Generator sequence: Difference between revisions
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Given a choice of equave ''E'' and an GS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence GS(w<sub>1</sub>, ..., w<sub>''r''</sub>) where each w<sub>''i''</sub> is a sequence of ''k'' = ''k''(''i'') intervals, ''y''<sub>''i''1</sub>, ..., ''y''<sub>''ik''</sub>, where ''y''<sub>''i''1</sub> + ... + ''y''<sub>''ik''</sub> ≡ ''x''<sub>''i''</sub> modulo ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6) is a uniform splitting of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any 2/1-equivalent WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of GS(3/2), corresponding to a unique [[pergen]] with a 3/2 period. | Given a choice of equave ''E'' and an GS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence GS(w<sub>1</sub>, ..., w<sub>''r''</sub>) where each w<sub>''i''</sub> is a sequence of ''k'' = ''k''(''i'') intervals, ''y''<sub>''i''1</sub>, ..., ''y''<sub>''ik''</sub>, where ''y''<sub>''i''1</sub> + ... + ''y''<sub>''ik''</sub> ≡ ''x''<sub>''i''</sub> modulo ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6) is a uniform splitting of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any 2/1-equivalent WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of GS(3/2), corresponding to a unique [[pergen]] with a 3/2 period. | ||
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== Generator sequence properties == | |||
A generator sequence can be analyzed in terms of its length and variety. | |||
=== Length === | |||
The ''length'' of a generator sequence ''s'' is the length at which the GS repeats; it is the smallest ''n'' > 0 such that ''s''[''k'' + ''n''] = ''s''[''k'']. It is known that a length-2 WFGS gives rise to regular SV3 scales; see [[Ternary scale theorems]]. If a generator sequence is shorter than ''n'' − 1 then it is called ''good''{{idiosyncratic}}. | |||
=== Generator variety === | |||
If a scale has a [[generator sequence]] or is a union of multiple offset copies of the same generator sequence, then the ''generator variety'' is the number of generators in that sequence, not including the closing interval. | |||
There is no known simple relationship between a scale's [[step variety]] and its generator variety. MOS scales have step variety 2 and generator variety 1, and [[MOS substitution]] scales (including all regular SV3 scales) have step variety 3 and generator variety 2. However, not all scales with generator variety ''n'' have step variety at most ''n'' + 1: Consider a 13-note scale with a well-formed generator sequence GS('''x''', '''y''', '''y''', '''x''', '''y''') and suppose one scale step is reached via 4 generators. Then the word of stacked generators is '''xyyxyxyyxyxyz''' including the final closing generator '''z''', and there are 4 step sizes in terms of intervals in the generator interval class (10-steps because (4 generators) * (10 steps) = 40 steps ≡ 1 mod 13): | |||
# 2'''x''' + 2'''y''', | |||
# '''x''' + 3'''y''', | |||
# 2'''x''' + '''y''' + '''z''', | |||
# '''x''' + 2'''y''' + '''z'''. | |||
More trivially, a MOS generator when not stacked so that the resulting scale is a MOS size (equal to the denominator of a semiconvergent to the generator) will not have variety 2. | |||
== JI scales obtained from guided generator sequences == | == JI scales obtained from guided generator sequences == | ||