Messiaen MLT Applied to Microtonal Gamuts, Part LII

Native MLT Modes in Microtonal Equal Temperaments

Modes of limited transposition built from each temperament's own step vocabulary

No 12-TET reference. C = chromatic step (smallest interval of the ET). 2C = double chromatic, etc.

The Principle

Modes of limited transposition exist in any ET where n is not prime. A mode repeats every d steps (its period) across the octave, where d divides n. The fewer transpositions, the more internally symmetric and coloristically concentrated the pitch set. These modes can replace the full-gamut approach for individual layers, yielding a prismatic, Messiaen-like color logic within the polytempic polymicrotonal architecture.

Step Size Reference

All interval sizes in cents. Each ET's chromatic step (C) and multiples thereof. No 12-TET equivalents named.

ET	C (1 step)	2C	3C	5C	7C
24-TET	50.0¢	100.0¢	150.0¢	250.0¢	350.0¢
28-TET	42.9¢	85.7¢	128.6¢	214.3¢	300.0¢
36-TET	33.3¢	66.7¢	100.0¢	166.7¢	233.3¢
42-TET	28.6¢	57.1¢	85.7¢	142.9¢	200.0¢
72-TET	16.7¢	33.3¢	50.0¢	83.3¢	116.7¢

Pattern I: [C · 2C] repeating — period = 3 steps

Alternating chromatic and double chromatic. Period of 3 steps means the octave divides into equal cells of [C, 2C]. Works only where 3 divides n. Each ET produces a completely different-sounding scale from this same structure.

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	8 reps · 16 notes	C=50¢ 2C=100¢	0, 50, 150, 200, 250, 350, 400, 450, 550, 600, 650, 750, 800, 850, 950, 1000	8 transpositions
28-TET	n/a — 3 ∤ 28
36-TET	12 reps · 24 notes	C=33¢ 2C=67¢	0, 33, 100, 133, 200, 233, 300, 333, 400, 433, 500, 533, 600, 633, 700, 733, 800, 833 …	12 transpositions
42-TET	14 reps · 28 notes	C=29¢ 2C=57¢	0, 29, 86, 114, 171, 200, 229, 286, 343, 371, 429, 457, 514, 543, 600, 629 …	14 transpositions
72-TET	24 reps · 48 notes	C=17¢ 2C=33¢	0, 17, 50, 67, 100, 117, 150, 167, 200, 217, 250, 267, 300, 317 …	24 transpositions; best as structural frame

The same [C · 2C] alternation: in 24-TET oscillates between 50¢ and 100¢ — a fine shimmer. In 36-TET between 33¢ and 67¢ — finer still. In 42-TET between 29¢ and 57¢ — approaching the threshold of distinct pitch perception. The structure is constant; the sonic world entirely different in each ET.

Pattern II: [C · C · 2C] repeating — period = 4 steps

Two chromatics followed by a double chromatic. Period of 4 steps. Works where 4 divides n — notably not 42-TET (since 4 ∤ 42).

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	6 reps · 18 notes	C=50¢ C=50¢ 2C=100¢	0, 50, 100, 200, 250, 300, 400, 450, 500, 600, 650, 700, 800, 850, 900, 1000, 1050, 1100	6 transpositions
28-TET	7 reps · 21 notes	C=43¢ C=43¢ 2C=86¢	0, 43, 86, 171, 214, 257, 343, 386, 429, 514, 557, 600, 686, 729, 771, 857, 900, 943, 1029, 1071, 1114	7 transpositions — 7-fold symmetric
36-TET	9 reps · 27 notes	C=33¢ C=33¢ 2C=67¢	0, 33, 67, 133, 167, 200, 267, 300, 333, 400, 433, 467, 533, 567, 600, 667, 700, 733 …	9 transpositions
42-TET	n/a — 4 ∤ 42
72-TET	18 reps · 54 notes	C=17¢ C=17¢ 2C=33¢	0, 17, 33, 67, 83, 100, 133, 150, 167, 200, 217, 233, 267, 283, 300 …	18 transpositions; dense, useful as sub-gamut

In 28-TET this pattern produces the only 7-fold symmetric mode available from [C, C, 2C] — 21 notes with the 7-prime as structural axis. The two chromatics (43¢ each) followed by a double (86¢) creates a characteristic lurch native to 28-TET that has no analog in any other common temperament.

Pattern III: [C · 3C] repeating — period = 4 steps

Chromatic followed by triple chromatic. A wider step after a narrow one. Same period-4 family as Pattern II — same ET availability — but a more dramatically asymmetric cell.

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	6 reps · 12 notes	C=50¢ 3C=150¢	0, 50, 200, 250, 400, 450, 600, 650, 800, 850, 1000, 1050	6 transpositions
28-TET	7 reps · 14 notes	C=43¢ 3C=129¢	0, 43, 171, 214, 343, 386, 514, 557, 686, 729, 857, 900, 1029, 1071	7 transpositions — 7-fold symmetric
36-TET	9 reps · 18 notes	C=33¢ 3C=100¢	0, 33, 133, 167, 267, 300, 400, 433, 533, 567, 667, 700, 800, 833, 933, 967, 1067, 1100	9 transpositions
42-TET	n/a — 4 ∤ 42
72-TET	18 reps · 36 notes	C=17¢ 3C=50¢	0, 17, 67, 83, 133, 150, 200, 217, 267, 283, 333, 350, 400, 417, 467 …	18 transpositions

The narrow–wide gesture: a single chromatic step followed by a leap three times its size. In 28-TET: 43¢ then 129¢ — a fine inflection then a near-neutral-third leap, seven times around the octave. In 36-TET: 33¢ then 100¢ — even finer inflection, wider leap proportionally. Each ET makes the same gesture at a different magnification.

Pattern IV: [C · 2C · 3C] repeating — period = 6 steps

Three ascending step sizes in sequence: chromatic, double, triple. Period of 6 steps. Works where 6 divides n — not 28-TET (6 ∤ 28). The most Messiaen-like structure: a cell of three intervals climbing in size.

42-TET uniquely: Pattern IV produces 7-fold symmetry from the [C · 2C · 3C] ascending cell — the prime 7 governing the Messiaen-type mode. This conjunction is unavailable in any other common ET.

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	4 reps · 12 notes	C=50¢ 2C=100¢ 3C=150¢	0, 50, 150, 300, 350, 450, 600, 650, 750, 900, 950, 1050	4 transpositions — highly concentrated
28-TET	n/a — 6 ∤ 28
36-TET	6 reps · 18 notes	C=33¢ 2C=67¢ 3C=100¢	0, 33, 100, 200, 233, 300, 400, 433, 500, 600, 633, 700, 800, 833, 900, 1000, 1033, 1100	6 transpositions
42-TET	7 reps · 21 notes	C=29¢ 2C=57¢ 3C=86¢	0, 29, 86, 171, 200, 257, 343, 371, 429, 514, 543, 600, 686, 714, 771, 857, 886, 943, 1029, 1057, 1114	7 transpositions — 7-fold symmetric
72-TET	12 reps · 36 notes	C=17¢ 2C=33¢ 3C=50¢	0, 17, 50, 100, 117, 150, 200, 217, 250, 300, 317, 350, 400, 417, 450, 500 …	12 transpositions

In 42-TET, [C · 2C · 3C] gives 7-fold symmetry — the prime 7 appears as the structural axis through this climbing-cell logic. In 24-TET the same pattern gives only 4 transpositions (more limited, more coloristically concentrated — closer to Messiaen’s actual modes). 42-TET is the ET where Messiaen’s own mode-3 logic and septimal symmetry coincide.

Pattern V: [C · C · 2C · 2C] repeating — period = 6 steps

Two chromatics then two double chromatics — a denser opening cluster, then a wider pair. Same 6-step period availability as Pattern IV.

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	4 reps · 16 notes	C C 2C 2C = 50 50 100 100¢	0, 50, 100, 200, 300, 400, 450, 500, 600, 700, 800, 850, 900, 1000, 1100	4 transpositions
28-TET	n/a — 6 ∤ 28
36-TET	6 reps · 24 notes	C C 2C 2C = 33 33 67 67¢	0, 33, 67, 133, 200, 233, 267, 333, 400, 433, 467, 533, 600, 633, 667, 733, 800, 833 …	6 transpositions
42-TET	7 reps · 28 notes	C C 2C 2C = 29 29 57 57¢	0, 29, 57, 114, 171, 200, 229, 286, 343, 371, 400, 457, 514, 543, 571, 629, 686 …	7 transpositions — 7-fold symmetric
72-TET	12 reps · 48 notes	C C 2C 2C = 17 17 33 33¢	0, 17, 33, 67, 100, 117, 133, 167, 200, 217, 233, 267 …	12 transpositions

The cluster-then-expand gesture: two tight chromatic steps, then two double-chromatic steps widening out. In 42-TET 7-fold symmetry again: 28 notes across the octave with the septimal axis structuring the repetition.

Pattern VI: [2C · 5C] repeating — period = 7 steps

Double chromatic followed by quintuple chromatic. Period = 7 steps. Works only where 7 divides n — exclusively 28-TET and 42-TET among the five recommended ETs. Structurally unavailable in 24, 36, or 72-TET.

Exclusive to 28-TET and 42-TET. This is the only MLT pattern whose availability is determined by the prime 7. It cannot exist in 24, 36, or 72-TET — not as a practical matter but as a mathematical impossibility. The 7-fold symmetric mode lives only in the ETs that contain 7 as a structural divisor of the octave.

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	n/a — 7 ∤ 24
28-TET	4 reps · 8 notes	2C=86¢ 5C=214¢	0, 86, 300, 386, 600, 686, 900, 986	4 transpositions — highly reduced: only 8 pitches
36-TET	n/a — 7 ∤ 36
42-TET	6 reps · 12 notes	2C=57¢ 5C=143¢	0, 57, 200, 257, 400, 457, 600, 657, 800, 857, 1000, 1057	6 transpositions
72-TET	n/a — 7 ∤ 72

The 28-TET realization is particularly striking: only 8 pitches, 4 transpositions, built entirely from 86¢ (double chromatic) and 214¢ (quintuple chromatic) — two intervals with no 12-TET names or equivalents. In 42-TET: 57¢ and 143¢, twelve pitches. Both are native to the 7-prime axis and unreachable from any ET not divisible by 7.

Pattern VII: [C · 2C · 2C · 3C] repeating — period = 8 steps

Four-interval cell ascending then widening: C, 2C, 2C, 3C. Period = 8 steps. Works where 8 divides n — only 24-TET and 72-TET among the five.

ET	Repetitions	Intervals	Cents (selected)	Notes
24-TET	3 reps · 12 notes	C 2C 2C 3C = 50 100 100 150¢	0, 50, 150, 250, 400, 450, 550, 650, 800, 850, 950, 1050	3 transpositions — maximum color concentration
28-TET	n/a — 8 ∤ 28
36-TET	n/a — 8 ∤ 36
42-TET	n/a — 8 ∤ 42
72-TET	9 reps · 36 notes	C 2C 2C 3C = 17 33 33 50¢	0, 17, 50, 83, 133, 150, 183, 233, 267, 283, 317, 367, 400, 417, 450 …	9 transpositions

In 24-TET this pattern achieves only 3 transpositions from 12 pitches — close to Messiaen’s own degree of limitation, but with a native quarter-tone vocabulary. The cell [50, 100, 100, 150¢] has an internal momentum: narrow opening, two equal steps, then a widening close. No 12-TET mode has this exact shape.

All intervals expressed in each ET's own step units. Cent values rounded. "n/a" indicates the period does not divide the ET evenly — the pattern is structurally unavailable, not merely impractical. Designed for use as layer-specific pitch gamuts within the polytempic polymicrotonal system.