Meantone family: Difference between revisions

Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪), and tridecimal meantone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second.  

==== Fokkertone ====

Comma list: 66/65, 81/80, 99/98, 105/104

Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}

* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}

* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}

: unchanged-interval (eigenmonzo) basis: 2.11/9

{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}

Badness (Sintel): 0.746

Comma list: 66/65, 81/80, 99/98, 105/104, 120/119

Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}

* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}

{{Optimal ET sequence|legend=0| 12f, 31 }}

Badness (Sintel): 1.02

Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119

Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}

* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}

Badness (Sintel): 1.10

Grosstone maps 13/8 to the double-diminished seventh (C–B♭♭♭).  

Comma list: 81/80, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}

* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}

* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}

: eigenmonzo basis (unchanged-interval basis): 2.13/7

* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)

 

Badness (Sintel): 1.07

Comma list: 81/80, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}

* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}

{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}

Badness (Sintel): 1.06

Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}

* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}

{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}

Badness (Sintel): 1.07

Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪).  

Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.  

Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.  

Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension takes the tritone as 16/11, which it barely sounds like. The first alternative, domineering, takes the same step as 11/8, which it barely sounds like either. Domination tempers out 77/75 and identifies 11/8 with the augmented third; arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension [[neutrominant]], splitting the fifth as well as the chromatic semitone in two like in all [[rastmic clan|rastmic]] temperaments.