Meantone family: Difference between revisions

Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].

[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).

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.  

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

==== 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𝄪𝄪).  

Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}

Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}

Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}

Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}

Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}

Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}

Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}

Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}

Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}

{{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.  

Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}

Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}

Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}

Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}

Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}

Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}

=== Meanenneadecal ===

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.

Subgroup: 2.3.5.7.11

Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}

* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}

* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)

* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]

 

Badness (Sintel): 0.708

 

Subgroup: 2.3.5.7.11.13

 

Comma list: 45/44, 56/55, 78/77, 81/80

 

 

 

 

Comma list: 45/44, 56/55, 78/77, 81/80, 120/119

Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}

* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}

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

Badness (Sintel): 1.17

Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119

Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}

* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}

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

Badness (Sintel): 1.23

==== Vincenzo ====

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}

* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}

{{Optimal ET sequence|legend=0| 7d, 12, 19 }}

Badness (Sintel): 1.02

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}

* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}

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

Badness (Sintel): 1.30

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}

* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}

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

Badness (Sintel): 1.36

=== Bimeantone ===

11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  

 

Comma list: 81/80, 126/125, 245/242

 

: mapping generators: ~63/44, ~3

* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}

* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}

{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}

Badness (Sintel): 1.26

==== 13-limit ====

Comma list: 81/80, 105/104, 126/125, 245/242

Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}

* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}

{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}

Badness (Sintel): 1.19

==== 17-limit ====

Comma list: 81/80, 105/104, 126/125, 189/187, 221/220

Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}

* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}

{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}

Badness (Sintel): 1.15

==== 19-limit ====

Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220

Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}

* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}

{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}

Badness (Sintel): 1.08

=== Trimean ===

{{See also| No-sevens subgroup temperaments #Superpine }}

Comma list: 81/80, 126/125, 1344/1331

Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}

: mapping generators: ~2, ~11/10

* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}

{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}

Badness (Sintel): 1.68

Comma list: 81/80, 126/125, 144/143, 364/363

Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}

* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}

{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}

Badness (Sintel): 1.46

Comma list: 81/80, 126/125, 144/143, 189/187, 221/220

Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}

* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}

{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}

Badness (Sintel): 1.28

== Flattone ==

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]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 525/512

{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}

[[Optimal tuning]]s:  

: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}

* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}

* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}

: [[projection map]]: [{{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 }}]

[[Tuning ranges]]:  

* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)

* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]

* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]

[[Algebraic generator]]: Squarto, the positive root of 8''x''² - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}

[[Badness]] (Sintel): 0.976

=== 11-limit ===

This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].

Comma list: 45/44, 81/80, 385/384

Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}

Optimal tuning:  

* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}

Tuning ranges:  

* 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]

{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}

Badness (Sintel): 1.12

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 81/80

Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}

* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}

* 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]

Badness (Sintel): 0.920

== 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 }}

 

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

: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}

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

: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}

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

[[Badness]] (Sintel): 2.43

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}}

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

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

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

=== Domineering ===

Comma list: 36/35, 45/44, 64/63

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

 

 

 

 

 

 

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

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

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

== Sharptone ==

Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.

However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).

The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.  

[[Comma list]]: 21/20, 28/27

{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}

* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}

: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}

: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}

{{Optimal ET sequence|legend=1| 5, 7d, 12d }}

[[Badness]] (Sintel): 0.629

=== Meanertone ===

Comma list: 21/20, 28/27, 33/32

Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}

* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}

{{Optimal ET sequence|legend=0| 5, 7d, 12de }}

 

Badness (Sintel): 0.832

 

== Supermean ==

Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].  

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 81/80, 672/625

 

{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}

: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}

: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}

 

{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}

 

[[Badness]] (Sintel): 3.40

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 56/55, 81/80, 132/125

 

Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}

 

Optimal tunings:  

* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}

Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}

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.

Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}

Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}

Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}