Generator sequence: Difference between revisions

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To exclude the case when the generator is a 1-step or a (len(scale) − 1)-step, the modifier ''non-step''{{idiosyncratic}} can be used.

Given a choice of equave ''E'' and an AGS ''S'' = AGS(''x''1, ..., ''x''''r''), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence AGS(w1, ..., w''r'') where each w''i'' is a sequence of ''k'' = ''k''(''i'') intervals, ''y''''i''1, ..., ''y''''ik'', where ''y''''i''1 + ... + ''y''''ik'' ≡ ''x''''i'' modulo ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, AGS(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 AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of AGS(3/2), corresponding to a [[pergen]].

Given a choice of equave ''E'' and an AGS ''S'' = AGS(''x''1, ..., ''x''''r''), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence AGS(w1, ..., w''r'') where each w''i'' is a sequence of ''k'' = ''k''(''i'') intervals, ''y''''i''1, ..., ''y''''ik'', where ''y''''i''1 + ... + ''y''''ik'' ≡ ''x''''i'' modulo ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, AGS(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 AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of AGS(3/2), corresponding to a unique [[pergen]].

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 To exclude the case when the generator is a 1-step or a (len(scale) &minus; 1)-step, the modifier ''non-step''{{idiosyncratic}} can be used.
-Given a choice of equave ''E'' and an AGS ''S'' = AGS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence AGS(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, AGS(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 AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of AGS(3/2), corresponding to a [[pergen]].
+Given a choice of equave ''E'' and an AGS ''S'' = AGS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence AGS(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, AGS(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 AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of AGS(3/2), corresponding to a unique [[pergen]].
 <!-- todo: Non-WF GSes-->