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The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.  
The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.


== Meantone (12&19, 2.3.5) ==
== Meantone (12&19, 2.3.5) ==
Line 54: Line 54:
{{see also| Meantone }}
{{see also| Meantone }}
{{Wikipedia|Septimal meantone temperament}}
{{Wikipedia|Septimal meantone temperament}}
The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the augmented fourth, and [[21/16]], C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 78: Line 78:
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]


[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, [[Tel:503.4257|503.4257]] cents. The recurrence converges quickly.
[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.


{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
Line 107: Line 107:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]


Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or [[Tel:696.9529|696.9529]] cents.
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


Vals: {{Val list| 12, 19e, 31, 105, 136b, 167be, 198be }}
Vals: {{Val list| 12, 19e, 31, 105, 136b, 167be, 198be }}
Line 566: Line 566:
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at [[Tel:506.3239|506.3239]] cents, equal to (1 + sqrt (19))/4.
Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.


{{Val list|legend=1| 7, 19, 26, 45 }}
{{Val list|legend=1| 7, 19, 26, 45 }}
Line 1,269: Line 1,269:
Mapping generators: ~7/5, ~16/11
Mapping generators: ~7/5, ~16/11


POTE generator: ~11/8 = [[Tel:552.5303|552.5303]]
POTE generator: ~11/8 = 552.5303


Vals: {{Val list| 24d, 26, 50d }}
Vals: {{Val list| 24d, 26, 50d }}
Line 1,282: Line 1,282:
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]


POTE generator: ~11/8 = [[Tel:552.5324|552.5324]]
POTE generator: ~11/8 = 552.5324


Vals: {{Val list| 24d, 26, 50d }}
Vals: {{Val list| 24d, 26, 50d }}
Line 1,295: Line 1,295:
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]


POTE generator: ~11/8 = [[Tel:552.6558|552.6558]]
POTE generator: ~11/8 = 552.6558


Vals: {{Val list| 24d, 26 }}
Vals: {{Val list| 24d, 26 }}
Line 1,308: Line 1,308:
Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]
Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]


POTE generator: ~11/8 = [[Tel:552.6382|552.6382]]
POTE generator: ~11/8 = 552.6382


Vals: {{Val list| 24d, 26 }}
Vals: {{Val list| 24d, 26 }}
Line 1,319: Line 1,319:
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 81/80, [[Tel:300125/294912|300125/294912]]
[[Comma list]]: 81/80, 300125/294912


[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
Line 1,327: Line 1,327:
{{Multival|legend=1| 4 16 -10 16 -27 -68 }}
{{Multival|legend=1| 4 16 -10 16 -27 -68 }}


[[POTE generator]]: ~48/35 = [[Tel:552.2206|552.2206]]
[[POTE generator]]: ~48/35 = 552.2206


{{Val list|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
{{Val list|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
Line 1,340: Line 1,340:
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]


POTE generator: ~11/8 = [[Tel:552.0929|552.0929]]
POTE generator: ~11/8 = 552.0929


Vals: {{Val list| 24, 26, 50 }}
Vals: {{Val list| 24, 26, 50 }}
Line 1,353: Line 1,353:
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]


POTE generator: ~11/8 = [[Tel:552.1498|552.1498]]
POTE generator: ~11/8 = 552.1498


Vals: {{Val list| 24, 26, 50 }}
Vals: {{Val list| 24, 26, 50 }}
Line 1,366: Line 1,366:
Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]
Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]


POTE generator: ~11/8 = [[Tel:552.1579|552.1579]]
POTE generator: ~11/8 = 552.1579


Vals: {{Val list| 24, 26, 50 }}
Vals: {{Val list| 24, 26, 50 }}
Line 1,379: Line 1,379:
Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]


POTE generator: ~11/8 = [[Tel:552.1196|552.1196]]
POTE generator: ~11/8 = 552.1196


Vals: {{Val list| 24, 26, 50 }}
Vals: {{Val list| 24, 26, 50 }}
Line 1,400: Line 1,400:
Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]


[[Gencom]]: [2 11/9; [[Tel:81/80 121|81/80 121]]/120]
[[Gencom]]: [2 11/9; 81/80 121/120]


[[POTE generator]]: ~11/9 = [[Tel:348.0938|348.0938]]
[[POTE generator]]: ~11/9 = 348.0938


{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
Line 1,419: Line 1,419:
Gencom mapping: [{{val|1 1 0 0 2 4}}, {{val|0 2 8 0 5 -1}}]
Gencom mapping: [{{val|1 1 0 0 2 4}}, {{val|0 2 8 0 5 -1}}]


Gencom: [2 11/9; [[Tel:66/65 81/80|66/65 81/80]] 121/120]
Gencom: [2 11/9; 66/65 81/80 121/120]


POTE generator: ~11/9 = [[Tel:348.9155|348.9155]]
POTE generator: ~11/9 = 348.9155


Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
Line 1,456: Line 1,456:
* 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]
* 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]


[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, [[Tel:348.6067|348.6067]] cents. Corresponding recurrence converges quickly.
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.


{{Val list|legend=1| 7, 24, 31 }}
{{Val list|legend=1| 7, 24, 31 }}
Line 1,596: Line 1,596:
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 81/80, [[Tel:5898240/5764801|5898240/5764801]]
[[Comma list]]: 81/80, 5898240/5764801


[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
Line 1,657: Line 1,657:
[[POTE generator]]: ~8/7 = 232.193
[[POTE generator]]: ~8/7 = 232.193


[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or [[Tel:232.0774|232.0774]] cents.
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,783: Line 1,783:
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is [[Tel:425.9311|425.9311]] cents.
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


{{Val list|legend=1| 14c, 17c, 31 }}
{{Val list|legend=1| 14c, 17c, 31 }}
Line 1,824: Line 1,824:
Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]
Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]


POTE generator: ~9/7 = [[Tel:425.7516|425.7516]]
POTE generator: ~9/7 = 425.7516


Vals: {{Val list| 14cf, 17c, 31f }}
Vals: {{Val list| 14cf, 17c, 31f }}
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