Talk:Pitch space

Linear pitch space
The linear pitch space formula seems wrong. It says,

"A fundamental frequency f is mapped to a real number p according to the equation



p = 69 + \log_2 {(f/440)} $$

This creates a linear space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and middle C is assigned the number 60."

I assume 440 is the frequency in Hertz of the 'A' above middle C. Plug that in the formula and you get

69 + \log_2 {(440/440)} = 69 + \log_2 {(1)} = 69 + 0 = 69. $$

Plug in 880 for the 'A' an octave higher and you get

69 + \log_2 {(880/440)} = 69 + \log_2 {(2)} = 69 + 1 = 70. $$ According to the description, the higher 'A' should correspond to 69 + 12 = 81. So the formula should be corrected to



69 + 12*\log_2 {(f/440)} $$

If no one corrects my interpretation in the next few days, I'll change the entry. Gyan 22:33, 28 March 2006 (UTC)


 * That's right, thanks -- sorry about the typo. I'd probably prefer to leave out the "*" for multiplication, which looks computery rather than mathy, but that's not very important.  Tymoczko 02:12, 1 April 2006 (UTC)


 * This appears to have been changed back. I suspect that some editors believe that the "69" edit is vandalism. I will try to correct it again; this has been wrong for a very long time now. Geoff (talk) 03:07, 2 April 2008 (UTC)

Post-Schoenbergian
the number 1954 in connection with Arnold Schoenberg is somewhat unlikely. he died three years before. Arno


 * It may have been published posthumously, though (whatever it was). --Camembert


 * Schoenberg, A. (1954) Structural functions of harmony. Norton, New York. Hyacinth 01:35, 25 Apr 2004 (UTC)

Tonnetz chart
Lerdahl incorrectly depicts the Tonnetz of Hugo Riemann, so I removed the chart. If we can find a name to go with it, it belongs in Modulatory space in any case. The 3-limit Partch tonality diamond seems kind of silly, and belongs in Tonality diamond if anywhere.

The correct Tonnetz uses the triaxially symmetrical hexagonal lattice, as can be seen from this:


 * 

209.68.147.75 20:05, 14 December 2005 (UTC)

Hugo Riemann's Tonnetz, which models modulatory space as a graph or lattice, is:
 * Perfect fifths are the horizontal axis, major thirds the vertical. First proposed by Euler, later used, not always in just intonation, by Hermann von Helmholtz (1863/1885), Arthur von Oettingen (1866), Renate Imag (1970), Longuet-Higgins (1962), Shepard (1982) "harmonic map".

Harry Partch's Tonality Diamond is similar:

$$ \begin{matrix} \; & \frac{3}{3} & \; \\ \frac{4}{3} & \; & \frac{3}{2} \\ \; & \frac{2}{2} & \; \end{matrix}$$
 * 3-limit just intonation

Gene Ward Smith 20:06, 14 December 2005 (UTC)


 * What's wrong with the tonality diamond? Is it silly because of the small limit? Hyacinth 09:45, 2 April 2006 (UTC)

Fibered pitch space
I restored fibered pitch space, as no reason was given for removing it. Gene Ward Smith 00:03, 5 January 2006 (UTC)


 * Original research? Hyacinth 11:04, 29 March 2006 (UTC)

Substantial revisions
I've substantially reworked this entry to try to bring it in line with contemporary music-theoretical thinking. I tried to distinguish more clearly the linear models, including their helical variants, from the discrete higher-dimensional lattices. I'm restoring the picture of the two-dimensional lattice, as this is quite helpful. I hope nobody minds.

I removed the section on "fibered" spaces because this strikes me as inappropriate and distracting for an introductory general-purpose article. Furthermore, it's not clear that the helical models are "fibered" in an ordinary mathematical sense -- at the very least, they're continuous spaces with discrete fibers, which is getting a bit exotic for my taste.

As I mentioned in the article, the material about Lerdahl should really be moved to another article, as it is not a pitch space as defined by the article.

Tymoczko (signature added by Hyacinth)


 * What source did you use for the current definition of pitch space in the article? See below. Hyacinth 08:55, 1 April 2006 (UTC)


 * "Pitch space" and "pitch class space" are very common music theoretical terms. ("Pitch space" gets 23,000 hits from Google.)  In general, theorists use the following convention: in "X space," the points represent Xs.  In "pitch space" the points are pitches, in "pitch class space" the points are pitch classes, and in "chord space," the points are chords.  It's unfortunate that Lerdahl doesn't follow this convention, though most people do.  Tymoczko 20:46, 1 April 2006 (UTC)


 * Okay, cite one. Hyacinth 09:16, 2 April 2006 (UTC)


 * I think you have the wrong idea about citations. We're working on an encyclopedia article, not a high-school research paper.  It's OK to define basic technical terms and definitions without referring to some other source that does so.  Check out any of the Wikipedia articles on mathematics -- for instance "Group" (or for that matter, "addition").  They do not cite elementary textbooks -- they simply provide the relevant definitions.  In this case, "pitch space" and "pitch class space" are terms that are familiar to anybody who thinks about music professionally.  A citation would simply be distracting.  Tymoczko 21:25, 2 April 2006 (UTC)


 * I've added a bibliographic reference to Joe Straus, which is the most reliable source for basic post-tonal music theory. Tymoczko 22:08, 2 April 2006 (UTC)


 * Wikipedia isn't written for professionals. Hyacinth 10:46, 3 April 2006 (UTC)


 * Of course not. My point is that it's not written like a high-school term paper, and that basic concepts do not need to be endlessly and distractingly cited.  Terms like "pitch space" and "pitch class space," and "addition" and "group," are widely used and widely understood.  What is needed is a clear exposition of their meaning (for nonprofessionals), not a series of pedantic and pointless references to elementary texts.  Nobody writes "2 + 2 = 4 (Williams, Elementary Arithmetic, p. 1)."  The reference is not needed.  Check out the Wikipedia article on "Cell (biology)" if you want an example.  How many citations are there in that article? Tymoczko 15:40, 4 April 2006 (UTC)


 * Check out Cell (biology) now. Hyacinth (talk) 20:39, 10 July 2008 (UTC)

Linear and helical pitch space
"Care must be taken when interpreting these models however, as it is not clear how to interpret 'distance' in the three-dimensional space containing the helix; nor is it clear how to interpret points in the three-dimensional space not contained on the helix itself."

I'm not sure what this sentence means. My understanding is that representing pitch space as a helix is just embedding a one-dimensional space in three dimensions for conceptual convenience. Therefore the only meaningful "distance" would be along the helix, and the interpretation of points not contained on the helix would be clear: such points are not a part of the space. However, I haven't read the work from which this model is drawn, so I hesitate to make authoritative statements that might be original research. Perhaps it would be best simply to remove this sentence? Geoff (talk) 03:21, 2 April 2008 (UTC)


 * If I had a suitable picture I would add it. Imagine the linear pitch space wound on the surface of a cylinder so that all the octaves of C are one turn of the helix apart above each other. Then C# would be 30° (= 360°/12) to the right on the cylinder. Again with all the octaves of C# above each other. And so on.


 * I think that the sentence in question is correct. However one needs a bit of a mathematical mind to understand it. I will try it in this way: In linear pitch space the musical notes live on a equally spaced line. In helix pitch space the same notes live on the surface of a cylinder on the musical helix. The distance between two notes is measured on the surface of this cylinder... TomyDuby (talk) 20:18, 10 July 2008 (UTC)