The investigations on this page were induced by a section in The Topos of Music - chapter 11.3.4, to be precise. There, the motif classification that I described here is applied to Z12 x Z5 (five-rhythm) and then compared to Z12 x Z12. The all-over number of motifs in Z12 x Z5 is smaller (since the basic set is), but the number of isomorphy classes is larger (in case of 3 elements: 45 classes for Z12 x Z5, 26 for Z12 x Z12) - this means: Z12 x Z12 offers more possibilities to form isomorphic motifs. The question is raised whether the human ability to recognize motifs could somehow be connected to the isomorphy class behaviour - since, as is known, musicians often have problems with five-rhythms.
In the book, the question is treated mainly from the rhythm viewpoint, which corresponds to european music tradition with its 12 tones - but from the mathematic viewpoint, this is not necessary at all! You could as well vary the number of pitches per octave - and the hypothesis raised in chapter 11.3.4 would then mean that motifs in five-rhythm, which, in 12-tone equal temperament, are more difficult to classify then those in three- or four-rhythm, would look easier if the octave is divided into 5 or 10 equal parts.
This made me apply the method of my thesis once more and calculate the isomorphy classes of motifs in Z10 x Z10 and Z5 x Z5. The case Z10 x Z10 is interesting because it is comparatively close to Z12 x Z12, and Z5 x Z5 is because a division of the octave into five equal parts yields a scale that is clearly identifiable as pentatonic, which means it is possibly relevant for existing music, too.
The results can be seen below. In Z10 x Z10, there are 10 classes of 3-element motifs - the difference to Z12 x Z12 is in a comparable ratio to the mentioned case Z12 x Z12 vs. Z12 x Z5! This means that Z10 x Z10 can serve as a real model case for a test of the hypothesis.
So I started to make music in 5-tet and 10-tet - grateful for the tuning functionality of modern synthesizers! However, the strange sounds of 10-tet were quite a challenge that I have not mastered so far... But there are others who have! It was again through the internet that I learned that there is indeed a kind of "scene" dedicated to microtonal music - found, among other places, in the Yahoo forum tuning-math. I once asked a question concerning this hypothesis there; the answers I got were mostly tending to the negative. One argument (by Paul Erlich) I think especially considerable: a division of the octave into 5 equal parts yields, as said, a pentatonic scale - and pentatonic scales are found in many music styles around the world. The mentioned hypothesis would mean that we could expect an analogous tendency to five-rhythms. In Paul's opinion, such a tendency does not exist; I think it might at least be worth a systematic inquiry...
My experiments with five-tone equal temperament were a little more successful than with 10-tone - as a first result I present my composition "Asîmchômsaia". As for the question above, though, I have not found the answer yet (I still have problems with the five-rhythm - in any tuning...).
Motif classification in Z10 x Z10
Isomorphy classes of motifs with 2 elements are, as shown in Z12 x Z12, determined by the gcd of the parameter differences, up to invertible elements. Possible values are: 1 (isomorphic to 3, 7, 9), 2 (isomorphic to 4, 6, 8) and 5, so we have 3 classes.
For motifs with 3 elements, there are 10 classes, see the table below. The invariants "volume" and "subset classification" are, unfortunately, not sufficient to determine the class in this case (in contrary to the case Z12 x Z12).
Isomorphy classes of 3-element motifs in Z10 x Z10
| Representant | Kernel | Volume | Subset classes |
|---|---|---|---|
| (0,0), (1,0), (0,1) | 0 x 0 | 1 | 1,1,1 |
| (0,0), (1,0), (0,6) | 0 x Z2*(0,1) | 2 | 1,1,2 |
| (0,0), (6,0), (0,6) | 0 x Z2 x Z2 | 2 | 2,2,2 |
| (0,0), (5,0), (0,1) | Z5*(1,0) x 0 | 5 | 1,1,5 |
| (0,0), (1,0), (9,6) | Z5*(1,1) x 0 | 2 | 1,1,2 |
| (0,0), (5,0), (0,6) | Z5*(1,0) x Z2*(0,1) | 0 | 1,2,5 |
| (0,0), (6,0), (4,5) | Z5*(1,1) x Z2*(1,0) | 0 | 1,1,2 |
| (0,0), (1,0), (9,0) | Z5*(1,1) x Z2*(1,1) | 2 | 1,1,2 |
| (0,0), (6,0), (4,0) | Z5*(1,1) x Z2 x Z2 | 2 | 2,2,2 |
| (0,0), (5,0), (0,5) | Z5 x Z5 x 0 | 5 | 5,5,5 |
Motif classification in Z5 x Z5
Z5 x Z5 is again a quantum leap simpler than Z10 x Z10 (possibly even too simple...) On the one hand, the basic space is, of course, much smaller, and on the other hand 5 is prime and hence Z5 a field. This means that there is exactly one class of 2-element motifs since all non-vanishing numbers are invertible.
The reasoning for 3-element motifs goes as follows: Z5 being a field and hence Z5 x Z5 a vector space, every motif with non-vanishing volume can be mapped to {(0,0), (0,1), (1,0)} via an affine transformation. Those with volume 0 can be mapped to the x axis, where there are, up to translations, 2 variants ({0,1,2} und {0,1,3}); these are isomorphic as well. Hence we have 2 classes of motifs with 3 elements, and the volume is sufficient to determine the class.
For the classification of motifs with 4 elements I went back to the module-theoretic method. Module classes there are 8 - but only for 4 of them there exist musical motifs. Compared to Z12, everything is quite limited here...
Isomorphy classes of 4-element motifs in Z5 x Z5
Isomorphy classes of 4-element motifs in Z5 x Z5
| Representant | Kernel | Volume | Subset classes | (0,0), (1,0), (2,0), (0,1) | Z5*(1,0,0) | 1 | 0,1,1,1 |
|---|---|---|---|
| (0,0), (0,1), (1,0), (4,4) | Z5*(1,1,1) | 1 | 1,1,1,1 |
| (0,0), (0,1), (1,0) | (1,1) Z5*(1,1,4) | 1 | 1,1,1,1 |
| (0,0), (1,0), (2,0), (3,0) | Z5*(1,0,4) x Z5*(0,1,3) | 0 | 0,0,0,0 |
For the kernels 0, Z5*(1,0,0) x Z5*(0,1,0), Z5*(0,1,1) x Z5*(0,4,1) and Z5xZ5xZ5, there is no motif class.
My composition "Asîmchômsaia", in Z5 x Z5
Composing in Z5 x Z5 is a quite different experience than with 12 tones. The stock of pitches being so limited, there is not much to do harmonically; on the other side, it is precisely this limitedness that makes the application of real two-dimensional motif transformations possible without too much calculation. There are quite a few general affine transformations present in "Asîmchômsaia".
In case you do not care about mathematics: no problem, you needn't - the piece can well be listened to without it (well, so I hope at least...). In the other case, some remarks concerning the "making of" can be found on the bottom of this page.
Asîmchômsaia (SoundClick) - Asîmchômsaia (mx3.ch) - Asîmchômsaia (Jamendo)
About the "making of"
The piece is about 2' 40" long and can be subdivided into 4 parts:
- First, after a short percussion intro, there is a simple melody, in a Koto-like sound (the piece has a slightly east-easian or oriental flavor, suiting to the pentatonic scale). This melody, BTW, has been composed in the "traditional" way (without math! It is a coincidence that it is 10 bars long...).
- Then the accompaniment starts (Glockenspiel- or marimba-like sound). The accompaniment patterns are all orbits, i.e. they were created starting with one tone and repatedly applying a certain affine transformation until the starting point is reached again. The affine transformations that are used were derived from the main melody via selecting certain 3-element motifs of it and then calculating transformations mapping those motifs to each other. The percussion patterns, BTW, were created in exactly the same way.
Afterwards, the beginning theme is repeated, reinforced by a guitar sound and slightly enlarged. The additional parts are - guess what - parts of the original transformed under some of the affine transformations introduced above. - Another voice is introduced (horn sound). It plays an enlarged version of the main theme, whith enlargement factor about 2.5 (every measure corresponds to about two beats of the original - it is not a mathematical strict transformation this time). The other instruments continue to play orbit patterns. Koto and guitar (in unison here) play special orbits: all of them were created with the same affine transformation, starting in every measure with the tones the horn plays in the same measure.
- In the final section, the orbit idea is pushed to its extreme. The basis is again an affine transformation, whose orbits are all concatenated. The resulting melody is played by all instruments in a canon, with the orbits successively adding up until the whole space Z5 x Z5 is filled.