User:CompactStar/Meantone family: Difference between revisions

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Created page with "THIS IS A TESTING PAGE FOR TEMPLATES The 5-limit parent comma of the '''meantone family''' is the syntonic comma, 81/80. This is the one they all temper out. The..."
 
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Undecimal meantone maps the [[11/8]] to the double augmented third (C-Ex), and tridecimal meantone maps the [[13/8]] to the double augmented fifth (C-Gx). Note 13/11 is conflated with 6/5, represented by the minor third. Note also 11/10~13/12 is the double augmented unison; 12/11, double diminished third; and 14/13, minor second.  
Undecimal meantone maps the [[11/8]] to the double augmented third (C-Ex), and tridecimal meantone maps the [[13/8]] to the double augmented fifth (C-Gx). Note 13/11 is conflated with 6/5, represented by the minor third. Note also 11/10~13/12 is the double augmented unison; 12/11, double diminished third; and 14/13, minor second.  


Subgroup: 2.3.5.7.11
{{Temperament data|subgroup=2.3.5.7.11|comma_list=81/80, 99/98, 126/125|generators=2/1,3/2}}
 
Comma list: 81/80, 99/98, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.1676
 
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: eigenmonzo (unchanged-interval) basis: 2.11/9
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
 
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
 
{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b }}
 
Badness: 0.017027


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