Meantone family: Difference between revisions

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]].

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

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.

Liese splits the [[3/1|perfect twelfth]] into three generators of ~10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. It may be described as {{nowrap| 17c & 19 }}; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55.  

The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.

Lithium is named after the 3rd element for having a 3rd-octave period (and also for lithium's molar mass of 6.9 g/mol since 69edo supports it). Its ploidacot is triploid monocot. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.

Squares splits the [[6/1|6th harmonic]] into four subminor sixths of [[11/7]]~[[14/9]] (or splits a [[8/3|perfect eleventh]] into four supermajor thirds of [[9/7]]~[[14/11]]), and uses it for a generator. It may be described as {{nowrap| 14c & 17c }}; its ploidacot is beta-tetracot. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8-, 11-, and 14-note mos scales available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].

Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^1/20, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.

The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament.

Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as {{nowrap| 12 & 26 }}; its ploidacot is diploid monocot. It tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.

Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.  

Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}

: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}

: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}

{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}

[[Badness]] (Sintel): 0.324

 

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}

: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}

: error map: {{val| 0.000 -8.047 -10.680 +12.133 }}

[[Badness]] (Sintel): 0.326

Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}

Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}

* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}

Badness (Sintel): 0.561

 

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}

: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}

: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}

[[Badness]] (Sintel): 0.451

Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}

Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}

* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}

Badness (Sintel): 1.40