Ploidacot/Gamma-pentacot: Difference between revisions

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Created page with "{{Breadcrumb}} {{Infobox ploidacot|Ploids=1|Shears=3|Cots=5|Pergen=[P8, P11/5]|Forms=7, 11, 18, 25|Title=Gamma-pentacot|Wedgie=5}} '''Gamma-pentacot''' is a temperament archetype where the generator is a subneutral third, five of which make a perfect eleventh of 8/3, and the period is a 2/1 octave. Gamma-pentacot temperaments typically generate the 4L 3s and 7L 4s MOS scales, and they split the chromatic semitone into five equal parts, creating..."
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{{Breadcrumb}}
{{Breadcrumb}}
{{Infobox ploidacot|Ploids=1|Shears=3|Cots=5|Pergen=[P8, P11/5]|Forms=7, 11, 18, 25|Title=Gamma-pentacot|Wedgie=5}}
{{Infobox ploidacot|Ploids=1|Shears=3|Cots=5|Pergen=[P8, P11/5]|Forms=7, 11, 18, 25, 32|Title=Gamma-pentacot|Wedgie=5}}
'''Gamma-pentacot''' is a temperament archetype where the generator is a subneutral third, five of which make a perfect eleventh of [[8/3]], and the period is a [[2/1]] octave. Gamma-pentacot temperaments typically generate the [[4L 3s]] and [[7L 4s]] MOS scales, and they split the chromatic semitone into five equal parts, creating "supraminor", "subneutral", "supraneutral", and "submajor" intervals.
'''Gamma-pentacot''' is a temperament archetype where the generator is a subneutral third, five of which make a perfect eleventh of [[8/3]], and the period is a [[2/1]] octave. Gamma-pentacot temperaments typically generate the [[4L 3s]], [[7L 4s]], and [[7L 11s]] MOS scales, and they split the chromatic semitone into five equal parts, creating "supraminor", "subneutral", "supraneutral", and "submajor" intervals.


Gamma-pentacot temperaments often generate [[7L 11s]], 7L 18s, 7L 25s, 7L 32s, and 7L 39s as chromatic scales, and for particularly sharp tunings 39L 7s.
Gamma-pentacot temperaments often generate 7L 18s, 7L 25s, 7L 32s, and 7L 39s as chromatic scales, and for particularly flat tunings 18L 7s, 25L 7s, 32L 7s, or 39L 7s.


== Intervals and notation ==
== Intervals and notation ==
Line 24: Line 24:
| 951.339
| 951.339
| ^^A
| ^^A
|  
| sub-semiaugmented sixth
|-
|-
| −28
| −28
| 90.948
| 90.948
| vC#
| vC#
|  
| subaugmented unison
|-
|-
| −27
| −27
| 430.557
| 430.557
| ^E
| ^E
|  
| supermajor third
|-
|-
| −26
| −26
| 770.166
| 770.166
| vvG#
| vvG#
|  
| supra-semiaugmented fifth
|-
|-
| −25
| −25
Line 49: Line 49:
| 249.384
| 249.384
| ^^D
| ^^D
|  
| sub-semiaugmented second
|-
|-
| −23
| −23
| 588.993
| 588.993
| vF#
| vF#
|  
| subaugmented fourth
|-
|-
| −22
| −22
| 928.602
| 928.602
| ^A
| ^A
|  
| supermajor sixth
|-
|-
| −21
| −21
| 68.211
| 68.211
| vvC#
| vvC#
|  
| supra-semiaugmented unison
|-
|-
| −20
| −20
Line 74: Line 74:
| 747.429
| 747.429
| ^^G
| ^^G
|  
| sub-semiaugmented fifth
|-
|-
| −18
| −18
| 1087.038
| 1087.038
| vB
| vB
|  
| submajor seventh
|-
|-
| −17
| −17
| 226.647
| 226.647
| ^D
| ^D
|  
| supermajor second
|-
|-
| −16
| −16
| 566.256
| 566.256
| vvF#
| vvF#
|  
| supra-semiaugmented fourth
|-
|-
| −15
| −15
Line 99: Line 99:
| 45.474
| 45.474
| ^^C
| ^^C
|  
| sub-semiaugmented unison
|-
|-
| −13
| −13
| 385.083
| 385.083
| vE
| vE
|  
| submajor third
|-
|-
| −12
| −12
| 724.692
| 724.692
| ^G
| ^G
|  
| super-fifth
|-
|-
| −11
| −11
| 1064.301
| 1064.301
| vvB
| vvB
|  
| supraneutral seventh
|-
|-
| −10
| −10
Line 124: Line 124:
| 543.519
| 543.519
| ^^F
| ^^F
|  
| sub-semiaugmented fourth
|-
|-
| −8
| −8
| 883.128
| 883.128
| vA
| vA
|  
| submajor sixth
|-
|-
| −7
| −7
| 22.737
| 22.737
| ^C
| ^C
|  
| super-unison
|-
|-
| −6
| −6
| 362.346
| 362.346
| vvE
| vvE
|  
| supraneutral third
|-
|-
| −5
| −5
Line 149: Line 149:
| 1041.564
| 1041.564
| ^^Bb
| ^^Bb
|  
| subneutral seventh
|-
|-
| −3
| −3
| 181.173
| 181.173
| vD
| vD
|  
| submajor second
|-
|-
| −2
| −2
| 520.782
| 520.782
| ^F
| ^F
|  
| super-fourth
|-
|-
| −1
| −1
| 860.391
| 860.391
| vvA
| vvA
|  
| supraneutral sixth
|-
|-
| 0
| 0
Line 174: Line 174:
| 339.609
| 339.609
| ^^Eb
| ^^Eb
|  
| subneutral third
|-
|-
| 2
| 2
| 679.218
| 679.218
| vG
| vG
|  
| sub-fifth
|-
|-
| 3
| 3
| 1018.827
| 1018.827
| ^Bb
| ^Bb
|  
| supraminor seventh
|-
|-
| 4
| 4
| 158.436
| 158.436
| vvD
| vvD
|  
| supraneutral second
|-
|-
| 5
| 5
Line 199: Line 199:
| 837.654
| 837.654
| ^^Ab
| ^^Ab
|  
| subneutral sixth
|-
|-
| 7
| 7
| 1177.263
| 1177.263
| vC
| vC
|  
| sub-octave
|-
|-
| 8
| 8
| 316.872
| 316.872
| ^Eb
| ^Eb
|  
| supraminor third
|-
|-
| 9
| 9
| 656.481
| 656.481
| vvG
| vvG
|  
| supra-semidiminished fifth
|-
|-
| 10
| 10
Line 224: Line 224:
| 135.699
| 135.699
| ^^Db
| ^^Db
|  
| subneutral second
|-
|-
| 12
| 12
| 475.308
| 475.308
| vF
| vF
|  
| sub-fourth
|-
|-
| 13
| 13
| 814.917
| 814.917
| ^Ab
| ^Ab
|  
| supraminor sixth
|-
|-
| 14
| 14
| 1174.526
| 1174.526
| vvC
| vvC
|  
| supra-semidiminished octave
|-
|-
| 15
| 15
Line 249: Line 249:
| 633.744
| 633.744
| ^^Gb
| ^^Gb
|  
| sub-semidiminished fifth
|-
|-
| 17
| 17
| 973.353
| 973.353
| vBb
| vBb
|  
| subminor seventh
|-
|-
| 18
| 18
| 112.962
| 112.962
| ^Db
| ^Db
|  
| supraminor second
|-
|-
| 19
| 19
| 452.571
| 452.571
| vvF
| vvF
|  
| supra-semidiminished fourth
|-
|-
| 20
| 20
Line 274: Line 274:
| 1131.789
| 1131.789
| ^^Cb
| ^^Cb
|  
| sub-semidiminished octave
|-
|-
| 22
| 22
| 271.398
| 271.398
| vEb
| vEb
|  
| subminor third
|-
|-
| 23
| 23
| 611.007
| 611.007
| ^Gb
| ^Gb
|  
| supradiminished fifth
|-
|-
| 24
| 24
| 950.616
| 950.616
| vvBb
| vvBb
|  
| supra-semidiminished seventh
|-
|-
| 25
| 25
Line 299: Line 299:
| 429.834
| 429.834
| ^^Fb
| ^^Fb
|  
| sub-semidiminished fourth
|-
|-
| 27
| 27
| 769.443
| 769.443
| vAb
| vAb
|  
| subminor sixth
|-
|-
| 28
| 28
| 1109.052
| 1109.052
| ^Cb
| ^Cb
|  
| supradiminished octave
|-
|-
| 29
| 29
| 248.661
| 248.661
| vvEb
| vvEb
|  
| supra-semidiminished third
|-
|-
| 30
| 30
Line 325: Line 325:
An obvious interpretation for gamma-pentacot is [[amity]], 5/4 is equated to 4 octaves minus 13 generators, and 7/4 is equated to 17 generators minus 4 octaves. Other interpretations include [[sixix]], which interprets 6/5 as a generator.
An obvious interpretation for gamma-pentacot is [[amity]], 5/4 is equated to 4 octaves minus 13 generators, and 7/4 is equated to 17 generators minus 4 octaves. Other interpretations include [[sixix]], which interprets 6/5 as a generator.


[[Category:Ploidacot]]
[[Category:Ploidacots|Gamma-pentacot]]

Latest revision as of 23:46, 7 January 2026

< Ploidacot
Gamma-pentacot
Pergen [P8, P11/5]
Numeral form 3-sheared 5-cot
Pure generator size 339.61 ¢
Pure period size 1200 ¢
Forms 7, 11, 18, 25, 32
Characteristic multival entry 5

Gamma-pentacot is a temperament archetype where the generator is a subneutral third, five of which make a perfect eleventh of 8/3, and the period is a 2/1 octave. Gamma-pentacot temperaments typically generate the 4L 3s, 7L 4s, and 7L 11s MOS scales, and they split the chromatic semitone into five equal parts, creating "supraminor", "subneutral", "supraneutral", and "submajor" intervals.

Gamma-pentacot temperaments often generate 7L 18s, 7L 25s, 7L 32s, and 7L 39s as chromatic scales, and for particularly flat tunings 18L 7s, 25L 7s, 32L 7s, or 39L 7s.

Intervals and notation

While there is no agreed-upon notation system for gamma-pentacot, the notation provided here is based on interpreting the generator as a subneutral third, and allowing for an ^ or v to stand for 1/5 of a chromatic semitone, so ^^^C and vvC# are enharmonic.

Gamma-pentacot intervals (assuming pure fifth and octave)
# Cents Notation Name
−30 611.730 F# augmented fourth
−29 951.339 ^^A sub-semiaugmented sixth
−28 90.948 vC# subaugmented unison
−27 430.557 ^E supermajor third
−26 770.166 vvG# supra-semiaugmented fifth
−25 1109.775 B major seventh
−24 249.384 ^^D sub-semiaugmented second
−23 588.993 vF# subaugmented fourth
−22 928.602 ^A supermajor sixth
−21 68.211 vvC# supra-semiaugmented unison
−20 407.820 E major third
−19 747.429 ^^G sub-semiaugmented fifth
−18 1087.038 vB submajor seventh
−17 226.647 ^D supermajor second
−16 566.256 vvF# supra-semiaugmented fourth
−15 905.865 A major sixth
−14 45.474 ^^C sub-semiaugmented unison
−13 385.083 vE submajor third
−12 724.692 ^G super-fifth
−11 1064.301 vvB supraneutral seventh
−10 203.910 D major second
−9 543.519 ^^F sub-semiaugmented fourth
−8 883.128 vA submajor sixth
−7 22.737 ^C super-unison
−6 362.346 vvE supraneutral third
−5 701.955 G perfect fifth
−4 1041.564 ^^Bb subneutral seventh
−3 181.173 vD submajor second
−2 520.782 ^F super-fourth
−1 860.391 vvA supraneutral sixth
0 0.000 C perfect unison
1 339.609 ^^Eb subneutral third
2 679.218 vG sub-fifth
3 1018.827 ^Bb supraminor seventh
4 158.436 vvD supraneutral second
5 498.045 F perfect fourth
6 837.654 ^^Ab subneutral sixth
7 1177.263 vC sub-octave
8 316.872 ^Eb supraminor third
9 656.481 vvG supra-semidiminished fifth
10 996.090 Bb minor seventh
11 135.699 ^^Db subneutral second
12 475.308 vF sub-fourth
13 814.917 ^Ab supraminor sixth
14 1174.526 vvC supra-semidiminished octave
15 294.135 Eb minor third
16 633.744 ^^Gb sub-semidiminished fifth
17 973.353 vBb subminor seventh
18 112.962 ^Db supraminor second
19 452.571 vvF supra-semidiminished fourth
20 792.180 Ab minor sixth
21 1131.789 ^^Cb sub-semidiminished octave
22 271.398 vEb subminor third
23 611.007 ^Gb supradiminished fifth
24 950.616 vvBb supra-semidiminished seventh
25 90.225 Db minor second
26 429.834 ^^Fb sub-semidiminished fourth
27 769.443 vAb subminor sixth
28 1109.052 ^Cb supradiminished octave
29 248.661 vvEb supra-semidiminished third
30 588.270 Gb diminished fifth

Temperament interpretations

An obvious interpretation for gamma-pentacot is amity, 5/4 is equated to 4 octaves minus 13 generators, and 7/4 is equated to 17 generators minus 4 octaves. Other interpretations include sixix, which interprets 6/5 as a generator.

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