rainbow pattern.

Units of measurement or equal temperament is made pink (12 is red, 24 is red/pink, > is pink).

Higher harmonics are gray.

Hyacinth (talk) 12:04, 16 July 2015 (UTC)

One problem is that the text becomes hard to read if the background colors are not soft enough. Especially for people who are color blind. SharkD  Talk  06:14, 13 November 2016 (UTC)

What should soft color and hard color link to? Hyacinth (talk) 23:56, 6 April 2017 (UTC)

Made a guess. Hyacinth (talk) 00:32, 7 April 2017 (UTC)

Other tuning images

Is there any interest in me creating more of images like this, but with different tunings? SharkD  Talk  01:38, 14 November 2016 (UTC)

See: WP:BOLD and WP:CAREFUL. Hyacinth (talk) 01:10, 5 February 2020 (UTC)

Superparticular ratio

There is a column in the table to mark superparticular ratios. The corresponding WP page states they are of importance in music theory, but that page does not mention them. I cannot possibly see any musical relevance of these ratios, except the small ones that naturally occur in limit-tunings. Therefore I propose to eliminate this column. −Woodstone (talk) 14:14, 1 November 2020 (UTC)

Inconsistency in mathematical data in 2 rows

Hi, I am parsing this wikipedia table for a microtonal project with (Python/Pandas, JavaScript/Mathjs), So I am adding a numerical ratio field (that makes much more sense from the polychromatic/microtonal perspective of my project) after looking the data there are Inconsistencies, things that doesn't match between the cents and the ratio expressed as a formula.

I prefer to just warn about this that change it myself, I am only reporting for the mathemathical inconsistency I don't know if there are other issues or what are the right values. (I am sorry for the format) the end number is the Ratio, also the fractions are represented in a way that Pandas or Python could be get the value with pd.eval or eval

ID Cents
24	23.46	B♯+++	531441/524288	3**(12)/2**(19)	Pythagorean comma,[10][11] ditonic comma	3	1.013643265
316	1223.46	B♯+++	531441/524288	3**(12)/2**(18) Pythagorean augmented seventh	3	1.013643265
531441/524288=1.013643265
3**(12)/2**(18)=2.02728652954102
531441/262144 = 2.02728652954102 <== so this is the right value

237	833.09		5**(1/2)+1/2	φ/1		Golden ratio (833 cents scale)		2.736067977
238	833.11		233/144		233/2**(4)×3**(2)	Golden ratio approximation (833 cents scale)	233	1.618055556
233/144 = 1.61805555555556 = 233/2**(4)×3**(2)
5**(1/2)+1/2 = 2.736067977 = 1742.524889 Cents, this formula maybe wrong, also why two rows pointing to the golden ration?

For the purpose of tuning, notes are conventionally considered octave repeating, and intervals are therefore reduced by factors two to stay under 2. So the second one does not really belong in the table (just as the other few over 2). These seem to be given because they have special name. That is not a good idea, because there is not end to it in view of wider intervals, many of which are common in jazz harmony. They do not add essentially different interval types, only their names.

The golden ratio is (5^1/2+1)/2 or 833.09 cents. You have the brackets wrong. The other one is a rational approximation to the golden ratio. Both are curiosities and have no musical meaning.

−Woodstone (talk) 15:52, 16 July 2021 (UTC)

Agree, ratios should be given between the octave only.

I don't see any logical reason to have two golden ratios in this list or to have it without the right mathemathical notation as you did mention (5^1/2+1)/2 instead of 5^1/2+1/2 ::-49.230.136.53 (talk) 09:00, 18 July 2021 (UTC)

2-limit

Unisons and octaves should be 2-limit, not 3-limit, according to http://www.tonalsoft.com/enc/l/limit.aspx — Omegatron (talk) 01:00, 27 December 2022 (UTC)