Diaschismic: Difference between revisions

{{Infobox regtemp

| Title = Diaschismic

| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17

| Comma basis = [[2048/2025]] (5-limit); <br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)

| Edo join 1 = 46 | Edo join 2 = 58

| Mapping = 2; 1 -2 -8 -12 -15 1

| Generators = 3/2 | Generators tuning = 703.9 | Optimization method = CWE

| Odd limit 2 = 17-limit 21 | Mistuning 2 = ??? | Complexity 2 = 46

The canonical [[extension]] to the [[7-limit]] lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]], as well as the argent comma, [[5120/5103]]. This equates the [[64/63|septimal comma]] with the [[81/80|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals.  

A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp.  

Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup|2.3.5.17-]][[subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.  

An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.  

See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic|#Septimal diaschismic]] for technical data.  

Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 generator steps, which represent the generic comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.  

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.

== Chords and harmony ==

{{See also| Chords of diaschismic }}

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