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In music, 'Modes of limited transposition' are modes that have a limited availability of transpositions. Unlike a major scale that has $12$ possible unique transpositions, the seven modes of limited transposition have fewer than $12$ possibilities of transposition.

French Composer Olivier Messiaen coined this idea, and he wrote, "Their series is closed, it is mathematically impossible to find others, at least in our tempered system of $12$ semitones." I want to try and figure out why. Might anyone have any idea how I could use mathematics to prove this statement?

Edit: See Link in comments for ideas surrounding Group Theory

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  • $\begingroup$ Looking at the seven possible modes en.wikipedia.org/wiki/… one can see case-by-case that his holds. Don't know if there is any deeper analysis that would apply ? $\endgroup$ Mar 2, 2021 at 14:30
  • $\begingroup$ @TomCollinge I wonder if there is any group theory that could apply to this matter... arxiv.org/pdf/1407.5757.pdf Page 25 onwards of this article raises interesting ideas surrounding mathematical interpretation $\endgroup$
    – Nipster
    Mar 2, 2021 at 14:33
  • $\begingroup$ So, the question would be how to prove mathematically that there are no more possible modes. I'll be interested to see the answer....... $\endgroup$ Mar 2, 2021 at 15:08

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The reference you provided, arxiv.org/pdf/1407.5757.pdf, includes some uses of arithmetic modulo $12$ that suggest a handy way of identifying a mode of limited transposition.

First, identify the twelve pitch classes of conventional equal temperament with the numbers in $\mathbb Z_{12}$. A scale is an ordered list of numbers modulo $12$ obtained from the list $(t, t+1, t+2, \ldots, t+11)$ for some $t \in \mathbb Z_{12}$ by deleting zero or more elements (other than $t$) from that list of $12$ numbers. A transposition of a scale is the image of the scale under a function $x \mapsto x+k$ for some constant $k \in \mathbb Z_{12}$; a mode of a scale is a cyclic permutation (which may be the identity permutation) of the elements of the list.

For example, if we identify the pitch class C with $0 \in \mathbb Z_{12}$, the C major scale is $(0,2,4,5,7,9,11)$; the A major scale is a transposition of the C major scale, $(9,11,1,2,4,6,8)$; and the natural A minor scale (aka A Aeolian) is a mode of the C major scale, $(9,11,0,2,4,5,7)$.

The subset of elements of $\mathbb Z_{12}$ of a scale is the pitch collection of that scale; all modes of that scale have the same pitch collection.

A mode of limited transposition is a set of scales consisting of one scale and all modes of all its transpositions, such that the set of pitch collections belonging to this entire set of scales has fewer than $12$ members.

(The word "mode" is used in two very different ways here. We call A Aeolian a "mode" of C major because you can play A Aeolian simply by starting at the sixth note of C major and playing the same cycle of pitch classes. In contrast, Messaien's Mode $1$ and Mode $5$ are distinct "modes" because even though they have the same number of notes, there is no way to make Mode $5$ sound like Mode $1$ simply by starting on a different note in the scale. In the first sense, "mode" means "can use the same notes," whereas in the second sense, "mode" means "cannot use the same notes." In order to distinguish the two meanings, I will henceforth use the second sense of "mode" only in the complete phrase "mode of limited transposition.")

These definitions are more mathematical than musical. They allow the following to be scales:

\begin{align} & (0), \\ & (0, 6), \\ & (0, 4, 8), \\ & (0, 3, 6, 9). \end{align}

The first "scale" isn't much of a scale at all; you play it on a piano by striking the C key in each successive octave. The second "scale" goes up by augmented fourths -- C, F$\sharp$, C, F$\sharp$, C -- also not what you would usually consider a musical scale. The next two scales are the notes of an augmented triad and a diminished seventh chord, respectively, and would sound more like arpeggios than like scales. But it would take some tweaking of the definitions to eliminate these as scales.

According to the definitions, the last three of these scales belong to modes of limited transposition. Messiaen does not count them, perhaps because (as already noted) they do not appear to be scales in a musical sense.

But if we accept all of these scales as scales (and the last three as modes of limited transposition) and are willing to continue to accept other musically strange scales, it is not hard to show the following:

A scale belongs to a mode of limited transposition if and only if it has the same pitch collection as its image under $x \mapsto x+k$ where $k$ is congruent to a proper divisor of $12$ (that is, where $k \in \{1,2,3,4,6\} \subset \mathbb Z_{12}$).

To give a notion of how this might be proved, consider that in order for the transpositions of a scale to have fewer than $12$ pitch collections, at least two of those transpositions must have the same pitch collection. We can transpose those two scales to the original scale and the original scale transposed by $x \mapsto x+n$ for some $n\in\mathbb Z\setminus\{0\}$; then all the transpositions produced by composing $x \mapsto x+n$ one or more times have the same pitch collection as the original scale, and one of these transpositions is $x\mapsto x+k$ for $k$ congruent to a proper divisor of $12$.

So we can identify a mode of limited transposition (abbreviated MOLT in the text below) by a function $x\mapsto x+k$ for $k$ congruent to a proper divisor of $12$ and a subsequence of the sequence $(0, 1, 2, \ldots, k-1)$; the remainder of the scale is repeatedly applying $x\mapsto x+k$ to this sequence until the next application would produce the original sequence, and concatenating the resulting sequences onto the original subsequence. Moreover, without loss of generality we can require $0$ to be in the subsequence, because a MOLT will always contain a scale starting at $0$.

We merely need to watch for duplicates, because in some cases a MOLT can be generated in more than one way according to the preceding paragraph. In order to help avoid duplication, I will use the convention of always selecting the mode of the representative scale that starts with the longest run of semitones. Note that this convention means I always delete $k-1$ from the subsequence.

Starting with $k = 1$, the following MOLT has only one unique pitch collection:

  • $(0),\ x\mapsto x+1$ produces the scale $(0,1,2,3,4,5,6,7,8,9,10,11)$ (the chromatic scale).

For $k = 2$, the following MOLT has two pitch collections:

  • $(0),\ x\mapsto x+2$ produces the scale $(0,2,4,6,8,10)$. This is Messiaen's Mode $1.$

We do not count $(0,1),\ x\mapsto x+2$ since it produces the same MOLT as $(0),\ x\mapsto x+1$.

For $k = 3,$ each MOLT has three pitch collections:

  • $(0),\ x\mapsto x+3$. This is the diminished seventh chord already described.
  • $(0,1),\ x\mapsto x+3$. This is Messiaen's Mode $2.$

For $k = 4,$ each MOLT has four pitch collections:

  • $(0),\ x\mapsto x+4$. This is the augmented triad already described.
  • $(0,1),\ x\mapsto x+4$.
  • $(0,1,2),\ x\mapsto x+4$. This is a mode of Messiaen's Mode $3$ (that is, $(0,2,3),\ x\mapsto x+4$).

Note that $(0,2),\ x\mapsto x+4$ duplicates $(0),\ x\mapsto x+2.$

For $k = 6,$ each MOLT has six pitch collections. For scales with no semitones:

  • $(0),\ x\mapsto x+6$; augmented fourths.
  • $(0,2),\ x\mapsto x+6$ (a mode of $(0,4),\ x\mapsto x+6$).

Note that $(0,3),\ x\mapsto x+6$ is the same scale as $(0),\ x\mapsto x+3$. For runs of one semitone we have:

  • $(0,1),\ x\mapsto x+6$.
  • $(0,1,3),\ x\mapsto x+6$.
  • $(0,1,4),\ x\mapsto x+6$.

Note that $(0,1,3,4),\ x\mapsto x+6$ is the same scale as $(0,1),\ x\mapsto x+3.$ For runs of two semitones we have

  • $(0,1,2),\ x\mapsto x+6$. This is a mode of Messiaen's Mode $5$ (that is, $(0,1,5),\ x\mapsto x+6$).
  • $(0,1,2,4),\ x\mapsto x+6$. This is a mode of Messiaen's Mode $6$ (that is, $(0,2,4,5),\ x\mapsto x+6$).

For runs of three or more semitones:

  • $(0,1,2,3),\ x\mapsto x+6$. This is a mode of Messiaen's Mode $4$ (that is, $(0,1,2,5),\ x\mapsto x+6$).
  • $(0,1,2,3,4),\ x\mapsto x+6$. This is Messiaen's Mode $7.$

And that's all there are, $16$ modes of limited transposition in total. There may be additional properties a MOLT should have that the other nine do not; I just don't know specifically what they are. And it may be, as suggested in your source, that Messaien was simply wrong about the mathematical impossibility of finding a MOLT he did not count, even if his additional criteria are applied.


Note that the classification of modes of limited transposition above does not distinguish $(0,1,3,4,6,7,9,10)$ from $(0,2,3,5,6,8,9,11).$ I consider both of these to be Messaien's Mode $2,$ because if you simply transpose $(0,1,3,4,6,7,9,10)$ up by a whole step ($x\mapsto x+2$) and start playing at pitch $0$ of the new collection of pitches, you play $(0,2,3,5,6,8,9,11).$ The reason I think these are not distinct modes of limited transposition are because a MOLT is supposed not to have a tonic (so it should not matter which note of the scale is played first) and because Messaien never identifies two distinct "modes of limited transposition" that can be mapped one to the other in this way.

But if we distinguish a different mode within a MOLT for each transposition that gives a different set of pitch classes, with the restriction that one of those pitch classes must be $0$, then we get the following additional modes:

From $(0,1),\ x\mapsto x+3$, the additional mode $(0,2),\ x\mapsto x+3$.

From $(0,1),\ x\mapsto x+4$, the additional mode $(0,3),\ x\mapsto x+4$.

From $(0,1,2),\ x\mapsto x+4$, the additional modes $(0,1,3),\ x\mapsto x+4$ and $(0,2,3),\ x\mapsto x+4$.

From $(0,2),\ x\mapsto x+6$, the additional mode $(0,4),\ x\mapsto x+6$.

From $(0,1),\ x\mapsto x+6$, the additional mode $(0,5),\ x\mapsto x+6$.

From $(0,1,3),\ x\mapsto x+6$, the additional modes $(0,2,5),\ x\mapsto x+6$ and $(0,3,4),\ x\mapsto x+6$).

From $(0,1,4),\ x\mapsto x+6$, the additional modes $(0,3,5),\ x\mapsto x+6$ and $(0,2,3),\ x\mapsto x+6$.

From $(0,1,2),\ x\mapsto x+6$, the additional modes $(0,1,5),\ x\mapsto x+6$ and $(0,4,5),\ x\mapsto x+6$.

From $(0,1,2,4),\ x\mapsto x+6$, the additional modes $(0,1,3,5),\ x\mapsto x+6$ and $(0,2,4,5),\ x\mapsto x+6$ and $(0,2,3,4),\ x\mapsto x+6$.

From $(0,1,2,3),\ x\mapsto x+6$, the additional modes $(0,1,2,5),\ x\mapsto x+6$ and $(0,1,4,5),\ x\mapsto x+6$ and $(0,3,4,5),\ x\mapsto x+6$.

From $(0,1,2,3,4),\ x\mapsto x+6$, the additional modes $(0,1,2,3,5),\ x\mapsto x+6$ and $(0,1,2,4,5),\ x\mapsto x+6$ and $(0,1,3,4,5),\ x\mapsto x+6$ and $(0,2,3,4,5),\ x\mapsto x+6$.

I count $22$ additional modes this way. That is, there are a total of $38$ distinct modes (in the sense of "starting on a different degree of the scale") among the modes of limited transposition.

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    $\begingroup$ So a mode of limited transposition is one that has a non-trivial stabilizer under the action of $\Bbb{Z}_{12}$. Thanks for explaining this in a language I can understand! I may use this as an exercise one of these years :-) $\endgroup$ Mar 3, 2021 at 7:53
  • $\begingroup$ Thank you for such an exceptional answer. May I ask why $(0,2,3)$ with the mapping $x \mapsto x+6$ would be omitted, for example? Would these not form extra MOTL? $\endgroup$
    – Nipster
    Mar 11, 2021 at 13:16
  • $\begingroup$ Or alternatively, another example would be $(0,2,3)$ under the mapping $x \mapsto x+4$ would generate the scale $(0,2,3,4,6,7,8,10,11)$ $\endgroup$
    – Nipster
    Mar 11, 2021 at 13:32
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    $\begingroup$ In fact $(0,1,3),\ x\mapsto x+6$ is listed under $k = 6$, the second case of "runs of one semitone". The sequence $(0,2,3),\ x\mapsto x+6$ is listed as $(0,1,4),\ x\mapsto x+6$. I also explicitly called out $(0,2,3),\ x\mapsto x+4$ as an alternate spelling of $(0,1,2),\ x\mapsto x+4.$ I do not count scales as distinct MOLT when you can get a transposition of one scale from the other just by starting at a different place in the scale. $\endgroup$
    – David K
    Mar 11, 2021 at 13:46

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