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'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?

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  • By 'mathematically', I think he meant 'by using patterns of tones/semitones', rather than pure maths. While the term 'modes' is technically correct, I wish he'd have come up with a different term - there's enough modes, as we know them, without adding to the list...
    – Tim
    Mar 2 at 13:53
  • @Tim Can you explain a bit more what you mean by your comment?
    – DPJDPJ
    Mar 2 at 14:28
  • Messiaen used the term mode because that's what a set of notes, with tone/semitone spaces, in particular order, is. It's just that there are many, many modes that already exist, such as church, Ambrosian, modern, all with their own special 'foibles', and differs, so when Mesiaen decided to embark on his thesis, adding his own 'modes' to what already existed muddied things (for me). Even calling them scales may have simplified things. Just my view...
    – Tim
    Mar 2 at 15:00
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Well, to give a boring answer to how it can be proved – since there are only finitely many subsets of the 12 tones (namely, 4096), you can just list them all and check exhaustively. But we can be a bit more efficient than that.

There's not really such a thing as limited transposition. What characterises these scales is that some of the transpositions will be enharmonically equivalent to the original. In other words, there's a transpositional symmetry in the scale. (Mathematically, you could say the group of transpositions contains nontrivial equivalences classes with respect to action on those scales.)

In effect, this boils down to these scales splitting up into several equal parts. Because 12 has the integer factors 2, 3, 4 and 6, that means you have the following options, omitting scales that are equivalent by global transposition and omitting the scales that have additional finer-grained symmetry:

  • Size-2 blocks: s0=□□, s1=■□, s2=■■
  • Size-3 blocks: □□□≃s0, s3=■□□, s4=■■□, ■■■=s2
  • Size-4 blocks: □□□□≃s0, s5=■□□□, s6=■■□□, ■□■□=s1, s7=■■■□, ■■■■=s2
  • Size-6 blocks: □□□□□□≃s0, s8=■□□□□□, s9=■■□□□□, s10=■□■□□□, ■□□■□□=s3, s11=■■■□□□, s12=■■□■□□, s13=■■□□■□, ■□■□■□≃s1, s14=■■■■□□, s15=■■■□■□, ■■□■■□≃s4, s16=■■■■■□, ■■■■■■≃s1

Putting it back together again, that means we have these 17 note-sets which have a transpositional symmetry:

□□□□□□□□□□□□, ■□□□□□■□□□□□, ■■□□□□■■□□□□, ■□□□■□□□■□□□, ■□■□□□■□■□□□, ■■■□□□■■■□□□, ■□□■□□■□□■□□, ■■□■□□■■□■□□, ■■□□■■□□■■□□, ■□■■□□■□■■□□, ■■■■□□■■■■□□, ■□■□■□■□■□■□, ■■■□■□■■■□■□, ■■□■■□■■□■■□, ■■■□■■■□■■■□, ■■■■■□■■■■■□, ■■■■■■■■■■■■

From these we can manually filter out those that Messiaen considers “modes”, as those that have no steps larger than two semitones and aren't simply the full chromatic scale:

■□■□■□■□■□■□, ■■■□■□■■■□■□, ■■□■■□■■□■■□, ■■■□■■■□■■■□, ■■■■■□■■■■■□

For reasons that are a bit mysterious to me, he also includes ■■■■□□■■■■□□ and ■■■□□□■■■□□□, though these have bigger steps in them.

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    I wonder if using Cayley tables would be clearer. Maybe clearer to the mathematical mind but less clear to the musical one. Mar 2 at 16:02
  • I know that this answer is incredibly useful, but from a mathematical perspective (I don't have too much of a music theory background) I somewhat struggle to comprehend it!
    – DPJDPJ
    Mar 3 at 1:06
  • @DPJDPJ so, what is not clear? –I mean, I didn't really use any group theory here since most people on the site don't have too much of a maths background. This could of course be expressed more elegantly by talking about subgroups, but I daresay the stupid manual enumeration approach is clear enough as well? Mar 3 at 9:11
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There are several papers on the subject. With any limited set of tones (12 in the Western Case) and choosing a subset (7 for example), there are only a finite number of combinations that fit. Here's an example. https://watermark.silverchair.com/11-2-187.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAApcwggKTBgkqhkiG9w0BBwagggKEMIICgAIBADCCAnkGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMQNoU_4sw2QMGddJfAgEQgIICSo0Fg44tm037cyi7Ycd0eU3Ua3NTss0FSuFqA5fOt5ULGv1Aqv1TCgkX7QbjKn4hF156Bdsdpl8H5v5OrreDEIuDOwuTYMbrKaP_gnD_i9ZTnPQfldpjKSa2LJATRveFtt6K9BCIIKQzgBjgSs68SfeLeD_Ziw2zTR01NJ4lOMg5Dls4-i1BbBGbY-x6yk_ZF2d0oeCJ57zTXDmPr35-hiQv62Vlmu_szzLDaJJVAO5WDox3L9J1OPJHXbA41HodeNm--BEmjYk_cE21KT2jiInz9bz_yl9K-ZQ9JTbnobRSAESB3aIXMI_ttetHsevIgMJco-GfkIwLEF66j4nl-LVFh_U7U7PqT67PdPMMuTPn9KVGM8i6OptH1aU4xbdM3BL1dQYNnnqw1Mivqxh9mgNHTMCNtfIfpeb2C-PMAdJsIVhCRH4mPjVJBSnvUBa1iSirhJ4pR8-qrVvmaWe71LnLPBk2yzWJ3WoWIVnj6hgqpZyxeDExT0Oy9NpkrrlIyHcUEpObtNLmK3HHj4rcGykQDmxiRyEIVCfayUByqhD7DGO06azXPSY3jwBa_xJq8LAsTvUjO5qorvSWQ_1zpHZsOzXHU74lC0SxRaLdv6ac7sUr2obZnO44C5k_FX7TzRYce8_BeMQkXUCravvxxaw3sakZiNmpOAr9QTAodUwYFORMRAnEgbBE3VbNwFsgL4RnH2cX7pWOHzF8GeHa-kFuH4XlQR4NJsubGf96nX4AO4PDkEEq4kAp2hgi2Zhv2-N_KZYZHsPPrPM (That's long enough to take one to a particular atom in the document or maybe a particular electron on Arcturus). For Western music, we consider a closed 8-tone scale (the last note an octave above the first). There are 12 half-step intervals to select from. There is a constraint that the intervals should span the octave. Two obvious choices are 12 half steps or 6 whole steps which yield the chromatic and whole-tone scale respectively. WIth 7 notes (the most common), one needs 5 whole steps and 2 half steps. Still, there are many arrangements. The major scale and its cyclic rotations are one possibility. While not derived this way historically, these are the most "balanced" inf various way. The link leads to one discussion on the subject. There are more (but I can't remember the exact links).

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