Ploidacot/Gamma-hexacot: Difference between revisions

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Line 4: Line 4:


== Intervals and notation ==
== Intervals and notation ==
Due to dividing the eighteenth into so many steps, standard notation becomes almost useless for gamma-hexacot. Regardless, notation has been provided for where gamma-hexacot intervals align with standard monocot intervals (which use [[chain-of-fifths notation]]).
While there is no agreed-upon notation system for gamma-hexacot, the following is based on interpreting the generator as a grave fourth, and allowing for an ^ or v to stand for 1/6 of a diatonic semitone, so ^^^C and vvvDb are enharmonic.


{| class="wikitable"
{| class="wikitable"
Line 21: Line 21:
| −23
| −23
| 890.828
| 890.828
|  
| vA
|  
|  
|-
|-
| −22
| −22
| 173.835
| 173.835
|  
| vvD
|  
|  
|-
|-
| −21
| −21
| 656.843
| 656.843
|  
| ^^^F#/vvvG
|  
|  
|-
|-
| −20
| −20
| 1139.850
| 1139.850
|  
| ^^B
|  
|  
|-
|-
| −19
| −19
| 422.858
| 422.858
|  
| ^E
|  
|  
|-
|-
Line 51: Line 51:
| −17
| −17
| 188.873
| 188.873
|  
| vD
|  
|  
|-
|-
| −16
| −16
| 671.880
| 671.880
|  
| vvG
|  
|  
|-
|-
| −15
| −15
| 1154.888
| 1154.888
|  
| ^^^B/vvvC
|  
|  
|-
|-
| −14
| −14
| 437.895
| 437.895
|  
| ^^E
|  
|  
|-
|-
| −13
| −13
| 920.903
| 920.903
|  
| ^A
|  
|  
|-
|-
Line 81: Line 81:
| −11
| −11
| 686.918
| 686.918
|  
| vG
|  
|  
|-
|-
| −10
| −10
| 1169.925
| 1169.925
|  
| vvC
|  
|  
|-
|-
| −9
| −9
| 452.933
| 452.933
|  
| ^^^E/vvvF
|  
|  
|-
|-
| −8
| −8
| 935.940
| 935.940
|  
| ^^A
|  
|  
|-
|-
| −7
| −7
| 218.948
| 218.948
|  
| ^D
|  
|  
|-
|-
Line 111: Line 111:
| −5
| −5
| 1184.963
| 1184.963
|  
| vC
|  
|  
|-
|-
| −4
| −4
| 467.970
| 467.970
|  
| vvF
|  
|  
|-
|-
| −3
| −3
| 950.978
| 950.978
|  
| ^^^A/vvvBb
|  
|  
|-
|-
| −2
| −2
| 233.985
| 233.985
|  
| ^^D
|  
|  
|-
|-
| −1
| −1
| 716.993
| 716.993
|  
| ^G
|  
|  
|-
|-
Line 141: Line 141:
| 1
| 1
| 483.007
| 483.007
|  
| vF
|  
|  
|-
|-
| 2
| 2
| 966.015
| 966.015
|  
| vvBb
|  
|  
|-
|-
| 3
| 3
| 249.022
| 249.022
|  
| ^^^D/vvvEb
|  
|  
|-
|-
| 4
| 4
| 732.030
| 732.030
|  
| ^^G
|  
|  
|-
|-
| 5
| 5
| 15.037
| 15.037
|  
| ^C
|  
|  
|-
|-
Line 171: Line 171:
| 7
| 7
| 981.052
| 981.052
|  
| vBb
|  
|  
|-
|-
| 8
| 8
| 264.060
| 264.060
|  
| vvEb
|  
|  
|-
|-
| 9
| 9
| 747.067
| 747.067
|  
| ^^^G/vvvAb
|  
|  
|-
|-
| 10
| 10
| 30.075
| 30.075
|  
| ^^C
|  
|  
|-
|-
| 11
| 11
| 513.082
| 513.082
|  
| ^F
|  
|  
|-
|-
Line 201: Line 201:
| 13
| 13
| 279.097
| 279.097
|  
| vEb
|  
|  
|-
|-
| 14
| 14
| 762.105
| 762.105
|  
| vvAb
|  
|  
|-
|-
| 15
| 15
| 45.112
| 45.112
|  
| ^^^C/vvvDb
|  
|  
|-
|-
| 16
| 16
| 528.120
| 528.120
|  
| ^^F
|  
|  
|-
|-
| 17
| 17
| 1011.127
| 1011.127
|  
| ^Bb
|  
|  
|-
|-
Line 231: Line 231:
| 19
| 19
| 777.142
| 777.142
|  
| vAb
|  
|  
|-
|-
| 20
| 20
| 60.150
| 60.150
|  
| vvDb
|  
|  
|-
|-
| 21
| 21
| 543.157
| 543.157
|  
| ^^^F/vvvGb
|  
|  
|-
|-
| 22
| 22
| 1026.165
| 1026.165
|  
| ^^Bb
|  
|  
|-
|-
| 23
| 23
| 309.172
| 309.172
|  
| ^Eb
|  
|  
|-
|-

Latest revision as of 07:39, 26 January 2026

< Ploidacot
Gamma-hexacot
Pergen [P8, ccP4/6]
Numeral form 3-sheared 6-cot
Pure generator size 483.01 ¢
Pure period size 1200 ¢
Forms 17, 22, 27, 32, 37
Characteristic multival entry 6

Gamma-hexacot is a temperament archetype where the generator is a grave fourth of about 482–484 ¢, six of which make 16/3 (perfect eighteenth, two octaves above a perfect fourth), and the period is a 2/1 octave. Gamma-hexacot temperaments also include all alpha-dicot and tricot intervals. Gamma-hexacot temperaments typically generate the 5L 7s, 5L 12s, and 5L 17s MOS scales.

Intervals and notation

While there is no agreed-upon notation system for gamma-hexacot, the following is based on interpreting the generator as a grave fourth, and allowing for an ^ or v to stand for 1/6 of a diatonic semitone, so ^^^C and vvvDb are enharmonic.

Gamma-hexacot intervals (assuming pure fifth and octave)
# Cents Notation Name
−24 407.820 E major third
−23 890.828 vA
−22 173.835 vvD
−21 656.843 ^^^F#/vvvG
−20 1139.850 ^^B
−19 422.858 ^E
−18 905.865 A major sixth
−17 188.873 vD
−16 671.880 vvG
−15 1154.888 ^^^B/vvvC
−14 437.895 ^^E
−13 920.903 ^A
−12 203.910 D major second
−11 686.918 vG
−10 1169.925 vvC
−9 452.933 ^^^E/vvvF
−8 935.940 ^^A
−7 218.948 ^D
−6 701.955 G perfect fifth
−5 1184.963 vC
−4 467.970 vvF
−3 950.978 ^^^A/vvvBb
−2 233.985 ^^D
−1 716.993 ^G
0 0.000 C perfect unison
1 483.007 vF
2 966.015 vvBb
3 249.022 ^^^D/vvvEb
4 732.030 ^^G
5 15.037 ^C
6 498.045 F perfect fourth
7 981.052 vBb
8 264.060 vvEb
9 747.067 ^^^G/vvvAb
10 30.075 ^^C
11 513.082 ^F
12 996.090 Bb minor seventh
13 279.097 vEb
14 762.105 vvAb
15 45.112 ^^^C/vvvDb
16 528.120 ^^F
17 1011.127 ^Bb
18 294.135 Eb minor third
19 777.142 vAb
20 60.150 vvDb
21 543.157 ^^^F/vvvGb
22 1026.165 ^^Bb
23 309.172 ^Eb
24 792.180 Ab minor sixth

Temperament interpretations

An obvious interpretation for gamma-hexacot is hemiseven, where the generator is 45/34, two of which make 7/4, three make 15/13 above an octave, four make 26/17~32/21 above an octave, six make 4/3 above two octaves, seven make 30/17 above two octaves, seventeen make 9/5 above six octaves, and nineteen make 11/7 above seven octaves.

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