Talk:Pitch class space

Lerdahl

Now that the Lerdahl stuff has been incorporated into this article, I added a little more detail explaining the relation between the chromatic circular model, and Lerdahl's 2D "bar graph."

Nice work. However, I removed the following:

• "The individual levels are also supposed to represent melodic 'alphabets' that capture the kind of melodic motions typically found in tonal music. Thus the model asserts that tonal melodies frequently arpeggiate the tonic triad (level three) but only infrequently move by perfect fourth. This is certainly an accurate description of classical tonality, though in other tonal styles, such as jazz, motion by perfect fourth is reasonably common."

Lerdahl isn't claiming that the octave is the most common melodic motion in tonal music (that would make boring music). Knopp's quote clarifies (below the "graph"): "Tonal space...is the abstract nexus of possible normative harmonic connections in a system, as opposed to the actual series of temporal connections in a realized work, linear or otherwise." Hyacinth 09:43, 2 April 2006 (UTC)

Have you read this book? Lerdahl explicitly endorses the claim that the levels provide melodic alphabets. Page 44: "they hypothesize that listeners structure tonal sequences by means of hierarchically organized alphabets. The superordinate alphabet is the octave; then come triads and seventh chords, then the diatonic scale, and finally the chromatic ..." The point is note that the octave is most common, but that the 5 levels provide the alphabets. I'm reverting to the earlier version. Tymoczko 14:35, 2 April 2006 (UTC)

• "The individual levels are also supposed to represent melodic "alphabets" that capture the kind of melodic motions typically found in tonal music. Thus the model asserts that tonal melodies frequently arpeggiate the tonic triad (level three) but only infrequently move by perfect fourth. This is certainly an accurate description of classical tonality, though in other tonal styles, such as jazz, motion by perfect fourth is reasonably common."

I replaced the first sentence with the quote. The quote mentions alphabets and explains that listeners model relationships this way, not that all tonal music actually arpeggiates these levels, as the Knopp explains. Thus I removed the second sentence again. The model includes plenty of fourths, including all six in the chromatic scale. Thus, and that it replies to the previous removed sentence, I removed the third sentences again. Hyacinth 18:38, 2 April 2006 (UTC)

Look, your alteration misses the point completely. The point is that Lerdahl's pitch space does two separate things: 1) indicate the differing "importance" of the different notes; and 2) provide "melodic alphabets" governing the melodic motion. These two are in principle separable. Furthermore, the model does not include perfect fourths as a melodic alphabet. There is no level at which fourths appear consecutively. Melodic motion is supposed to take place by using consecutive pitches at a given level: thus C-E-G or C-D-E-F-G, etc. There isn't room for fourths in the model. Tymoczko 19:40, 2 April 2006 (UTC)

Actually, I take it back -- I think you're right about the perfect fourths. Lerdahl would say that melodic motion is cognized in terms of one of the five levels, and perfect fourths would be cognized either with reference to the diatonic or chromatic level. So, thanks for pressing the issue -- I learned something! Tymoczko 02:41, 3 April 2006 (UTC)

Original research

Sigh. This is what I apparently didn't have in mind when I posted the original research tag on Modulatory space. The term "pitch class space" appears to have been coined on Wikipedia by Gene Ward Smith. While I think the term is perfectly reasonable, and both pitch class and pitch space are accepted terms, my sources on pitch space don't mention it once. Hyacinth 09:49, 2 April 2006 (UTC)

What is going on here? We have a reasonable article about an accepted music theoretical term? Where is the original research? I have sources (Boretz, "Meta Variations") using the term "pitch class space" that date back to 1972. I'm removing the original research tag. Tymoczko 14:38, 2 April 2006 (UTC)

The tag is actually a request for you to cite or quote him. Hyacinth 18:56, 2 April 2006 (UTC)

Circular pitch class space

Why does the article say pitch class space is a circle? As with a lot of stuff here, I think this is very limited, both historicially and in terms of focusing exclusively on Western practice. It takes as the norm Western practice for the last century, and treats it as a universal law of nature. Is this NPOV? Gene Ward Smith 17:23, 8 April 2006 (UTC)

You might have a look at the article on pitch class before we discuss this further. That's a little more detailed, and might answer some of your questions. For what it's worth, octave equivalence is not a Western invention, nor is log-frequency space. Log-frequency space is a line, and a line modulo a translation is a circle. That part is just math. If you want to model the Balinese Gamelan's nearly equiheptatonic scale in pitch class space, that's not a problem -- just take the points (0, 1.7, 3.4, 5.1, 6.9, 8.6, 10.3). If you want to model just intonation in pitch class space, that's also doable. Tymoczko 02:31, 9 April 2006 (UTC)

R/Z is a circle, but that doesn't mean it's the only way to look at pitch classes. More natural for meantone, for example, is to view it as a chain of fifths. However, my main objection to the article is that it seems to view 12-equal as a law of nature, which it isn't. Gene Ward Smith 21:38, 11 April 2006 (UTC)

Are you thinking of the rare musics without octave equivalence? Hyacinth 07:57, 9 April 2006 (UTC)

No, I am thinking of the extremely common musics which do not use equal temperament. That includes, for instance, Western music over a large period of time. In both Pythagorean and meantone tunings, the pitch class space is a line, not a circle. In non-Western tunings, you might get a sort of a circle but not an equal one, based on octave-equivalencing a particular non-equal scale.

Gene Ward Smith

Of course you can represent just intonational lattices using a circle: you're just taking steps of various sizes as you go around, and you never return exactly to where you began. However, you do come infinitesimally close to where you began. It's precisely the circular structure of pitch class space that explains why you ascend by a million or so perfect fifths, and end up perceptually very very very very close to the pitch class from which you started. (NB: perceptually, you end up very close to where you started, not a million steps away!) You need to distinguish the underlying space (generated by octave equivalence and the translation invariance of pitch perception in log frequency space) from the pitch structures built within it (the lattices of equal temperament, just intonation, and whatnot). Tymoczko 22:46, 11 April 2006 (UTC)