Unque: Created page with "'''Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!''' 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 litt..."

2026-02-09T19:17:31Z

Created page with "'''Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!''' 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 litt..."

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'''Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!'''

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, [[2/1|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-."
{| class="wikitable sortable mw-collapsible"
|+2.3.7 Intervals
!Ratio
![[Subgroup monzos and vals|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
| rowspan="2" |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
|
|}

== Chords, Scales, and Counterpoint ==

User:Unque/2.3.7 Composition Theory - Revision history