Diasem: Difference between revisions

Line 190:

== Important properties ==

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

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

* > 4\7 if L > m + s

* = 4\7 if L = m + s

Line 242:

## LMLLSLMLS (LH Dark Dorian) > LMLSLMLLS (LH Dark Aeolian) > MLLSLMLSL (LH Phrygian) > MLSLMLLSL (LH Locrian)

This provides a clear motivation for the diatonic-based mode names.

=== Interpretation as negative-s blackdye ===

Consider right-hand diasem in a tuning with positive step sizes. There exists a way of superimposing a left-hand diasem mode on the right hand diasem so that the right-hand diasem and the left-hand diasem overlap in 8 notes, yielding a scale of 10 notes, possibly after changing the mode of right-hand diasem. For example, superimposing LMLSLMLSL (RH Aeolian) and LMLSLMLLS (LH Dark Aeolian) gives LMLSLMLS(L-S)S (in fifth-based notation on C: C D Ebv Fv F G Abv Bbv Bb Cv C, where v denotes lowering by s). Note that the union is achiral; we have removed the chirality of diasem by taking the "best" possible union of LH and RH diasem. This new scale has two chains of perfect fifths each spanning 5 notes:

Chain 1: '''Bb''' (L-S)SLMLS '''F''' LMLS(L-S)S '''C''' LMLSL '''G''' MLS(L-S)SL '''D'''

Chain 2: '''Ab''' LS(L-S)SLM '''Ebv''' LSLML '''Bbv''' S(L-S)SLML '''Fv''' SLMLS(L-S) '''Cv'''

Notice that we have two generator chains of equal length. To give this scale a generator-offset structure we can treat the small 2-step (L+S) as the offset of the 10-note scale, and treat L+S as one scale step. This gives the following ordering: C Cv D Ebv F Fv G Abv Bb Bbv C, or in step sizes, -s L+s M L+s -s L+s M L+s -s L+s. This is formally a [[blackdye]] (sL'ML's'L'ML's'L') pattern, albeit with a negative step size s' = -s.

== In JI and similar tunings ==

@@ Line 190: / Line 190: @@
 == Important properties ==
-Any diasem scale has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:
+Any diasem scale with positive step sizes has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:
 * > 4\7 if L > m + s
 * = 4\7 if L = m + s
@@ Line 242: / Line 242: @@
 ## LMLLSLMLS (LH Dark Dorian) > LMLSLMLLS (LH Dark Aeolian) > MLLSLMLSL (LH Phrygian) > MLSLMLLSL (LH Locrian)
 This provides a clear motivation for the diatonic-based mode names.
+=== Interpretation as negative-s blackdye ===
+Consider right-hand diasem in a tuning with positive step sizes. There exists a way of superimposing a left-hand diasem mode on the right hand diasem so that the right-hand diasem and the left-hand diasem overlap in 8 notes, yielding a scale of 10 notes, possibly after changing the mode of right-hand diasem. For example, superimposing LMLSLMLSL (RH Aeolian) and LMLSLMLLS (LH Dark Aeolian) gives LMLSLMLS(L-S)S (in fifth-based notation on C: C D Ebv Fv F G Abv Bbv Bb Cv C, where v denotes lowering by s). Note that the union is achiral; we have removed the chirality of diasem by taking the "best" possible union of LH and RH diasem. This new scale has two chains of perfect fifths each spanning 5 notes:
+Chain 1: '''Bb''' (L-S)SLMLS '''F''' LMLS(L-S)S '''C''' LMLSL '''G''' MLS(L-S)SL '''D'''
+Chain 2: '''Ab''' LS(L-S)SLM '''Ebv''' LSLML '''Bbv''' S(L-S)SLML '''Fv''' SLMLS(L-S) '''Cv'''
+Notice that we have two generator chains of equal length. To give this scale a generator-offset structure we can treat the small 2-step (L+S) as the offset of the 10-note scale, and treat L+S as one scale step. This gives the following ordering: C Cv D Ebv F Fv G Abv Bb Bbv C, or in step sizes, -s L+s M L+s -s L+s M L+s -s L+s. This is formally a [[blackdye]] (sL'ML's'L'ML's'L') pattern, albeit with a negative step size s' = -s.
 == In JI and similar tunings ==