Hemimean clan: Difference between revisions

The generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning. The name was given by [[Petr Pařízek]] in 2011.<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group (archive) | ''Suggested names for the unclasified temperaments'']</ref>

Emka may be described as the 37 & 50 temperament, which tempers out the hemimean comma and 84035/82944 (quinzo-ayo). It was named by [[Petr Pařízek]] in 2011.<ref name="petr's long post"/>

Tremka may be described as the 50 & 111 temperament. The name ''tremka'' was devised by [[Petr Pařízek]] in 2011, initially used for the [[no-sevens subgroup temperaments #Tremka|no-7's version]], especially in the 2.3.5.11.13 subgroup<ref name="petr's long post"/>, but extending to the full 13-limit or higher does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.

In the no-3's [[11-limit]], there is a natural extension for prime 11 by equating the sharply tuned [[25/16]] with [[11/7]], tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])³, tempering out [[1375/1372]].  

 

In the no-3's 19-limit extension mediantone, this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.

Comma list: 176/175, 1375/1372

{{Mapping|legend=2| 1 0 -3 -7 | 0 2 5 9 }}

Optimal tunings:  

* WE: ~2 = 1199.3680{{c}}, ~28/25 = 194.4533{{c}}

{{Optimal ET sequence|legend=0| 6, 19e, 25, 31, 37 }}

Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension of didacus if prime 3 is not included.

{{Mapping|legend=2| 1 0 -3 -7 13 | 0 2 5 9 -8 }}

Optimal tunings:  

* WE: ~2 = 1199.3069{{c}}, ~28/25 = 194.4820{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 194.5685{{c}}

{{Optimal ET sequence|legend=0| 6, 25, 31, 37 }}

Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on ''meantone'', as the context one is most likely to first discover this logic is when the tone also represents 9/8~10/9.

In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[meantone family #Undecimal meantone (huygens)|undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of 19/17~28/25 as ([[3/2]])²/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.

Comma list: 176/175, 221/220, 476/475, 640/637, 1375/1372

{{Mapping|legend=2| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}

Optimal tunings:  

* WE: ~2 = 1199.3213{{c}}, ~19/17 = 194.8469{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~19/17 = 194.9266{{c}}

Roulette is an alternative no-3's 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)² to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])³ is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.

Comma list: 170/169, 176/175, 476/475, 640/637, 1375/1372

{{Mapping|legend=2| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}

Optimal tunings:  

* WE: ~2 = 1199.7001{{c}}, ~19/17 = 194.2060{{c}}

{{Optimal ET sequence|legend=0| 6g, 31, 37, 68, 105 }}

== Rectified hebrew (2.5.7.13) ==

Rectified hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11{{clarify}}. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three. Notably, it is the no-3 no-11 restriction of [[Sycamore family #Septimal sycamore|sycamore]].

{{Mapping|legend=2| 1 2 2 3 | 0 6 15 13 }}

Optimal tunings:  

* WE: ~2 = 1199.8702{{c}}, ~26/25 = 64.6016{{c}}

{{Optimal ET sequence|legend=0| 18, 19, 37, 93, 130 }}

== Isra (2.9.5.7) ==

Isra (''iss-RAH'') results from taking every other generator of [[meantone family #Septimal meantone|septimal meantone]], or from didacus if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to [[luna family #Luna|luna]], a restriction of [[#Hemithirds|hemithirds]], which is in turn an extension of didacus.

Subgroup: 2.9.5.7

Comma list: 81/80, 126/125

{{Mapping|legend=2| 1 0 -4 -13 | 0 1 2 5 }}

Optimal tunings:  

* WE: ~2 = 1201.2358{{c}}, ~9/8 = 193.1885{{c}}

{{Optimal ET sequence|legend=0| 6, 19, 25, 31, 56b′, 87b′ }}

Tutone is every other step of undecimal meantone, or undecimal didacus with the generator interpreted as 9/8.

Subgroup: 2.9.5.7.11

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

{{Mapping|legend=2| 1 0 -4 -13 -25 | 0 1 2 5 9 }}

{{Mapping|legend=4| 1 0 -4 -13 -25 | 0 1/2 2 5 9 }}

Optimal tunings:  

* WE: ~2 = 1200.7636{{c}}, ~9/8 = 194.0608{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 194.0629{{c}}

{{Optimal ET sequence|legend=0| 6, 19e, 25, 31, 68b′, 99b′ }}

Badness (Sintel): 0.281

Subgroup: 2.9.5.7.11

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

{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}

{{Mapping|legend=4| 1 0 -4 -13 -6 | 0 1/2 2 5 3 }}

Optimal tunings:  

* WE: ~2 = 1199.6946{{c}}, ~9/8 = 192.4513{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 192.4166{{c}}

{{Optimal ET sequence|legend=0| 6, 19, 25e }}

Badness (Sintel): 0.354

Subgroup: 2.9.5.7.13

Comma list: 65/64, 81/80, 91/90

{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}

{{Mapping|legend=4| 1 0 -4 -13 0 10 | 0 1/2 2 5 0 -2 }}

Optimal tunings:  

* WE: ~2 = 1203.3456{{c}}, ~9/8 = 191.5914{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 191.4621{{c}}

{{Optimal ET sequence|legend=0| 6, 7, 13, 19, 25f, 44df }}

Badness (Sintel): 0.325