Meantone family: Difference between revisions

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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.

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.

Subgroup: 2.3.5.7

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

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

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

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* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]

Algebraic generator: Traverse, the positive real root of ''x''4 + 2''x'' - 13, or 696.9529 cents.

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

Vals: {{Val list| 12, 19e, 31, 105, 136b, 167be, 198be }}

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* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]

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

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

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Mapping generators: ~7/5, ~16/11

POTE generator: ~11/8 = 552.5303

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

Vals: {{Val list| 24d, 26, 50d }}

Line 1,282:

Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]

POTE generator: ~11/8 = 552.5324

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

Vals: {{Val list| 24d, 26, 50d }}

Line 1,295:

Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]

POTE generator: ~11/8 = 552.6558

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

Vals: {{Val list| 24d, 26 }}

Line 1,308:

Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]

POTE generator: ~11/8 = 552.6382

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

Vals: {{Val list| 24d, 26 }}

Line 1,319:

Subgroup: 2.3.5.7

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

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

[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]

Line 1,327:

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

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

Line 1,340:

Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]

POTE generator: ~11/8 = 552.0929

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

Vals: {{Val list| 24, 26, 50 }}

Line 1,353:

Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]

POTE generator: ~11/8 = 552.1498

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

Vals: {{Val list| 24, 26, 50 }}

Line 1,366:

Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]

POTE generator: ~11/8 = 552.1579

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

Vals: {{Val list| 24, 26, 50 }}

Line 1,379:

Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]

POTE generator: ~11/8 = 552.1196

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

Vals: {{Val list| 24, 26, 50 }}

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Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]

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

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

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

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

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Gencom mapping: [{{val|1 1 0 0 2 4}}, {{val|0 2 8 0 5 -1}}]

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

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

POTE generator: ~11/9 = 348.9155

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

Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}

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* 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]

[[Algebraic generator]]: Mohabis, real root of 3''x''3 - 3''x''2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

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

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Subgroup: 2.3.5.7

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

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

[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]

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[[POTE generator]]: ~8/7 = 232.193

[[Algebraic generator]]: Rabrindanath, largest real root of ''x''8 - 3''x''2 + 1, or 232.0774 cents.

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

[[Minimax tuning]]:

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: [[Eigenmonzo]]s (unchanged intervals): 2, 5

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

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

Line 1,824:

Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]

POTE generator: ~9/7 = 425.7516

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

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

@@ Line 54: / Line 54: @@
 {{see also| Meantone }}
 {{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 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.
+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.
 Subgroup: 2.3.5.7
@@ Line 78: / Line 78: @@
 * 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, 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, [[Tel:503.4257|503.4257]] cents. The recurrence converges quickly.
 {{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]
-Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
+Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or [[Tel:696.9529|696.9529]] cents.
 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]
-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.
+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.
 {{Val list|legend=1| 7, 19, 26, 45 }}
@@ Line 1,269: / Line 1,269: @@
 Mapping generators: ~7/5, ~16/11
-POTE generator: ~11/8 = 552.5303
+POTE generator: ~11/8 = [[Tel:552.5303|552.5303]]
 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 }}]
-POTE generator: ~11/8 = 552.5324
+POTE generator: ~11/8 = [[Tel:552.5324|552.5324]]
 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 }}]
-POTE generator: ~11/8 = 552.6558
+POTE generator: ~11/8 = [[Tel:552.6558|552.6558]]
 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 }}]
-POTE generator: ~11/8 = 552.6382
+POTE generator: ~11/8 = [[Tel:552.6382|552.6382]]
 Vals: {{Val list| 24d, 26 }}
@@ Line 1,319: / Line 1,319: @@
 Subgroup: 2.3.5.7
-[[Comma list]]: 81/80, 300125/294912
+[[Comma list]]: 81/80, [[Tel:300125/294912|300125/294912]]
 [[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 }}
-[[POTE generator]]: ~48/35 = 552.2206
+[[POTE generator]]: ~48/35 = [[Tel:552.2206|552.2206]]
 {{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 }}]
-POTE generator: ~11/8 = 552.0929
+POTE generator: ~11/8 = [[Tel:552.0929|552.0929]]
 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 }}]
-POTE generator: ~11/8 = 552.1498
+POTE generator: ~11/8 = [[Tel:552.1498|552.1498]]
 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 }}]
-POTE generator: ~11/8 = 552.1579
+POTE generator: ~11/8 = [[Tel:552.1579|552.1579]]
 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 }}]
-POTE generator: ~11/8 = 552.1196
+POTE generator: ~11/8 = [[Tel:552.1196|552.1196]]
 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]]: [2 11/9; 81/80 121/120]
+[[Gencom]]: [2 11/9; [[Tel:81/80 121|81/80 121]]/120]
-[[POTE generator]]: ~11/9 = 348.0938
+[[POTE generator]]: ~11/9 = [[Tel:348.0938|348.0938]]
 {{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: [2 11/9; 66/65 81/80 121/120]
+Gencom: [2 11/9; [[Tel:66/65 81/80|66/65 81/80]] 121/120]
-POTE generator: ~11/9 = 348.9155
+POTE generator: ~11/9 = [[Tel:348.9155|348.9155]]
 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]
-[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
+[[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.
 {{Val list|legend=1| 7, 24, 31 }}
@@ Line 1,596: / Line 1,596: @@
 Subgroup: 2.3.5.7
-[[Comma list]]: 81/80, 5898240/5764801
+[[Comma list]]: 81/80, [[Tel:5898240/5764801|5898240/5764801]]
 [[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
@@ Line 1,657: / Line 1,657: @@
 [[POTE generator]]: ~8/7 = 232.193
-[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
+[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or [[Tel:232.0774|232.0774]] cents.
 [[Minimax tuning]]:
@@ Line 1,783: / Line 1,783: @@
 : [[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 425.9311 cents.
+[[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.
 {{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 }}]
-POTE generator: ~9/7 = 425.7516
+POTE generator: ~9/7 = [[Tel:425.7516|425.7516]]
 Vals: {{Val list| 14cf, 17c, 31f }}