Modal UDP notation: Difference between revisions

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Given a monotone periodic scale ''S'', suppose it is also a [[MOS]] or DE scale. Let the generator {{nowrap|''S''(''m'') {{=}} ''g''}} be such that {{nowrap|''g'' ≥ ''S''(''i'' + ''m'') − ''S''(''i'')}} for all ''i''. If ''Q'' is the period of ''S'', let ''u'' be the largest integer such that {{nowrap|0 ≤ ''u'' < ''Q''}} and {{nowrap|''S''(''m''{{dot}}''u'') {{=}} ''g''{{dot}}''u''}}, and ''d'' the largest integer such that {{nowrap|0 ≤ ''d'' < ''Q''}} and {{nowrap|''S''(−''m''{{dot}}''d'') {{=}} −''g''{{dot}}''d''}}. If {{nowrap|''S''(''P''{{dot}}''Q'') {{=}} octave}}, so that ''P'' is the number of periods to an octave, let {{nowrap|''U'' {{=}} ''P''{{dot}}''u''}} and {{nowrap|''D'' {{=}} ''P''{{dot}}''d''}}. Then the UDP notation for the given mode is is ''U''|''D''(''P''). If {nowrap|''P'' {{=}} 1}} we may omit it and just write ''U''|''D''.

For example, consider the quasiperiodic function {{nowrap|Ionian(i) {{=}} V[((''i'' + 3) mod 7) + 1] + 31 &lceil;{{~~frac~~|''n'' + 4|7}} − 49&rceil;}}, where {{nowrap|''V'' {{=}} [5, 10, 15, 18, 23, 28, 31]}}. This has period 7, and {{nowrap|Ionian(7) {{=}} 31}}, where the tuning is [[31edo]] so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41… corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…, and going down from 0, it gives 0, −3, −8, −13… corresponding to 0, −1, −2, −3…. This gives the Ionian, or major, mode of the diatonic scale. Then {{nowrap|Ionian(4) {{=}} 18}}, the fifth, and {{nowrap|18 ≥ Ionian(''i'' + 4) − Ionian(''i'')}} for all ''i''. We have {{nowrap|Ionian(4) {{=}} 18|Ionian(8) {{=}} 36|Ionian(12) {{=}} 54|Ionian(16) {{=}} 72}}, and {{nowrap|Ionian(20) {{=}} 90}}. However, {{nowrap|Ionian(4·6) {{=}} Ionian(24) {{=}} 106}}, which is less than {{nowrap|6{{dot}}18 {{=}} 108}}. Hence the largest value for which {{nowrap|Ionian(4{{dot}}''u'') {{=}} 18{{dot}}''u''}} is {{nowrap|''u'' {{=}} 5}}. Similarly, {{nowrap|Ionian(−4) {{=}} −18}}, but {{nowrap|Ionian(−8) {{=}} −34}}, not −36, and so {{nowrap|''d'' {{=}} 1}}. Since {{nowrap|Ionian(7) {{=}} 31}}, which is the octave, {{nowrap|''P'' {{=}} 1}}, so {{nowrap|''U'' {{=}} ''u'' {{=}} 5}}, {{nowrap|''D'' {{=}} ''d'' {{=}} 1}}, and the UDP notation for Ionian is 5|1(1), or simply 5|1.

For example, consider the quasiperiodic function {{nowrap|Ionian(i) {{=}} V[((''i'' + 3) mod 7) + 1] + 31 &lceil;{{sfrac|''n'' + 4|7}} − 49&rceil;}}, where {{nowrap|''V'' {{=}} [5, 10, 15, 18, 23, 28, 31]}}. This has period 7, and {{nowrap|Ionian(7) {{=}} 31}}, where the tuning is [[31edo]] so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41… corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…, and going down from 0, it gives 0, −3, −8, −13… corresponding to 0, −1, −2, −3…. This gives the Ionian, or major, mode of the diatonic scale. Then {{nowrap|Ionian(4) {{=}} 18}}, the fifth, and {{nowrap|18 ≥ Ionian(''i'' + 4) − Ionian(''i'')}} for all ''i''. We have {{nowrap|Ionian(4) {{=}} 18|Ionian(8) {{=}} 36|Ionian(12) {{=}} 54|Ionian(16) {{=}} 72}}, and {{nowrap|Ionian(20) {{=}} 90}}. However, {{nowrap|Ionian(4·6) {{=}} Ionian(24) {{=}} 106}}, which is less than {{nowrap|6{{dot}}18 {{=}} 108}}. Hence the largest value for which {{nowrap|Ionian(4{{dot}}''u'') {{=}} 18{{dot}}''u''}} is {{nowrap|''u'' {{=}} 5}}. Similarly, {{nowrap|Ionian(−4) {{=}} −18}}, but {{nowrap|Ionian(−8) {{=}} −34}}, not −36, and so {{nowrap|''d'' {{=}} 1}}. Since {{nowrap|Ionian(7) {{=}} 31}}, which is the octave, {{nowrap|''P'' {{=}} 1}}, so {{nowrap|''U'' {{=}} ''u'' {{=}} 5}}, {{nowrap|''D'' {{=}} ''d'' {{=}} 1}}, and the UDP notation for Ionian is 5|1(1), or simply 5|1.

== Rationale ==

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 Given a monotone periodic scale ''S'', suppose it is also a [[MOS]] or DE scale. Let the generator {{nowrap|''S''(''m'') {{=}} ''g''}} be such that {{nowrap|''g'' &ge; ''S''(''i'' + ''m'') − ''S''(''i'')}} for all ''i''. If ''Q'' is the period of ''S'', let ''u'' be the largest integer such that {{nowrap|0 &le; ''u'' &lt; ''Q''}} and {{nowrap|''S''(''m''{{dot}}''u'') {{=}} ''g''{{dot}}''u''}}, and ''d'' the largest integer such that {{nowrap|0 &le; ''d'' &lt; ''Q''}} and {{nowrap|''S''(−''m''{{dot}}''d'') {{=}} −''g''{{dot}}''d''}}. If {{nowrap|''S''(''P''{{dot}}''Q'') {{=}} octave}}, so that ''P'' is the number of periods to an octave, let {{nowrap|''U'' {{=}} ''P''{{dot}}''u''}} and {{nowrap|''D'' {{=}} ''P''{{dot}}''d''}}. Then the UDP notation for the given mode is is ''U''|''D''(''P''). If {nowrap|''P'' {{=}} 1}} we may omit it and just write ''U''|''D''.
-For example, consider the quasiperiodic function {{nowrap|Ionian(i) {{=}} V[((''i'' + 3) mod 7) + 1] + 31 &lceil;{{frac|''n'' + 4|7}} − 49&rceil;}}, where {{nowrap|''V'' {{=}} [5, 10, 15, 18, 23, 28, 31]}}. This has period 7, and {{nowrap|Ionian(7) {{=}} 31}}, where the tuning is [[31edo]] so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41… corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…, and going down from 0, it gives 0, −3, −8, −13… corresponding to 0, −1, −2, −3…. This gives the Ionian, or major, mode of the diatonic scale. Then {{nowrap|Ionian(4) {{=}} 18}}, the fifth, and {{nowrap|18 &ge; Ionian(''i'' + 4) − Ionian(''i'')}} for all ''i''. We have {{nowrap|Ionian(4) {{=}} 18|Ionian(8) {{=}} 36|Ionian(12) {{=}} 54|Ionian(16) {{=}} 72}}, and {{nowrap|Ionian(20) {{=}} 90}}. However, {{nowrap|Ionian(4·6) {{=}} Ionian(24) {{=}} 106}}, which is less than {{nowrap|6{{dot}}18 {{=}} 108}}. Hence the largest value for which {{nowrap|Ionian(4{{dot}}''u'') {{=}} 18{{dot}}''u''}} is {{nowrap|''u'' {{=}} 5}}. Similarly, {{nowrap|Ionian(−4) {{=}} −18}}, but {{nowrap|Ionian(−8) {{=}} −34}}, not −36, and so {{nowrap|''d'' {{=}} 1}}. Since {{nowrap|Ionian(7) {{=}} 31}}, which is the octave, {{nowrap|''P'' {{=}} 1}}, so {{nowrap|''U'' {{=}} ''u'' {{=}} 5}}, {{nowrap|''D'' {{=}} ''d'' {{=}} 1}}, and the UDP notation for Ionian is 5|1(1), or simply 5|1.
+For example, consider the quasiperiodic function {{nowrap|Ionian(i) {{=}} V[((''i'' + 3) mod 7) + 1] + 31 &lceil;{{sfrac|''n'' + 4|7}} − 49&rceil;}}, where {{nowrap|''V'' {{=}} [5, 10, 15, 18, 23, 28, 31]}}. This has period 7, and {{nowrap|Ionian(7) {{=}} 31}}, where the tuning is [[31edo]] so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41… corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…, and going down from 0, it gives 0, −3, −8, −13… corresponding to 0, −1, −2, −3…. This gives the Ionian, or major, mode of the diatonic scale. Then {{nowrap|Ionian(4) {{=}} 18}}, the fifth, and {{nowrap|18 &ge; Ionian(''i'' + 4) − Ionian(''i'')}} for all ''i''. We have {{nowrap|Ionian(4) {{=}} 18|Ionian(8) {{=}} 36|Ionian(12) {{=}} 54|Ionian(16) {{=}} 72}}, and {{nowrap|Ionian(20) {{=}} 90}}. However, {{nowrap|Ionian(4·6) {{=}} Ionian(24) {{=}} 106}}, which is less than {{nowrap|6{{dot}}18 {{=}} 108}}. Hence the largest value for which {{nowrap|Ionian(4{{dot}}''u'') {{=}} 18{{dot}}''u''}} is {{nowrap|''u'' {{=}} 5}}. Similarly, {{nowrap|Ionian(−4) {{=}} −18}}, but {{nowrap|Ionian(−8) {{=}} −34}}, not −36, and so {{nowrap|''d'' {{=}} 1}}. Since {{nowrap|Ionian(7) {{=}} 31}}, which is the octave, {{nowrap|''P'' {{=}} 1}}, so {{nowrap|''U'' {{=}} ''u'' {{=}} 5}}, {{nowrap|''D'' {{=}} ''d'' {{=}} 1}}, and the UDP notation for Ionian is 5|1(1), or simply 5|1.
 == Rationale ==