Meantone family: Difference between revisions
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{{See also| Archytas clan }} | {{See also| Archytas clan }} | ||
The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. 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]]. | 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. | 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. | ||
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== Supermean == | == 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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, 672/625 | [[Comma list]]: 81/80, 672/625 | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Main| Mohajira }} | {{Main| Mohajira }} | ||
Mohajira can be viewed as derived from mohaha which maps the interval half a [[chroma]] flat of | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Mohamaq == | == Mohamaq == | ||
Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as {{nowrap| 17c & 24 }}; its ploidacot is dicot, the same as mohajira. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span> | <span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span> | ||
Liese splits the | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{See also| No-sevens subgroup temperaments #Superpine }} | {{See also| No-sevens subgroup temperaments #Superpine }} | ||
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. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Lithium == | == Lithium == | ||
Lithium is named after the 3rd element for having a 3rd-octave period | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Main| Squares }} | {{Main| Squares }} | ||
Squares splits the | 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]]. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Jerome == | == Jerome == | ||
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the | Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Meantritone == | == Meantritone == | ||
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Injera == | == Injera == | ||
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 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. | 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. | ||
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name] | [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name] | ||
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{{Main| Teff }} | {{Main| Teff }} | ||
Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in | 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. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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== Pombe == | == Pombe == | ||
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. 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. | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Orphic == | == Orphic == | ||
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Cloudtone == | == Cloudtone == | ||
The cloudtone temperament | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||