Talk:Modulatory space

Needs fixing
This article is way too mathy, and seems a little idiosyncratic to me. The very title seems a little misconceived, since it's theoretically possible to "modulate" from one seven-note collection of pitches to another. Why should modulation necessarily be represented by pitch classes?

If it were up to me, this whole article would be retitled: something like "representations of pitch class spaces using discrete, just-intonation models." The abstract mathematical language would then be replaced with something more concrete and musical. - Tymoczko (signed by Hyacinth)

Original research
The term "modulatory space" appears to have been coined on Wikipedia by Gene Ward Smith. Hyacinth 10:58, 29 March 2006 (UTC)


 * I meant pitch class space. Talk:Pitch class space. Hyacinth 09:48, 2 April 2006 (UTC)


 * Do any of the references use the term "modulatory space", or is this really a neologism? Regarding the use of mathematical language: there's nothing wrong with using mathematics to study music, and I find the whole topic of the mathematics of musical tonality and tuning fascinating -- applying group theory to it is intriguing. However, I'm concerned that Wikipedia might not be the right place for it yet, unless it can be shown to already be in use in the outside world: sadly, if this is original research, we are going to have to delete it. -- The Anome 11:09, 29 March 2006 (UTC)


 * I personally am in favor of deleting this page. As far as I know, "modulatory space" really is a neologism -- I've certainly never encountered it anywhere else.  More troubling, to me, is the fact that the underlying viewpoint here is very idiosyncratic.  For instance, modulation occurs in all sorts of spaces -- twelve-tone equal temperament, the unequal fixed temperaments of earlier tuning systems -- in nothing like the way described here.  So this article really isn't about modulation, or the "spaces" in which modulation occurs.  It's about modulation as understood from a very particular just intonation standpoint. Tymoczko 02:24, 1 April 2006 (UTC)

I must take my above comment back as I created this article as a redirect. Obvsiously I either read it somewhere or the concept is so basic as to not be original research. Apologies to Gene Ward Smith. Hyacinth 08:50, 1 April 2006 (UTC)


 * I took the name "modulatory space" from the redirect, but not before checking that it had, in fact, been in occasional use--for instance in "Schoenberg's Modulatory Calculations" by Murray Dineen. Hence, I've changed the intro sentence to say it is not a term in regular use. As for too math, the topic is mathematical by its nature--"space". It could be modified to explain the concepts in less mathematical terms, but there are a lot of things in music theory which are mathematical. I think there is a bit of a double standard here--since music is a humanity, you can say "too mathematical" more than you would in, say, physics, where you'd just be laughed at.


 * The stuff on seven-limit lattices of chords has been around for a number of years now on the net, and has a web based exposition. Is there a rule which says that print publication is required? I'd like some feedback on that question. Gene Ward Smith 18:24, 1 April 2006 (UTC)


 * Did you mean to label the five limit section with the original research tag? If so it conventionally goes at the top of the section. Hyacinth 18:36, 1 April 2006 (UTC)

First, does Dineen use the term "modulatory space" in the sense discussed in the article? I'd be surprised if so. Schoenberg's modulatory space, at least as discussed in "Structural Funcitons of Harmony," is much more closely related to the chord spaces discussed in the Wikipedia article by that name. Second, a search of JSTOR and RILM, which archive a substantial portion of the published scholarly articles on music turned up no hits for "modulatory space."

In any case, to me the issue is not the term "modulatory space" but rather the content of the article. The article says that modulation occurs in spaces with certain features. But the claim itself is not true -- modulation can occur in a much wider variety of spaces. For example, equal-tempered music modulates, but does not fit into the categories laid out in this article.

I don't think that print publication is required, and I don't think that neologisms should always be avoided. I certainly believe that music theory can be modeled mathematically. I do think that web material needs to be looked at carefully -- some of what's on the web is very idiosyncratic, crankish, and unreliable.

My main problem is with the title and presentation of this article. It suggests it talks about "modulation" quite generally, but doesn't. A compromise is to retitle this article so that it specifically refers to modulation as understood by just intonation theorists, or to "pure interval lattices" or something of the sort. Right now it promises more than it delivers. Tymoczko 20:29, 1 April 2006 (UTC)

I read the Dineen article. In my view it doesn't relate to the content of this entry. The term "modulatory space" appears only in passing, and doesn't really suggest the approach taken here. Tymoczko 20:37, 1 April 2006 (UTC)

Original research
I'm putting the original research tag back on. If we move this article elsewhere, and make clear that it's about just intonation, I'd be up for removing it. Tymoczko 14:47, 2 April 2006 (UTC)


 * What does that have to do with original research? Why don't you just generalize the article? Hyacinth 16:37, 2 April 2006 (UTC)


 * I used the original research tag because of what struck me as an extremely idiosyncratic approach to the subject. "Modulatory spaces," as I would understand the (neologistic) term, would include spaces whose points are keys.  A lot of the material in here doesn't seem relevant to that -- what's discussed here are just-intonation based pitch class lattices.  So my first step in generalizing would be to remove most of what's currently in the article.  I'd then present material from Krumhansl on toroidal key-spaces, etc., that seems relevant.  Tymoczko 18:27, 2 April 2006 (UTC)


 * Are you saying this article is currently about pitch or pitch class space in just intonation? Hyacinth 10:51, 3 April 2006 (UTC)


 * The first thing I'm saying is that I don't really know what this article is about. That is a bad sign -- if it's not comprehensible to me, it's not clear who it's going to be comprehensible to.  Second, yes, much of the material here seems to be about pitch and pitch class lattices in just intonation, and how to represent modulation on these lattices.  However, given the rather overwhelming level of detail, I do not favor transferring this material into the pitch and pitch class articles.  The "pitch" article already has some information about these lattices.  This is why I thought the best policy might be to retitle the article -- "pitch and pitch class in just intonation" might be a reasonable title. Tymoczko 15:44, 4 April 2006 (UTC)


 * The article is about pitch class space in systems of intonation. In other words, it covers pitch classes in equal temperaments, in meantone intonation, in just intonation, or what have you. That is not covered in the article on pitch class space, and it needs an article somewhere with some name attached to it. To began with, what name? We could certainly move the article if we had a good place to move it, with the meaning of "octave equivalent notes in some system of tuning". Gene Ward Smith 00:43, 10 May 2006 (UTC)


 * I recommend removing the material on equal temperament and renaming the article: "Pitch Class Lattices in Just Intonation." I also recommend adding some material about octave equivalence.  As per our discussion on the "pitch class" talk space, it seems to me that the geometrical models discussed here do not represent octave equivalence very well: they give no indication that one can take many perfect-fifth steps and end up perceptually very close to where one started.  This is an important feature of human music perception regardless of what tuning system one favors.  I also favor adding some graphics to illustrate the various lattices that are being discussed. Tymoczko 22:04, 13 May 2006 (UTC)


 * Why? Gene already pointed out that this applies equally well to any regular temperament as to just intonation. Also, I strongly disagree with your assertion that "one can take many perfect-fifth steps and end up perceptually very close to where one started". That's not even what "octave equivalence" means. —Keenan Pepper 01:16, 14 May 2006 (UTC)


 * If you're talking about pitch classes, then you're ignoring what octave you're in. If you ascend by 12 just fifths, you end up 7 octaves and 23.46 cents away from where you started.  Ignoring the octaves, this means you're just 23.46 cents away from where you began.  You can get even closer by taking more perfect-fifth steps and ignoring more octaves.  None of the lattices discussed in this article take this into account -- on a standard just-intonation lattice, 12 perfect-fifth steps away from your starting point leaves you 12 lattice steps away from where you started.  There's no indication that these steps have brought you 23.46 cents away from where you began.  Further steps on the lattice leave you ever-farther from where you began.  In this sense, the models disregard octave equivalence -- there's no representation of the fact that 12 just fifths brings you to a point that is very nearly octave equivalent to the starting point.  Explicit representation of octaves -- say along an additional spatial dimension -- doesn't help, unless one wants to create a severely distorted, non-Euclidean space. Tymoczko 16:09, 14 May 2006 (UTC)


 * But "23.46 cents away" is distance in physical pitch space, not modulatory space. There shouldn't be any indication that you're "23.46 cents away", because that has nothing to do with modulatory space. —Keenan Pepper 16:52, 14 May 2006 (UTC)


 * Wait. My point is that modulatory space is deeply unsatisfactory because it does not represent important features of human music perception.  My point is that the notion of distance used in "modulatory space" is not relevant to any perceptually real notion of musical distance.  Every human being everywhere is going to hear pitches that are separated by < .000000001 cents as very similar -- indeed, indistinguishable.  And yet in modulatory space, these pitches can be arbitrarily far apart.  If you look back two paragraphs, I said that one can take "many perfect-fifth steps and end up perceptually very close to where one started."  By which I mean, adjusting for octaves, one can take many perfect fifth steps and end up < .000000001 cents from where one started.  I don't see where you disagree with this. Tymoczko 23:30, 17 May 2006 (UTC)


 * Yes, I'm quite familiar with continued fractions and I understand that chains of incommensurable intervals come arbitrarily close to closed cycles. You don't have to keep repeating that. My point is that that has nothing to do with the purpose of modulatory space, which is to help people understand harmonic relationships. You say it's "deeply unsatisfactory", but I could just as easily say that physical pitch space is "deeply unsatisfactory" because it doesn't reflect consonance and dissonance. Every human being can tell the difference between a 702 cent interval and a 720 cent interval, even though they might not be able to tell the difference between a 600 cent interval and a 620 cent interval. Physical pitch space doesn't reflect that, but that doesn't mean it's useless, that just means it's a space for looking at one aspect of music which ignores others. Modulatory space just looks at different aspects. —Keenan Pepper 02:00, 18 May 2006 (UTC)


 * We agree that different musical spaces represent different things. Q: What, precisely, are these just intonational lattices supposed to represent?  It can't be anything about perception, since we agree that there's no way to locate a pitch aurally on the lattice.  If I play a note, you have no idea where I am on the lattice.  If you arbitrarily choose a point, and I play another one, you have no idea whether I've played a note close to your chosen point or very far from it.  So what is the space modelling?  Perhaps it's modelling how some people conceive of musical relationships -- but if so, that should be spelled out and explained.  Who are the people who think in terms of this lattice?  There's nothing about any of this in this article.  Nor have I ever seen a really compelling explanation of why it's a good idea to think in terms of these sorts of lattices.  What do they do for us that we can't get in other ways?  (BTW, the point about consonance and dissonance seems wrong to me: these depend on the overtone structure of a sound, not the pure ratio of fundamental frequencies.  With the right overtone structure, 620 cents can be made to beat and sound wrong, while 600 cents sounds consonant and in tune.  See Sethares, "Tuning, Timbre, Spectrum, Scale."  For this reason, you don't want pitch spaces to represent consonance and dissonance.)  Tymoczko 18:00, 18 May 2006 (UTC)