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Reflect difference of 7-odd-limit diamond tradeoff and 9-odd-limit diamond tradeoff on some temperaments
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [685.714, 720.000] (4\7 to 3\5)
* [[Diamond monotone]] range: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
* [[Diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* [[Diamond tradeoff]] range: ~3/2 = [694.786, 701.955]
* Diamond monotone and tradeoff: [694.786, 701.955]
* Diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]


{{Val list|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
{{Val list|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [694.737, 700.000] (11\19 to 7\12)
* [[Diamond monotone]] range: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* [[Diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
* Diamond monotone and tradeoff: [694.786, 700.000]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 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, 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.
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [696.774, 700.000] (18\31 to 7\12)
* Diamond monotone range: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* Diamond tradeoff range: [691.202, 701.955]
* Diamond tradeoff range: ~3/2 = [691.202, 701.955]
* Diamond monotone and tradeoff: [696.774, 700.000]
* 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 696.9529 cents.
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [696.774, 697.674] (18\31 to 25\43)
* Diamond monotone range: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* Diamond tradeoff range: [691.202, 701.955]
* Diamond tradeoff range: ~3/2 = [691.202, 701.955]
* Diamond monotone and tradeoff: [696.774, 697.674]
* Diamond monotone and tradeoff: ~3/2 = [696.774, 697.674]


Vals: {{Val list| 12, 19ef, 31, 43, 74 }}
Vals: {{Val list| 12, 19ef, 31, 43, 74 }}
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [694.737, 696.774] (11\19 to 18\31)
* Diamond monotone range: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* Diamond tradeoff range: [691.202, 701.955]
* Diamond tradeoff range: ~3/2 = [691.202, 701.955]
* Diamond monotone and tradeoff: [694.737, 696.774]
* Diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]


Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [694.737, 696.774] (11\19 to 18\31)
* Diamond monotone range: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* Diamond tradeoff range: [691.202, 701.955]
* Diamond tradeoff range: ~3/2 = [691.202, 701.955]
* Diamond monotone and tradeoff: [694.737, 696.774]
* Diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]


Vals: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
Vals: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [694.737, 700.000] (11\19 to 7\12)
* Diamond monotone range: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* Diamond tradeoff range: [682.502, 704.377]
* Diamond tradeoff range: ~3/2 = [682.502, 704.377]
* Diamond monotone and tradeoff: [694.737, 700.000]
* Diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]


Vals: {{Val list| 7d, 12, 19, 31e, 50ee }}
Vals: {{Val list| 7d, 12, 19, 31e, 50ee }}
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[[Comma list]]: 81/80, 525/512
[[Comma list]]: 81/80, 525/512


[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
[[Mapping]]: [{{val| 1 0 -4 17 }}, {{val| 0 1 4 -9 }}]


{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
* [[Diamond monotone]] range: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* [[Diamond tradeoff]] range: [692.353, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* Diamond monotone and tradeoff: [692.353, 694.737]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [692.353, 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 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.
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
* Diamond monotone range: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* Diamond tradeoff range: [682.502, 701.955]
* Diamond tradeoff range: ~3/2 = [682.502, 701.955]
* Diamond monotone and tradeoff: [692.308, 694.737]
* Diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


Vals: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}
Vals: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
* Diamond monotone range: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* Diamond tradeoff range: [682.502, 701.955]
* Diamond tradeoff range: ~3/2 = [682.502, 701.955]
* Diamond monotone and tradeoff: [692.308, 694.737]
* Diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


Vals: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Vals: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [700.000, 720.000] (7\12 to 3\5)
* [[Diamond monotone]] range: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* [[Diamond tradeoff]] range: [694.786, 715.587]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* Diamond monotone and tradeoff: [700.000, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* Diamond monotone and tradeoff: ~3/2 = [700.000, 715.587]


{{Val list|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
{{Val list|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [700.000, 705.882] (7\12 to 10\17)
* Diamond monotone range: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* Diamond tradeoff range: [691.202, 715.587]
* Diamond tradeoff range: ~3/2 = [691.202, 715.587]
* Diamond monotone and tradeoff: [700.000, 705.882]
* Diamond monotone and tradeoff: ~3/2 = [700.000, 705.882]


POTE generator: ~3/2 = 703.254
POTE generator: ~3/2 = 703.254
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: 705.882 (10\17)
* Diamond monotone range: ~3/2 = 705.882 (10\17)
* Diamond tradeoff range: [691.202, 715.587]
* Diamond tradeoff range: ~3/2 = [691.202, 715.587]
* Diamond monotone and tradeoff: 705.882
* Diamond monotone and tradeoff: ~3/2 = 705.882


Vals: {{Val list| 12f, 17c, 29cdef }}
Vals: {{Val list| 12f, 17c, 29cdef }}
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{{main| Semaphore and Godzilla }}
{{main| Semaphore and Godzilla }}


Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo|19EDO]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [240.000, 257.143] (1\5 to 3\14)
* [[Diamond monotone]] range: ~7/6 = [240.000, 257.143] (1\5 to 3\14)
* [[Diamond tradeoff]] range: [231.174, 266.871]
* [[Diamond tradeoff]] range: ~7/6 = [231.174, 266.871]
* Diamond monotone and tradeoff: [240.000, 257.143]
* Diamond monotone and tradeoff: ~7/6 = [240.000, 257.143]


{{Val list|legend=1| 5, 14c, 19 }}
{{Val list|legend=1| 5, 14c, 19 }}
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [252.632, 257.143] (4\19 to 3\14)
* Diamond monotone range: ~7/6 = [252.632, 257.143] (4\19 to 3\14)
* Diamond tradeoff range: [231.174, 266.871]
* Diamond tradeoff range: ~7/6 = [231.174, 266.871]
* Diamond monotone and tradeoff: [252.632, 257.143]
* Diamond monotone and tradeoff: ~7/6 = [252.632, 257.143]


Vals: {{Val list| 14c, 19, 33cd, 52cd }}
Vals: {{Val list| 14c, 19, 33cd, 52cd }}
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Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: 694.737 (4\19)
* Diamond monotone range: ~7/6 = 252.632 (4\19)
* Diamond tradeoff range: [621.581, 737.652]
* Diamond tradeoff range: ~7/6 = [231.174, 289.210]
* Diamond monotone and tradeoff: 694.737
* Diamond monotone and tradeoff: ~7/6 = 252.632


Vals: {{Val list| 14cf, 19, 33cdff, 52cdff }}
Vals: {{Val list| 14cf, 19, 33cdff, 52cdff }}
Line 1,128: Line 1,133:


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [685.714, 700.000] (8\14 to 7\12)
* [[Diamond monotone]] range: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* [[Diamond tradeoff]] range: [688.957, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* Diamond monotone and tradeoff: [688.957, 700.000]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [688.957, 700.000]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
Line 1,151: Line 1,158:


Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: [685.714, 700.000] (8\14 to 7\12)
* Diamond monotone range: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* Diamond tradeoff range: [682.458, 701.955]
* Diamond tradeoff range: ~3/2 = [682.458, 701.955]
* Diamond monotone and tradeoff: [685.714, 700.000]
* Diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


Vals: {{Val list| 12, 14c, 26, 90bce, 116bcce }}
Vals: {{Val list| 12, 14c, 26, 90bce, 116bcce }}
Line 1,171: Line 1,178:


Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: 692.308 (15\26)
* Diamond monotone range: ~3/2 = 692.308 (15\26)
* Diamond tradeoff range: [682.458, 701.955]
* Diamond tradeoff range: ~3/2 = [682.458, 701.955]
* Diamond monotone and tradeoff: 692.308 (15\26)
* Diamond monotone and tradeoff: ~3/2 = 692.308


Vals: {{Val list| 12f, 14cf, 26, 38e }}
Vals: {{Val list| 12f, 14cf, 26, 38e }}
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: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo]]s: 2, 5
: [[Eigenmonzo]]s: 2, 5
[[Tuning ranges]]:
* [[Diamond monotone]] range: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit [[diamond tradeoff]]: ~128/105 = [345.601, 350.978]
* 7-odd-limit diamond monotone and tradeoff: ~128/105 = [347.393, 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, 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.
Line 1,311: Line 1,325:
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
: Eigenmonzos: 2, 5
: Eigenmonzos: 2, 5
[[Tuning ranges]]:
* Diamond monotone range: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* Diamond tradeoff range: ~11/9 = [344.999, 350.978]
* Diamond monotone and tradeoff: ~11/9 = [348.387, 350.000]


Vals: {{Val list| 7, 24, 31 }}
Vals: {{Val list| 7, 24, 31 }}
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