# Mode of Limited Transposition for Microtonal Temperament

There's a recent paper discussing the theory of limited transposition for general n-temperament scale: https://www.semanticscholar.org/paper/Generalizing-Messiaen-%E2%80%99-s-Modes-of-Limited-to-a-Barat%C3%A8/cd47bf9eaaa4f656449a4a78313a255a5784b787

The theory itself is trivial mathematically speaking. There's always the possibility of limited transposition if n is a composite number.

However, I am curious if this concept is explored compositionally? Is there an application of this concept to produce anything meaningful musically?

• Accompanying the paper is a web app that demonstrates the application of the theory explained at messiaen.lim.di.unimi.it . You can change various parameters to get different modes. The start and end points of the span of the mode can be changed. It includes audio output so you have a better understanding of the mode created.
– user70304
Aug 2, 2020 at 22:11
• I don't know how Messiaen uses Modes of Limited Transposition, but surely whatever approach he uses could be applied to any mode of limited transposition. If one believes that the limited transposition of a mode gives it a certain sound (e.g. atonality), then one could simply create microtonal scales with this property and play around with it. Aug 3, 2020 at 19:16
• This is good motivation to try composing something so I can come back and add a "yes" answer! Aug 26, 2022 at 13:46

Is there an application of [Modes of Limited Transposition in Microtonality] to produce anything meaningful musically?

The TL;DR answer is (or, seems to be) no, at least not explicitly.

However...

Messaien's Modes of Limited Transposition are a special case of Transpositional Symmetry, a concept from Musical Set Theory, which refers to any set of pitches which can be transposed (non-trivially) onto itself. For example, the (12-tet) chromatic scale can be transposed by any (12-tet) interval and still contain the same set of pitches.

Has anyone applied Transpositional Symmetry within Microtonality?

Not clear, without reading some pretty heady papers (for which I'll provide references at the end). However, Transpositional Symmetry is an important part of Serial Composition.

Okay. So what about Serialism more generally within Microtonal Music?

Here there is a bit of a problem, as (contemporary) Microtonal composition arose as something of a reaction against Serialism. The combination of the two seems to be a rarity.

Ben Johnston is the notable exception. From Wikipedia:

Johnston's early efforts in just composition drew heavily on the accomplishments of post-Webern serialism. His 7-limit String Quartet No. 4 "Amazing Grace", was commissioned by the Fine Arts Music Foundation of Chicago, and was first recorded by the Fine Arts Quartet on Nonesuch Records in 1980 (then reissued on Gasparo as GS205). His String Quartet No. 4, perhaps Johnston's best-known composition, has also been recorded by the Kronos Quartet. The Kepler Quartet (Sharan Leventhal, Eric Segnitz, Brek Renzelman, and Karl Lavine) also recorded the piece for New World Records, as part of a complete 10-quartet series documenting Johnston's entire cycle of string quartets. The Third Quartet was premiered as part of this series by the Concord String Quartet at Alice Tully Hall at Lincoln Center for the Performing Arts, on March 15, 1976, the composer's fiftieth birthday.

Rockwell, John. 1976. "Music: Concord Strings; Quartet Performs Pieces by Johnston, Foss and Rochberg at Tully Hall". The New York Times (March 17): 33.

This is where those "heady" papers come into play. The starting point (for me) was Daniel Huey's dissertation: "Harmony Harmony, Voice Leading, and Microtonal Syntax in Ben Johnston’s String Quartet No. 5". It does not address symmetry directly, but points to some other papers that might.

And last but not least, here are two other composers who seem to have integrated Microtonality and Serialism.