Meantone family: Difference between revisions

In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). Septimal meantone tempers out the common 7-limit commas [[126/125]], [[225/224]], and [[3136/3125]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.  

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

This extension is perhaps the most practical, as it maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.  

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𝄪𝄪).  

{{See also| Meantone vs meanpop }}

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.  

In flattone tunings, the fifth is typically even flatter than that of [[19edo]]. Here, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].

=== 13-limit ===

== Flattertone ==

Flattertone tunings are typically at least as flat as [[26edo]]. Here, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). [[26edo]] and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.

[[Comma list]]: 81/80, 1875/1792

{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}

{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}

[[Badness]] (Sintel): 2.43

==== 11-limit ====

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1375/1344

Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}

* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}

{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}

Badness (Sintel): 1.53

* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)

== Dominant ==

{{Main| Dominant (temperament) }}

{{See also| Archytas clan }}

The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh (C–Bb). The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].

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.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 64/63

{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}

: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}

: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}

[[Tuning ranges]]:  

* [[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]

* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}

[[Badness]] (Sintel): 0.524

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 64/63

Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}

 

* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}

Badness (Sintel): 0.799

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 64/63, 66/65

Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}

* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}

* 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]

 

==== Dominion ====

Comma list: 26/25, 36/35, 56/55, 64/63

Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}

* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17c }}

=== Domination ===

Comma list: 36/35, 64/63, 77/75

Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}

* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}

{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}

Badness (Sintel): 1.21

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}

Optimal tunings:  

* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}

{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}

Badness (Sintel): 1.13

=== Arnold ===

Comma list: 22/21, 33/32, 36/35

Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}

* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}

Badness (Sintel): 0.864

Mildtone tempers out 16128/15625 and finds the interval class of 7 at 22 generators up, as a triple-augmented fifth (C–G#x).  

Mohajira can be viewed as derived from mohaha which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31.

* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)