Diaschismic family: Difference between revisions

The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.

== Srutal (12&34, aka diaschismic) ==

* [[diamond monotone]] range: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 706.843]

* diamond monotone and tradeoff range: ~3/2 = [701.955, 706.843]

=== Seven limit extensions ===

== Srutal ==

* [[diamond monotone]] range: ~3/2 = [703.448, 705.882] (34\58 to 20\34)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 706.843]

* diamond monotone and tradeoff range: ~3/2 = [703.448, 705.882]

=== 11-limit ===

* [[diamond monotone]] range: ~3/2 = [704.348, 705.882] (27\46 to 20\34)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 706.843]

* diamond monotone and tradeoff range: ~3/2 = [704.348, 705.882]

=== 13-limit ===

* [[diamond monotone]] range: ~3/2 = [704.348, 705.882] (27\46 to 20\34)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 706.843]

* diamond monotone and tradeoff range: ~3/2 = [704.348, 705.882]

== Pajara ==

Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

* [[diamond monotone]] range: ~3/2 = [700.000, 720.000] (7\12 to 6\10)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 715.587]

* diamond monotone and tradeoff range: ~3/2 = [701.955, 715.587]

=== 11-limit ===

* [[diamond monotone]] range: ~3/2 = [700.000, 709.091] (7\12 to 13\22)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 715.587]

* diamond monotone and tradeoff range: ~3/2 = [701.955, 709.091]

==== 13-limit ====

* [[diamond monotone]] range: ~3/2 = [700.000, 709.091] (7\12 to 13\22)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 738.573]

* diamond monotone and tradeoff range: ~3/2 = [701.955, 709.091]

=== Pajarous ===

* [[diamond monotone]] range: ~3/2 = 709.091 (13\22)

* [[diamond tradeoff]] range: ~3/2 = [701.955, 715.803]

* diamond monotone and tradeoff range: ~3/2 = 709.091

==== 13-limit ====

==== Pajaro ====

=== Pajaric ===

==== 13-limit ====

=== Hemipaj ===

== Diaschismic ==

A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46&58. However described, diaschismic has a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.

=== 11-limit ===

=== 13-limit ===

=== 17-limit (Na"Naa') ===

== Keen ==

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&56 temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival|2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.

=== 11-limit ===

== Bidia ==

=== 11-limit ===

=== 13-limit ===

=== 17-limit ===

== Echidna ==

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&58 temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be add to {{val| 138 219 321 388 }}.

=== 11-limit ===

* 11-odd-limit

=== 13-limit ===

=== 17-limit ===

== Echidnic ==

=== 11-limit ===

=== 13-limit ===

=== 17-limit ===

=== Compositions ===

== Shrutar ==

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.

[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]

=== 11-limit ===

Mapping: [{{val| 2 1 9 -2 8 }}, {{val| 0 2 -4 7 -1 }}]

=== 13-limit ===

Mapping: [{{val| 2 1 9 -2 8 -10 }}, {{val| 0 2 -4 7 -1 16 }}]

Vals: {{Val list| 22, 24, 46, 68, 114 }}

Badness: 0.0281

=== 17-limit ===

Mapping: [{{val| 2 1 9 -2 8 -10 6 }}, {{val| 0 2 -4 7 -1 16 2 }}]

Vals: {{Val list| 22, 24, 46, 68, 114 }}

Badness: 0.0187

=== 19-limit ===

Mapping: [{{val| 2 1 9 -2 8 -10 6 -10 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]

Vals: {{Val list| 22, 24, 46, 68, 114, 182bef }}

Badness: 0.0175

== Sruti ==

{{Val list|legend=1| 24, 34d, 58, 150cd, 208cd, 266cd }}

[[Badness]]: 0.1174

=== 11-limit ===

Vals: {{Val list| 24, 34d, 58, 150cde, 208cde }}

Badness: 0.0415

=== 13-limit ===

Vals: {{Val list| 24, 34d, 58, 150cdef, 208cdef }}

Badness: 0.0238

== Anguirus ==

[[Badness]]: 0.0780

=== 11-limit ===

Badness: 0.0493

=== 13-limit ===

Badness: 0.0308

== Shru ==

[[POTE generator]]: ~64/63 = 50.135

{{Val list|legend=1| 22d, 24 }}

[[Badness]]: 0.1576

=== 11-limit ===

Vals: {{Val list| 22d, 24 }}

Badness: 0.0635

=== 13-limit ===

Badness: 0.0457