Xenharmonic Wiki

Meantone family: Difference between revisions

← Older editNewer edit →
VisualWikitext
Revert to -CTWE/KE. It was a readability hell
-diamond monotone and tradeoff (easily derivable and lack of significance); fix 13-limit ptolemy
Line 1,058: Line 1,058:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,068: Line 1,068:
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 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.
Line 1,091: Line 1,089:
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
Line 1,111: Line 1,108:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


{{Optimal ET sequence|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Line 1,118: Line 1,114:


Scales: [[flattone12]]
Scales: [[flattone12]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 65/64, 81/80, 105/104, 121/120
Mapping: {{mapping| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 346.910
{{Optimal ET sequence|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
Badness: 0.034316


== Dominant ==
== Dominant ==
Line 1,148: Line 1,131:


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 715.587]


{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
Line 1,167: Line 1,149:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 705.882]


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254
Line 1,187: Line 1,168:
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 705.882


{{Optimal ET sequence|legend=1| 12f, 17c, 29cdef }}
{{Optimal ET sequence|legend=1| 12f, 17c, 29cdef }}
Line 1,348: Line 1,328:


Badness: 0.021098
Badness: 0.021098


== Sharptone ==
== Sharptone ==
Line 1,446: Line 1,424:
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 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 1,474: Line 1,450:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
* 11-odd-limit diamond monotone and tradeoff: ~11/9 = [348.387, 350.000]


{{Optimal ET sequence|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}
Line 1,942: Line 1,917:
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 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]


{{Optimal ET sequence|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
{{Optimal ET sequence|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
Line 1,964: Line 1,937:
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


{{Optimal ET sequence|legend=1| 12, 14c, 26, 90bce, 116bcce }}
{{Optimal ET sequence|legend=1| 12, 14c, 26, 90bce, 116bcce }}
Line 1,982: Line 1,954:
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 692.308


{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 38e }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 38e }}
Line 2,272: Line 2,243:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Meantone family| ]] <!-- main article -->
[[Category:Meantone family| ]] <!-- main article -->
[[Category:Meantone|#]] <!-- key article -->
[[Category:Meantone| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Meantone_family"