Compton family: Difference between revisions

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The '''Compton family''' tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12edo, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
The '''compton family''' tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12edo, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.


== Compton ==
== Compton ==
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Mapping generators: ~256/243, ~5
Mapping generators: ~256/243, ~5


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.884 (~81/80 = 15.116)
[[Optimal tuning]] ([[POTE]]): ~256/243 = 1\12, ~5/4 = 384.884 (~81/80 = 15.116)


{{Val list|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}
{{Val list|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}
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[[Mapping]]: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]
[[Mapping]]: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]


[[Optimal tuning]] ([[POTE]]): ~5/4 = 383.7752 (~126/125 = 16.2248)
[[Optimal tuning]] ([[POTE]]): ~256/243 = 1\12, ~5/4 = 383.7752 (~126/125 = 16.2248)


{{Val list|legend=1| 12, 48d, 60, 72, 228, 300c, 372bc, 444bc }}
{{Val list|legend=1| 12, 48d, 60, 72, 228, 300c, 372bc, 444bc }}
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Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]
Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]


Optimal tuning (POTE): ~5/4 = 383.2660 (~100/99 = 16.7340)
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.2660 (~100/99 = 16.7340)


Optimal GPV sequence: {{Val list| 12, 48dee, 60e, 72 }}
Optimal GPV sequence: {{Val list| 12, 48dee, 60e, 72 }}
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Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]
Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]


Optimal tuning (POTE): ~5/4 = 383.9628 (~105/104 = 16.0372)
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.9628 (~105/104 = 16.0372)


Optimal GPV sequence: {{Val list| 12f, 48defff, 60eff, 72, 228f }}
Optimal GPV sequence: {{Val list| 12f, 48defff, 60eff, 72, 228f }}
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Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 0 0 1 2 3 4 0 }}]
Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 0 0 1 2 3 4 0 }}]


Optimal tuning (POTE): ~5/4 = 383.7500 (~105/104 = 16.2500)
Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 383.7500 (~105/104 = 16.2500)


Optimal GPV sequence: {{Val list| 12f, 60eff, 72 }}
Optimal GPV sequence: {{Val list| 12f, 60eff, 72 }}
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Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]
Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]


Optimal tuning (POTE): ~5/4 = 382.6116 (~100/99 = 17.3884)
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 382.6116 (~100/99 = 17.3884)


Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdef, 276cdeff }}
Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdef, 276cdeff }}
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Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 0 0 1 2 3 -2 0 }}]
Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 0 0 1 2 3 -2 0 }}]


Optimal tuning (POTE): ~5/4 = 382.5968 (~100/99 = 17.4032)
Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 382.5968 (~100/99 = 17.4032)


Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdefg, 276cdeffgg }}
Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdefg, 276cdeffgg }}
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Mapping generators: ~16/15, ~7
Mapping generators: ~16/15, ~7


[[Optimal tuning]] ([[POTE]]): ~64/63 = 26.790
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~64/63 = 26.790


{{Val list|legend=1| 12, 24, 36, 48c }}
{{Val list|legend=1| 12, 24, 36, 48c }}
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Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]
Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]


Optimal tuning (POTE): ~64/63 = 22.723
Optimal tuning (POTE): ~16/15 = 1\12, ~64/63 = 22.723


Optimal GPV sequence: {{Val list| 12, 36e, 48c, 108ccd }}
Optimal GPV sequence: {{Val list| 12, 36e, 48c, 108ccd }}
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Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]
Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]


Optimal tuning (POTE): ~64/63 = 27.864
Optimal tuning (POTE): ~16/15 = 1\12, ~64/63 = 27.864


Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
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Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]
Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]


Optimal tuning (POTE): ~36/35 = 32.776
Optimal tuning (POTE): ~16/15 = 1\12, ~36/35 = 32.776


Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
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Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]
Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]


Optimal tuning (POTE): ~36/35 = 37.232
Optimal tuning (POTE): ~16/15 = 1\12, ~36/35 = 37.232


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
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Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 0 0 0 1 1 1 0 }}]
Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 0 0 0 1 1 1 0 }}]


Optimal tuning (POTE): ~36/35 = 39.777
Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 39.777


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
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Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 0 0 0 1 1 1 0 0 }}]
Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 0 0 0 1 1 1 0 0 }}]


Optimal tuning (POTE): ~36/35 = 40.165
Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 40.165


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
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Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]
Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]


Optimal tuning (POTE): ~36/35 = 37.688
Optimal tuning (POTE): ~16/15 = 1\12, ~36/35 = 37.688


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
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Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]
Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]


Optimal tuning (POTE): ~36/35 = 38.097
Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 38.097


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
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Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]
Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]


Optimal tuning (POTE): ~36/35 = 38.080
Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 38.080


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
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Mapping generators: ~16/15, ~11
Mapping generators: ~16/15, ~11


[[Optimal tuning]] ([[POTE]]): ~45/44 = 34.977
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~45/44 = 34.977


{{Val list|legend=1| 12, 24d }}
{{Val list|legend=1| 12, 24d }}
Line 256: Line 256:
The hours temperament has a period of 1/24 octave and tempers out the [[cataharry comma]] (19683/19600) and the mirwomo comma (33075/32768). The name "hours" was so named for the following reasons – the period is 1/24 octave, and there are 24 hours per a day.
The hours temperament has a period of 1/24 octave and tempers out the [[cataharry comma]] (19683/19600) and the mirwomo comma (33075/32768). The name "hours" was so named for the following reasons – the period is 1/24 octave, and there are 24 hours per a day.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 33075/32768
[[Comma list]]: 19683/19600, 33075/32768
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Mapping generators: ~36/35, ~5
Mapping generators: ~36/35, ~5


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.033  
[[Optimal tuning]] ([[POTE]]): ~36/35 = 1\24, ~5/4 = 384.033  


{{Val list|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}
{{Val list|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}
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Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]
Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]


Optimal tuning (POTE): ~5/4 = 384.054
Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.054


Optimal GPV sequence: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
Optimal GPV sequence: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
Line 292: Line 292:
Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}]
Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}]


Optimal tuning (POTE): ~5/4 = 384.652
Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.652


Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}
Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}
Line 301: Line 301:
The decades temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the stearnsma (118098/117649). The name "decades" was so named for the following reasons – the period is 1/36 octave, and there are 36 decades (''ten days'') per a year (12 months × 3 decades per a month).  
The decades temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the stearnsma (118098/117649). The name "decades" was so named for the following reasons – the period is 1/36 octave, and there are 36 decades (''ten days'') per a year (12 months × 3 decades per a month).  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 118098/117649
[[Comma list]]: 1029/1024, 118098/117649
Line 311: Line 311:
{{Multival|legend=1| 0 36 0 57 0 -101 }}
{{Multival|legend=1| 0 36 0 57 0 -101 }}


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.764
[[Optimal tuning]] ([[POTE]]): ~49/48 = 1\36, ~5/4 = 384.764


{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
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Mapping: [{{val| 36 57 0 101 41 }}, {{val| 0 0 1 0 1 }}]
Mapping: [{{val| 36 57 0 101 41 }}, {{val| 0 0 1 0 1 }}]


Optimal tuning (POTE): ~5/4 = 384.150
Optimal tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.150


Optimal GPV sequence: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
Optimal GPV sequence: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
Line 331: Line 331:


== Omicronbeta ==
== Omicronbeta ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 225/224, 243/242, 441/440, 4375/4356
[[Comma list]]: 225/224, 243/242, 441/440, 4375/4356
Line 339: Line 339:
Mapping generators: ~100/99, ~13
Mapping generators: ~100/99, ~13


[[Optimal tuning]] ([[POTE]]): ~13/8 = 837.814
[[Optimal tuning]] ([[POTE]]): ~100/99 = 1\72, ~13/8 = 837.814


{{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}
{{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}

Revision as of 13:41, 21 December 2022

The compton family tempers out the Pythagorean comma, 531441/524288 = [-19 12⟩, and hence the fifths form a closed 12-note circle of fifths, identical to 12edo. While the tuning of the fifth will be that of 12edo, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

Compton

Compton tempers out the Pythagorean comma and has a period of 1\12, so it is the 12edo circle of fifths with an independent dimension for the harmonic 5. Equivalent generators are 5/4, 6/5, 10/9, 16/15 (the secor), 45/32, 135/128 and most importantly, 81/80. In terms of equal temperaments, it is the 12&72 temperament, and 72edo, 84edo or 240edo make for good tunings.

Subgroup: 2.3.5

Comma list: 531441/524288

Mapping: [⟨12 19 0], ⟨0 0 1]

Mapping generators: ~256/243, ~5

Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 384.884 (~81/80 = 15.116)

Template:Val list

Badness: 0.094494

Septimal compton

Septimal compton is also known as waage. In terms of the normal list, compton adds 413343/409600 = [-14 10 -2 1⟩ to the Pythagorean comma; however, it can also be characterized by saying it adds 225/224.

In either the 5- or 7-limit, 240edo is an excellent tuning, with 81/80 coming in at 15 cents exactly. In the 12edo, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this 72edo can be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 250047/250000

Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]

Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.7752 (~126/125 = 16.2248)

Template:Val list

Badness: 0.035686

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4356

Mapping: [⟨12 19 0 -22 -42], ⟨0 0 1 2 3]]

Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.2660 (~100/99 = 16.7340)

Optimal GPV sequence: Template:Val list

Badness: 0.022235

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 441/440

Mapping: [⟨12 19 0 -22 -42 -67], ⟨0 0 1 2 3 4]]

Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.9628 (~105/104 = 16.0372)

Optimal GPV sequence: Template:Val list

Badness: 0.021852

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 225/224, 289/288, 351/350, 441/440

Mapping: [⟨12 19 0 -22 -42 -67 49], ⟨0 0 1 2 3 4 0]]

Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 383.7500 (~105/104 = 16.2500)

Optimal GPV sequence: Template:Val list

Badness: 0.017131

Comptone

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 441/440, 1001/1000

Mapping: [⟨12 19 0 -22 -42 100], ⟨0 0 1 2 3 -2]]

Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 382.6116 (~100/99 = 17.3884)

Optimal GPV sequence: Template:Val list

Badness: 0.025144

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 289/288, 325/324, 441/440

Mapping: [⟨12 19 0 -22 -42 100 49], ⟨0 0 1 2 3 -2 0]]

Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 382.5968 (~100/99 = 17.4032)

Optimal GPV sequence: Template:Val list

Badness: 0.016361

Catler

In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of 12edo. Catler can also be characterized as the 12 & 24 temperament. 36edo or 48edo are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 9/7, 7/5, and most importantly, 64/63.

Subgroup: 2.3.5.7

Comma list: 81/80, 128/125

Mapping: [⟨12 19 28 0], ⟨0 0 0 1]]

Mapping generators: ~16/15, ~7

Optimal tuning (POTE): ~16/15 = 1\12, ~64/63 = 26.790

Template:Val list

Badness: 0.050297

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 128/125

Mapping: [⟨12 19 28 0 -26], ⟨0 0 0 1 2]]

Optimal tuning (POTE): ~16/15 = 1\12, ~64/63 = 22.723

Optimal GPV sequence: Template:Val list

Badness: 0.058213

Catlat

Subgroup: 2.3.5.7.11

Comma list: 81/80, 128/125, 540/539

Mapping: [⟨12 19 28 0 109], ⟨0 0 0 1 -2]]

Optimal tuning (POTE): ~16/15 = 1\12, ~64/63 = 27.864

Optimal GPV sequence: Template:Val list

Badness: 0.081909

Catcall

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 128/125

Mapping: [⟨12 19 28 0 8], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~16/15 = 1\12, ~36/35 = 32.776

Optimal GPV sequence: Template:Val list

Badness: 0.034478

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 81/80, 105/104

Mapping: [⟨12 19 28 0 8 11], ⟨0 0 0 1 1 1]]

Optimal tuning (POTE): ~16/15 = 1\12, ~36/35 = 37.232

Optimal GPV sequence: Template:Val list

Badness: 0.028363

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 56/55, 66/65, 81/80, 105/104

Mapping: [⟨12 19 28 0 8 11 49], ⟨0 0 0 1 1 1 0]]

Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 39.777

Optimal GPV sequence: Template:Val list

Badness: 0.023246

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95

Mapping: [⟨12 19 28 0 8 11 49 51], ⟨0 0 0 1 1 1 0 0]]

Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 40.165

Optimal GPV sequence: Template:Val list

Badness: 0.018985

Duodecic

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 81/80, 91/90, 128/125

Mapping: [⟨12 19 28 0 8 78], ⟨0 0 0 1 1 -1]]

Optimal tuning (POTE): ~16/15 = 1\12, ~36/35 = 37.688

Optimal GPV sequence: Template:Val list

Badness: 0.038307

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 56/55, 81/80, 91/90, 128/125

Mapping: [⟨12 19 28 0 8 78 49], ⟨0 0 0 1 1 -1 0]]

Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 38.097

Optimal GPV sequence: Template:Val list

Badness: 0.027487

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95

Mapping: [⟨12 19 28 0 8 78 49 51], ⟨0 0 0 1 1 -1 0 0]]

Optimal tuning (POTE): ~18/17 = 1\12, ~36/35 = 38.080

Optimal GPV sequence: Template:Val list

Badness: 0.020939

Duodecim

See also: Jubilismic clan § Duodecim

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 64/63

Mapping: [⟨12 19 28 34 0], ⟨0 0 0 0 1]]

Mapping generators: ~16/15, ~11

Optimal tuning (POTE): ~16/15 = 1\12, ~45/44 = 34.977

Template:Val list

Badness: 0.030536

Hours

The hours temperament has a period of 1/24 octave and tempers out the cataharry comma (19683/19600) and the mirwomo comma (33075/32768). The name "hours" was so named for the following reasons – the period is 1/24 octave, and there are 24 hours per a day.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 33075/32768

Mapping: [⟨24 38 0 123], ⟨0 0 1 -1]]

Wedgie: ⟨⟨ 0 24 -24 38 -38 -123 ]]

Mapping generators: ~36/35, ~5

Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.033

Template:Val list

Badness: 0.116091

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 9801/9800

Mapping: [⟨24 38 0 123 83], ⟨0 0 1 -1 0]]

Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.054

Optimal GPV sequence: Template:Val list

Badness: 0.036248

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 385/384

Mapping: [⟨24 38 0 123 83 33], ⟨0 0 1 -1 0 1]]

Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.652

Optimal GPV sequence: Template:Val list

Badness: 0.026931

Decades

The decades temperament has a period of 1/36 octave and tempers out the gamelisma (1029/1024) and the stearnsma (118098/117649). The name "decades" was so named for the following reasons – the period is 1/36 octave, and there are 36 decades (ten days) per a year (12 months × 3 decades per a month).

Subgroup: 2.3.5.7

Comma list: 1029/1024, 118098/117649

Mapping: [⟨36 57 0 101], ⟨0 0 1 0]]

Mapping generators: ~49/48, ~5

Wedgie: ⟨⟨ 0 36 0 57 0 -101 ]]

Optimal tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.764

Template:Val list

Badness: 0.108016

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1029/1024, 4000/3993

Mapping: [⟨36 57 0 101 41], ⟨0 0 1 0 1]]

Optimal tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.150

Optimal GPV sequence: Template:Val list

Badness: 0.043088

Omicronbeta

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 441/440, 4375/4356

Mapping: [⟨72 114 167 202 249 266], ⟨0 0 0 0 0 1]]

Mapping generators: ~100/99, ~13

Optimal tuning (POTE): ~100/99 = 1\72, ~13/8 = 837.814

Template:Val list

Badness: 0.029956

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