Diasem: Difference between revisions

'''Diasem''' is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with step pattern 5L 2M 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed diasem'' LMLSLMLSL and ''left-handed diasem'' LSLMLSLML; these are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales. The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset a subminor third or a supermajor second. [[21edo]] is the smallest edo to support a non-degenerate diasem.

Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[28/27]], and S represents [[64/63]].

"Diasem" is a name given by [[User:ks26|ks26]] (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "[[semaphore]]") since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[semiquartal]] (5L 4s); it is also a pun based on the [[diesis]], which appears as the small step in the scale in the [[31edo]] and [[36edo]] tunings.

|+ Comparison with semiquartal and diatonic in 62edo

| Semiquartal || 5L4s || 10\62, 3\62 || ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤

| Diasem || 5L2m2s || 10\62, 4\62, 2\62 || ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤

| Diatonic || 5L2s || 10\62, 6\62 || ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤

!rowspan=3|[[TAMNAMS|1-step]]

!rowspan=3|[[TAMNAMS|2-step]]

!rowspan=3|[[TAMNAMS|3-step]]

!rowspan=3|[[TAMNAMS|4-step]]

!rowspan=3|[[TAMNAMS|5-step]]

!rowspan=3|[[TAMNAMS|6-step]]

!rowspan=3|[[TAMNAMS|7-step]]

!rowspan=3|[[TAMNAMS|8-step]]

== Important properties ==

Any diasem scale has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:

Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. To minimize bias (e.g. towards certain tunings), this article names the modes after the [[semiquartal]] mode that the diasem mode is based on (the semiquartal modes are given in [[UDP notation]]).

| | LSLMLSLML <br/>LH Diasem 4|4<br/>LH Diasem Mixo

| | LMLSLMLSL <br/>RH Diasem 4|4<br/>RH Diasem Aeolian

| | SLMLSLMLL <br/>LH Diasem 0|8<br/>LH Diasem Brightened Aeolian

| | MLSLMLSLL <br/>RH Diasem 0|8<br/>RH Diasem Locrian

| | LMLSLMLLS <br/>LH Diasem 5|3<br/>LH Diasem Aeolian

| | LSLMLSLLM <br/>RH Diasem 5|3<br/>RH Diasem Ionian

| | MLSLMLLSL <br/>LH Diasem 1|7<br/>LH Diasem Locrian

| | SLMLSLLML <br/>RH Diasem 1|7<br/>RH Diasem Bright Dorian

| | LSLMLLSLM <br/>LH Diasem 6|2<br/>LH Diasem Ionian

| | LMLSLLMLS <br/>RH Diasem 6|2<br/>RH Diasem Dark Dorian

| | SLMLLSLML <br/>LH Diasem 2|6<br/>LH Diasem Bright Dorian

| | MLSLLMLSL <br/>RH Diasem 2|6<br/>RH Diasem Phrygian

| | LMLLSLMLS <br/>LH Diasem 7|1<br/>LH Diasem Dark Dorian

| | LSLLMLSLM <br/>RH Diasem 7|1<br/>RH Diasem Lydian

| | MLLSLMLSL <br/>LH Diasem 3|5<br/>LH Diasem Phrygian

| | SLLMLSLML <br/>RH Diasem 3|5<br/>RH Diasem Mixo

| | LLSLMLSLM <br/>LH Diasem 8|0<br/>LH Diasem Lydian

| | LLMLSLMLS <br/>RH Diasem 8|0<br/>RH Diasem Darkened Mixo

=== Dechiralized modes ===

By taking the union of the two chiralities for each diatonic mode, we obtain the achiral scale LSCSLMLSLM, where c = L &minus; s. This is a stretched version of the [[blackdye]] scale with the same step ratio (with the L in the "SLS" of blackdye made smaller). There are 8 dechiralized modes that result from changing an occurrence of SL and LS in it to SCS (on the same root as the original chiral modes).

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! style="text-align:center;" |Dechiralized modes

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The last two modes are exceptional because they begin and end in a C and thus do not result from a union-taking that preserves the tonic.

Like [[superpyth]], diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths. On the other hand, the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out [[81/80]] are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps.

! colspan="8" |Degrees of the mode LMLSLMLSL

| 23/16 and 39/32

=== Tuning examples ===

An example in the RH Diasem 7|1 mode LSLLMLSLM. ([[:File:Diasem Lydian Example Score.pdf|score]])

[[File:Diasem Lydian Example 41edo.mp3]] [[41edo]], L:M:S = 7:2:1<!--

An example in a Locrian-like diasem mode MLSLMLLSL. ([[:File:Diasem Locrian Example.pdf|score]])

[[File:Diasem Locrian Example 16edo.mp3]] [[16edo]], L:M:S = 2:2:1 (degenerate; this is [[basic]] [[[[superdiatonic]]]])

== Extensions ==

The diasem scale extends to a 14-note AG scale: LmLsLmLsL and LsLmLsLmL both extend to the scale mcmcmsmcmcmsmc (c = L &minus; m), with two 7-note mosses generated by the diasem's fifths separated by m. This scale is not chiral. This scale extends diasem like how [[blackdye]] is a 10-note non-chiral AG extension of the [[Zarlino]] scale's AG pattern 3L 2m 2s, LmsLmLs. The 14-note extension is one of:

Another extension is scscsmscscsmsc, with c = L - s (5L 2m 7s if c > m). 31edo diasem yields 5L 2m 7s with step ratio 4:2:1.  

Both these tunings, 5L 7m 2s and 5L 2m 7s, have been named '''crossdye''' ("crossed eyes" referring to the two copies of 5L 2s diatonic + "blackdye", courtesy of cellularAutomaton). 5L 7m 2s could be called ''chromatic crossdye'' and 5L 2m 7s could be called ''dietic crossdye''.

*[https://sevish.com/scaleworkshop/?name=Diasem%2026edo&data=46.15384615384615%0A92.3076923076923%0A138.46153846153845%0A184.6153846153846%0A230.76923076923077%0A276.9230769230769%0A323.0769230769231%0A369.2307692307692%0A415.38461538461536%0A461.53846153846155%0A507.6923076923077%0A553.8461538461538%0A600.%0A646.1538461538462%0A692.3076923076923%0A738.4615384615385%0A784.6153846153846%0A830.7692307692307%0A876.9230769230769%0A923.0769230769231%0A969.2307692307692%0A1015.3846153846154%0A1061.5384615384614%0A1107.6923076923076%0A1153.8461538461538%0A1200.&freq=440&midi=69&vert=9&horiz=1&colors=white%20black%20grey%20black%20white%20black%20white%20black%20grey%20black%20white%20white%20black%20grey%20black%20white%20black%20white%20black%20grey%20black%20white%20white%20black%20grey%20black&waveform=square&ampenv=organ Play 26edo diasem] - Sevish Scale Workshop

[[Category:Maximum variety 3 scales]]