Talk:Major sixth

Removed

I also removed the sentence "The ratios of both major and minor sixths are the ratios of pairs of consecutive numbers of the Fibonacci sequence: 5 and 8 for a minor third and 3 and 5 for a major third, the golden ratio lying between the minor sixth and the major sixth." -- because it seems irrelevant or not musically noteworthy. Correct me if I'm wrong. Pfly 18:29, 15 March 2007 (UTC)

Not a stub

this article is not a stub and should not be marked as one. all the information that is needed is here. ₡ŏņñöř 14:49, 29 April 2007 (UTC)

This article says...

The most common occurrence of the major sixth is between the third and (upper) root of minor chords.

Please prove that this is a more common occurrence than between the fifth and third of second inversion major chords. Georgia guy (talk) 14:21, 24 June 2011 (UTC)

Why? Also, see Template:Citation needed. Hyacinth (talk) 04:07, 26 June 2011 (UTC)

Redirect on example MIDI

The Play button on the example MIDI is wrongly redirecting. I suspect that the bot made an error. It redirects to the MIDI for a diminished seventh. — Preceding unsigned comment added by Kingsocarso (talk • contribs) 17:46, 14 March 2013 (UTC)

Pythagorean Tuning

Pythagorean intervals should not be included as part of just intonation. Just intonation requires that all intervals be pure. In Pythagorean tuning, only the fifths (and presumably fourths) are pure, and thus the ratios for any unrelated intervals should not be included for just intonation in the infobox. Ezhao02 (talk) 02:06, 5 June 2017 (UTC)

Actually, the verification for such things ought to be in the article itself (source citations in infoboxes should be avoided as far as possible). In this case, the Pythagorean major sixth is in fact cited in the text, to John Fonville's article on Ben Johnston's just intonation, with specific reference to his Sixth String Quartet, which uses 11-limit JI (neither 27 nor 16 have prime factors larger than 11).—Jerome Kohl (talk) 17:50, 5 June 2017 (UTC)

It seems to me that the very idea of a Pythagorean sixth is far-fetched, to say the least – that is to say that to call it "Pythagorean", even if it is formed by pure fifths and even if mentioned in an existing reference (J. Fonville), appears to claim something that almost certainly does not belong to Pythagoras' time − Ben Johnston notwithstanding. The same is true, as a matter of fact, of the major third. And to claim that this "Pythagorean sixth" may be obtained "by playing 27th harmonic and the next lowest octave of the fundamental frequency (16) together" is mere nonsense because nobody can play the 27th harmonic, on any instrument. This is taking one's fantasies for truths. (Similarly, to speak of "The nineteenth subharmonic" is nonsense, merely because subharmonics don't exist.)

I can show some sympathy to modern microtonal theory, but I think that its advocates should be aware of the metaphoric characted of their theory which, at any rate, has nothing to do with Pythagorean theory of two and a half millenaries ago and with which it should not claim any link, unless at the most metaphoric level. — Hucbald.SaintAmand (talk) 21:05, 5 June 2017 (UTC)

Major sixths did not exist in Pythagoras' time? I don't understand. Neither do I understand what playing 27th harmonics has to do with it. Please explain.—Jerome Kohl (talk) 05:02, 6 June 2017 (UTC)

We certainly have no music from the time of Pythagoras, Jerome, we cannot know whether major sixths existed by then. But that is not really the question, which merely is whether the Pythagoreans were aware of the construction of the major sixth and described it. The construction itself is not extremely complex: three pure fifths minus one octave [(3:2)³/2]. But the fact is that they were working with a monochord and interested in string ratios. They put a mobile bridge at 2/5 of the string length to show that two parts on one side sounded a fifth higher than three parts on the other side. There is no possibility to put a second bridge anywhere to illustrate another fifth above the first, and they merely were not interested – at least for what one can imagine today of what they did. (Greek numerals, which were the letters of their alphabet, Α=1, Β=2, Γ=3, Δ=4, etc., did not help even rather simple calculations, but discussing that would lead us too far.)

I am aware that the term "Pythagorean" may not refer to the 5th century B.C. and might merely mean "built of pure fifths" – as when we speak of the "Pythagorean" scale, the "Pythagorean" major third, etc. But I dislike this usage, because it plays on the ambiguity and insidiously makes an unjustified claim of Antiquity. You were tricked yourself, since your question was whether major sixths existed "in Pythagoras' time".

The mention of the 27th harmonic is in the article itself, mixing integers with harmonics and, after having said that the "Pythagorean sixth" corresponds to the ratio 27:16 (a ratio that the Antique Pythagoreans certainly never considered), writing (in terms that are not as clear as mines) that it can be obtained between the 27th and the 16th harmonics. The article refers to Helmholtz about this, and indeed Helmholtz believed things about Pythagoras that may no more be trusted today; but the references to Helmholtz in the article in fact belong to the additions by the English translator, Alexander Ellis. Ellis does mention the "27th harmonic", but I believe he was thinking of the 27th harmonic number, not the 27th harmonic partial. The article says "playing the 27th harmonic", which certainly concerns a harmonic partial, and certainly makes no sense. — Hucbald.SaintAmand (talk) 07:56, 6 June 2017 (UTC)

I see that I should have read the article more closely. I assumed you were objecting to the inclusion of the 27:16 ration in the infobox, and to Fonville's use of the term "Pythagorean". Further, I cannot believe that, even as early as the 5th century BC the Greeks used scales that avoided the major sixth. I think in this case your argument may be too subtle. Still, it is now clear to me what your objection is. Thank you.—Jerome Kohl (talk) 02:07, 7 June 2017 (UTC)

The source given did not say that a Pythagorean major sixth is considered just intonation. Ezhao02 (talk) 02:08, 7 June 2017 (UTC)

Thanks for checking before I could do so.—Jerome Kohl (talk) 02:23, 7 June 2017 (UTC)

I made a few additional changes just after yours, Jerome:

– I removed the reference to Fonville, which seemed unneeded in view of the fact that the name "Pythagorean major sixth" already appears in Ellis/Helmholtz. I don't think that quoting a "primary" source in this case, instead of a "secundary" one (that I didn't read), could be considered Original Research.

– I made it clearer that the reference to Helmholtz in fact is to Ellis' additions in his translation, also for the second reference about the septimal major sixth. I don't think Helmholtz ever mentioned septimal just intonation, which appears to have been an important concern of Ellis.

– I removed the idea of "playing" the 27th harmonic. It does not change much, but at least it removes what may be an overinterpretation of Ellis. What Ellis gives is merely a long table of "Intervals not exceeding one octave"; at 906 cents, he writes "The 27th harmonic, Pythagorean major sixth". In what sense he meant "the 27th harmonic" is unclear...

– I mentioned that the septimal major sixth is the inversion of the septimal minor third, in order to add a link to that article. All these articles about septimal tuning refer to Helmholtz, where they should mention Ellis. But that's not my job, I hate these microtonal people always rewriting the history of theory to suit their phantasms.

Hucbald.SaintAmand (talk) 18:02, 7 June 2017 (UTC)

Looks good to me, for what it is worth.—Jerome Kohl (talk) 18:25, 7 June 2017 (UTC)