User:Unque/2.3.7 Composition Theory
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The 2.3.7 subgroup is the series of just intonation intervals that can be created by combining the prime numbers 2, 3, and 7; unless otherwise specified, prime 2 will be considered by this page to act as a period or interval of equivalence, and thus we can describe most intervals via some combination of 3 and 7 with little ambiguity.
Intervals and Notation
Because any just intonation subgroup is infinitely dense, the ratios of the 2.3.7 subgroup cannot be enumerated by size; however, they can be unambiguously enumerated by odd limit.
The 2.3.7 subgroup can be unambiguously notated by using familiar Pythagorean diatonic notation as a basis, and adding one new pair of accidentals to represent some 7-limit formal chroma. For these purposes, I will use ^ and v to represent raising or lowering a note by 64/63. When naming intervals, alteration by one septimal chroma can be called "super-"/"sub-," and alteration by two septimal chromas can be called "ultra-"/"infra-."
| Ratio | Smonzo | Cents | Notation | Interval Name | Notes |
|---|---|---|---|---|---|
| 1/1 | [0, 0, 0⟩ | 0.000 | C | Unison | |
| 3/2 | [-1, 1, 0⟩ | 701.955 | G | Perfect Fifth | Generates the diatonic scale |
| 4/3 | [2, -1, 0⟩ | 498.045 | F | Perfect Fourth | |
| 7/4 | [-2, 0, 1⟩ | 968.826 | vB♭ | Subminor Seventh | |
| 8/7 | [3, 0, -1⟩ | 231.174 | ^D | Supermajor Second | |
| 7/6 | [-1, -1, 1⟩ | 266.871 | vE♭ | Subminor Third | |
| 12/7 | [2, 1, -1⟩ | 933.129 | ^A | Supermajor Sixth | |
| 9/8 | [-3, 2, 0⟩ | 203.910 | D | Major Second | |
| 16/9 | [4, -2, 0⟩ | 996.090 | B♭ | Minor Seventh | |
| 9/7 | [0, 2, -1⟩ | 435.084 | ^E | Supermajor Third | |
| 14/9 | [1, -2, 1⟩ | 764.916 | vA♭ | Subminor Sixth | |
| 21/16 | [-4, 1, 1⟩ | 470.781 | vF | Subfourth | |
| 32/21 | [5, -1, -1⟩ | 729.219 | ^G | Superfifth | |
| 27/16 | [-4, 3, 0⟩ | 905.865 | A | Major Sixth | |
| 32/27 | [5, -3, 0⟩ | 294.135 | E♭ | Minor Third | |
| 27/14 | [-1, 3, -1⟩ | 1137.039 | ^B | Supermajor Seventh | |
| 28/27 | [2, -3, 1⟩ | 62.961 | vD♭ | Subminor Second | |
| 49/32 | [-6, 0, 2⟩ | 737.652 | vvA♭ | Inframinor Sixth | |
| 64/49 | [7, 0, -2⟩ | 462.348 | ^^E | Ultramajor Third | |
| 49/48 | [-4, -1, 2⟩ | 35.697 | vvD♭ | Inframinor Second | |
| 96/49 | [5, 1, -2⟩ | 1164.303 | ^^B | Ultramajor Seventh | |
| 49/36 | [-2, -2, 2⟩ | 533.742 | vvG♭ | Infradiminished Fifth | |
| 72/49 | [3, 2, -2⟩ | 666.258 | ^^F♯ | Ultraäugmented Fourth | |
| 49/27 | [0, -3, 2⟩ | 1031.787 | vvC♭ | Infradiminished Unison | |
| 54/49 | [1, 3, -2⟩ | 168.213 | ^^C♯ | Ultrachroma | |
| 63/32 | [-5, 2, 1⟩ | 1172.736 | vC | Subunison | |
| 64/63 | [6, -2, -1⟩ | 27.264 | ^C | Superunison | Conventional "Septimal Comma" |
| 81/64 | [-6, 4, 0⟩ | 407.820 | E | Major Third | |
| 128/81 | [7, -4, 0⟩ | 792.180 | A♭ | Minor Sixth | |
| 81/49 | [0, 4, -2⟩ | 870.168 | ^^G♯ | Ultraäugmented Fifth | |
| 98/81 | [1, -4, 2⟩ | 329.832 | vvF♭ | Infradiminished Fourth | |
| 81/56 | [-3, 4, -1⟩ | 638.994 | ^F♯ | Superaugmented Fourth | |
| 112/81 | [4, -4, 1⟩ | 561.006 | vG♭ | Subdiminished Fifth |