Ploidacot/Beta-hexacot: Difference between revisions

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Created page with "{{Breadcrumb}} {{Infobox ploidacot|Ploids=1|Shears=2|Cots=6|Pergen=[P8, ccP5/6]|Forms=16, 23, 30, 37|Title=Beta-hexacot|Wedgie=6}} '''Beta-hexacot''' is a temperament archetype where the generator is an acute fourth of about 516–518¢, six of which make 6/1 (sixth harmonic, two octaves above a perfect fifth), and the period is a 2/1 octave. Beta-hexacot temperaments also include all dicot and omega-tricot intervals..."
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{{Breadcrumb}}
{{Breadcrumb}}
{{Infobox ploidacot|Ploids=1|Shears=2|Cots=6|Pergen=[P8, ccP5/6]|Forms=16, 23, 30, 37|Title=Beta-hexacot|Wedgie=6}}
{{Infobox ploidacot|Ploids=1|Shears=2|Cots=6|Pergen=[P8, ccP5/6]|Forms=16, 23, 30, 37|Title=Beta-hexacot|Wedgie=6}}
'''Beta-hexacot''' is a temperament archetype where the generator is an acute fourth of about 516–518¢, six of which make [[6/1]] (sixth harmonic, two octaves above a perfect fifth), and the period is a [[2/1]] octave. Beta-hexacot temperaments also include all [[Ploidacot/Dicot|dicot]] and [[Ploidacot/Omega-tricot|omega-tricot]] intervals. Beta-hexacot temperaments typically generate the [[7L 2s]], [[7L 9s]], and [[7L 16s]] MOS scales.
'''Beta-hexacot''' is a temperament archetype where the generator is an acute fourth of about 516–518{{c}}, six of which make [[6/1]] (sixth harmonic, two octaves above a perfect fifth), and the period is a [[2/1]] octave. Beta-hexacot temperaments also include all [[Ploidacot/Dicot|dicot]] and [[Ploidacot/Omega-tricot|omega-tricot]] intervals. Beta-hexacot temperaments typically generate the [[7L 2s]], [[7L 9s]], and [[7L 16s]] MOS scales.


== Intervals and notation ==
== Intervals and notation ==
Due to dividing the sixth harmonic into so many steps, standard notation becomes almost useless for beta-hexacot. Regardless, notation has been provided for where beta-hexacot intervals align with standard dicot intervals (which use [[neutral chain-of-fifths notation]]).
While there is no agreed-upon notation system for beta-hexacot, the following is based on interpreting the generator as an acute fourth, and allowing for an ^ or v to stand for 1/6 of a chromatic semitone, so {{nowrap|^^^Eb {{=}} vvvE {{=}} Ed}}.


{| class="wikitable"
{| class="wikitable"
Line 21: Line 21:
| −23
| −23
| 109.172
| 109.172
|  
| ^Db
|  
|  
|-
|-
| −22
| −22
| 626.165
| 626.165
|  
| ^^Gb
|  
|  
|-
|-
Line 36: Line 36:
| −20
| −20
| 460.150
| 460.150
|  
| vvF
|  
|  
|-
|-
| −19
| −19
| 977.142
| 977.142
|  
| vBb
|  
|  
|-
|-
Line 51: Line 51:
| −17
| −17
| 811.127
| 811.127
|  
| ^Ab
|  
|  
|-
|-
| −16
| −16
| 128.120
| 128.120
|  
| ^^Db
|  
|  
|-
|-
Line 66: Line 66:
| −14
| −14
| 1162.105
| 1162.105
|  
| vvC
|  
|  
|-
|-
| −13
| −13
| 479.097
| 479.097
|  
| vF
|  
|  
|-
|-
Line 81: Line 81:
| −11
| −11
| 313.082
| 313.082
|  
| ^Eb
|  
|  
|-
|-
| −10
| −10
| 830.075
| 830.075
|  
| ^^Ab
|  
|  
|-
|-
Line 96: Line 96:
| −8
| −8
| 664.060
| 664.060
|  
| vvG
|  
|  
|-
|-
| −7
| −7
| 1181.052
| 1181.052
|  
| vC
|  
|  
|-
|-
Line 111: Line 111:
| −5
| −5
| 1015.037
| 1015.037
|  
| ^Bb
|  
|  
|-
|-
| −4
| −4
| 332.030
| 332.030
|  
| ^^Eb
|  
|  
|-
|-
Line 126: Line 126:
| −2
| −2
| 166.015
| 166.015
|  
| vvD
|  
|  
|-
|-
| −1
| −1
| 683.007
| 683.007
|  
| vG
|  
|  
|-
|-
Line 141: Line 141:
| 1
| 1
| 516.993
| 516.993
|  
| ^F
|  
|  
|-
|-
| 2
| 2
| 1033.985
| 1033.985
|  
| ^^Bb
|  
|  
|-
|-
Line 156: Line 156:
| 4
| 4
| 867.970
| 867.970
|  
| vvA
|  
|  
|-
|-
| 5
| 5
| 184.963
| 184.963
|  
| vD
|  
|  
|-
|-
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| 7
| 7
| 18.948
| 18.948
|  
| ^C
|  
|  
|-
|-
| 8
| 8
| 535.940
| 535.940
|  
| ^^F
|  
|  
|-
|-
Line 186: Line 186:
| 10
| 10
| 369.925
| 369.925
|  
| vvE
|  
|  
|-
|-
| 11
| 11
| 886.918
| 886.918
|  
| vA
|  
|  
|-
|-
Line 201: Line 201:
| 13
| 13
| 720.903
| 720.903
|  
| ^G
|  
|  
|-
|-
| 14
| 14
| 37.895
| 37.895
|  
| ^^C
|  
|  
|-
|-
Line 216: Line 216:
| 16
| 16
| 1071.880
| 1071.880
|  
| vvB
|  
|  
|-
|-
| 17
| 17
| 388.873
| 388.873
|  
| vE
|  
|  
|-
|-
Line 231: Line 231:
| 19
| 19
| 222.858
| 222.858
|  
| ^D
|  
|  
|-
|-
| 20
| 20
| 739.850
| 739.850
|  
| ^^G
|  
|  
|-
|-
Line 246: Line 246:
| 22
| 22
| 573.835
| 573.835
|  
| vvF#
|  
|  
|-
|-
| 23
| 23
| 1090.828
| 1090.828
|  
| vB
|  
|  
|-
|-
Line 261: Line 261:


== Temperament interpretations ==
== Temperament interpretations ==
An obvious interpretation for beta-hexacot is [[gravity]], where the generator is [[27/20]], three of them make [[11/9]] above an octave, and six of them make a perfect fifth above two octaves. There are some extensions for the full 11-limit: marvo (65d & 72), zarvo (65 & 72), and gravid (58 & 65).
An obvious interpretation for beta-hexacot is [[gravity]], where the generator is [[27/20]], two of which make [[20/11]], three make {{nowrap|[[11/9]]~[[27/22]]}} above an octave, and six make a perfect fifth above two octaves. There are some extensions for the full 11-limit: marvo ({{nowrap|65d & 72}}), zarvo ({{nowrap|65 & 72}}), and gravid ({{nowrap|58 & 65}}).


[[Category:Ploidacots|Beta-hexacot]]
[[Category:Ploidacots|Beta-hexacot]]

Latest revision as of 12:29, 26 January 2026

< Ploidacot
Beta-hexacot
Pergen [P8, ccP5/6]
Numeral form 2-sheared 6-cot
Pure generator size 516.99 ¢
Pure period size 1200 ¢
Forms 16, 23, 30, 37
Characteristic multival entry 6

Beta-hexacot is a temperament archetype where the generator is an acute fourth of about 516–518 ¢, six of which make 6/1 (sixth harmonic, two octaves above a perfect fifth), and the period is a 2/1 octave. Beta-hexacot temperaments also include all dicot and omega-tricot intervals. Beta-hexacot temperaments typically generate the 7L 2s, 7L 9s, and 7L 16s MOS scales.

Intervals and notation

While there is no agreed-upon notation system for beta-hexacot, the following is based on interpreting the generator as an acute fourth, and allowing for an ^ or v to stand for 1/6 of a chromatic semitone, so ^^^Eb = vvvE = Ed.

Beta-hexacot intervals (assuming pure fifth and octave)
# Cents Notation Name
−24 792.180 Ab minor sixth
−23 109.172 ^Db
−22 626.165 ^^Gb
−21 1143.157 Cd semidiminished octave
−20 460.150 vvF
−19 977.142 vBb
−18 294.135 Eb minor third
−17 811.127 ^Ab
−16 128.120 ^^Db
−15 645.112 Gd semidiminished fifth
−14 1162.105 vvC
−13 479.097 vF
−12 996.090 Bb minor seventh
−11 313.082 ^Eb
−10 830.075 ^^Ab
−9 147.067 Dd neutral second
−8 664.060 vvG
−7 1181.052 vC
−6 498.045 F perfect fourth
−5 1015.037 ^Bb
−4 332.030 ^^Eb
−3 849.022 Ad neutral sixth
−2 166.015 vvD
−1 683.007 vG
0 0.000 C perfect unison
1 516.993 ^F
2 1033.985 ^^Bb
3 350.978 Ed neutral third
4 867.970 vvA
5 184.963 vD
6 701.955 G perfect fifth
7 18.948 ^C
8 535.940 ^^F
9 1052.933 Bd neutral seventh
10 369.925 vvE
11 886.918 vA
12 203.910 D major second
13 720.903 ^G
14 37.895 ^^C
15 554.888 Ft semiaugmented fourth
16 1071.880 vvB
17 388.873 vE
18 905.865 A major sixth
19 222.858 ^D
20 739.850 ^^G
21 56.843 Ct semiaugmented unison
22 573.835 vvF#
23 1090.828 vB
24 407.820 E major third

Temperament interpretations

An obvious interpretation for beta-hexacot is gravity, where the generator is 27/20, two of which make 20/11, three make 11/9~27/22 above an octave, and six make a perfect fifth above two octaves. There are some extensions for the full 11-limit: marvo (65d & 72), zarvo (65 & 72), and gravid (58 & 65).

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