Modal UDP notation - Revision history

ArrowHead294: /* Mathematical definition */

2025-03-12T13:15:03Z

Mathematical definition

← Older revision		Revision as of 13:15, 12 March 2025
Line 107:		Line 107:
	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''.		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 ==		== Rationale ==

ArrowHead294: /* Things to watch out for */

2025-03-12T13:14:25Z

Things to watch out for

← Older revision		Revision as of 13:14, 12 March 2025
Line 149:		Line 149:
	Requiring the generator to always be bright causes certain issues:		Requiring the generator to always be bright causes certain issues:
	* The bright and dark generators may "flip" based on the size of the generator and how many notes are in the corresponding MOS scale.		* The bright and dark generators may "flip" based on the size of the generator and how many notes are in the corresponding MOS scale.
	** The major pentatonic scale is a subset of the major scale. The genchain is simply shortened on either end. Thus one would expect the major pentatonic scale contained within Meantone[7] 5\|1 to be Meantone[5] 4\|0. But instead it's Meantone[5] 0\|4. This is because changing the size of the MOS scale often changes the generator to its octave inverse. Meantone[5]'s bright generator is the 4th not the 5th.		** The major pentatonic scale is a subset of the major scale. The genchain is simply shortened on either end. Thus one would expect the major pentatonic scale contained within Meantone[7] 5\|1 to be Meantone[5] 4\|0. But instead it's Meantone[5] 0\|4. This is because changing the size of the MOS scale often changes the generator to its octave inverse. Meantone[5]'s bright generator is the 4th not the 5th.
	** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700{{c}} for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700{{c}}), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700{{c}}), meaning that the bright generator is the fifth. As a result, Dominant[12] 7\|4 is ambiguous.		** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700{{c}} for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if the generator is flatter than 700{{cent}}), in which case the bright generator is the fourth, or [[5L 7s]] (if the generator is sharper than 700{{c}}), meaning that the bright generator is the fifth. As a result, "Dominant[12] 7\|4" is ambiguous.
	* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5\|2 is ambiguous.		* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5\|2 is ambiguous.

	Not all notations have these issues (see below).		Not all notations have these issues (see below).

ArrowHead294: /* Examples */

2025-03-12T13:13:28Z

Examples

← Older revision		Revision as of 13:13, 12 March 2025
Line 119:		Line 119:

	== Examples ==		== Examples ==
	For example, the proper generator for meantone[7] is the perfect fifth, because it's larger than the other specific interval it shares a class with, the diminished fifth. Consequentially, meantone[7]'s Ionian mode is 5\|1(1), which is 5\|1 for short, because it contains five chroma-positive generators up from the root and one down, as in the diagram F-[C]-G-D-A-E-B for C Ionian. This also means it has five "sharper" scale ~~degrees - the~~ second, third, fifth, sixth, and ~~seventh - and~~ one "flatter" scale ~~degree - the~~ fourth. If we want to sharpen the fourth to turn it into an augmented fourth, we arrive at 6\|0 or [C]-G-D-A-E-B-F#. Conversely, Aeolian mode, with only two sharp scale ~~degrees - the~~ second and ~~fifth - is~~ 2\|4. We can add accidentals as well, so that meantone's harmonic minor is 2\|4 #7.		For example, the proper generator for meantone[7] is the perfect fifth, because it's larger than the other specific interval it shares a class with, the diminished fifth. Consequentially, meantone[7]'s Ionian mode is 5\|1(1), which is 5\|1 for short, because it contains five chroma-positive generators up from the root and one down, as in the diagram {{nowrap\|{{dash\|F, [C], G, D, A, E, B\|s=hair}}}} for C Ionian. This also means it has five "sharper" scale degrees—the second, third, fifth, sixth, and seventh—and one "flatter" scale degree—the fourth. If we want to sharpen the fourth to turn it into an augmented fourth, we arrive at 6\|0 or {{nowrap\|{{dash\|[C], G, D, A, E, B, F♯\|s=hair}}}}. Conversely, Aeolian mode, with only two sharp scale degrees—the second and fifth—is 2\|4. We can add accidentals as well, so that meantone's harmonic minor is 2\|4 ♯7.

	The chroma-positive generator for porcupine[7] is the larger 7th, which is about ~11/6; as a consequence, porcupine[7]'s Lssssss mode is 6\|0, and sssLsss is 3\|3. Likewise, mavila[7]'s ssLsssL anti-Ionian is 1\|5, and Mavila[9]'s LLsLLLsLL "Olympian" mode is 4\|4.		The chroma-positive generator for porcupine[7] is the larger 7th, which is about ~11/6; as a consequence, porcupine[7]'s Lssssss mode is 6\|0, and sssLsss is 3\|3. Likewise, mavila[7]'s ssLsssL anti-Ionian is 1\|5, and Mavila[9]'s LLsLLLsLL "Olympian" mode is 4\|4.
Line 126:		Line 126:

	'''MOS scales'''		'''MOS scales'''

	* Meantone[7] Ionian, LLsLLLs: 5\|1		* Meantone[7] Ionian, LLsLLLs: 5\|1
	* Meantone[7] Aeolian, LsLLsLL: 2\|4		* Meantone[7] Aeolian, LsLLsLL: 2\|4
Line 143:		Line 142:

	'''MODMOS scales'''		'''MODMOS scales'''
			* The ascending melodic minor scale is 5\|1(1) ♭3, abbreviated 5\|1 ♭3 for short, but could also be 3\|3(1) ♯7, abbreviated 3\|3 ♯7 for short.
	* The ascending melodic minor scale is 5\|1(1) b3, abbreviated 5\|1 b3 for short, but could also be 3\|3(1) #7, abbreviated 3\|3 #7 for short.		* Paul Erlich's standard pentachordal major for Pajara[10] is 4\|4(2) ♯8, or alternatively 6\|2(2) ♭3.
	* Paul Erlich's standard pentachordal major for Pajara[10] is 4\|4(2) #8, or alternatively 6\|2(2) b3.		* Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6, is 6\|0 ♭7.
	* Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6, is 6\|0 b7.

	== Things to watch out for ==		== Things to watch out for ==

ArrowHead294: /* Mathematical definition */

2025-03-12T13:10:51Z

Mathematical definition

← Older revision		Revision as of 13:10, 12 March 2025
Line 105:		Line 105:
	A [[periodic scale]] ''S'' associates an interval ''S''(''i'') to every integer ''i'', such that there is a period (strictly, a quasiperiod) {{nowrap\|Q > 0}} and an interval of repetition ''R'' such that {{nowrap\|''S''(''i'' + ''Q'') {{=}} ''S''(''i'') + ''R''}}. ''Q'' is chosen so as to be minimal; there is no smaller period. ''S'' is monotone if {{nowrap\|''i'' < ''j''}} implies that {{nowrap\|''S''(''i'') < ''S''(''j'')}}.		A [[periodic scale]] ''S'' associates an interval ''S''(''i'') to every integer ''i'', such that there is a period (strictly, a quasiperiod) {{nowrap\|Q > 0}} and an interval of repetition ''R'' such that {{nowrap\|''S''(''i'' + ''Q'') {{=}} ''S''(''i'') + ''R''}}. ''Q'' is chosen so as to be minimal; there is no smaller period. ''S'' is monotone if {{nowrap\|''i'' < ''j''}} implies that {{nowrap\|''S''(''i'') < ''S''(''j'')}}.

	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 ~~<math>~~U\|D(P)~~</math>~~. If ~~<math>~~P = 1~~</math>~~ we may omit it and just write ~~<math>~~U\|D~~</math>~~.		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;{{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.

ArrowHead294: /* Mathematical definition */

2025-03-12T13:10:26Z

Mathematical definition

← Older revision		Revision as of 13:10, 12 March 2025
Line 107:		Line 107:
	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 <math>U\|D(P)</math>. If <math>P = 1</math> we may omit it and just write <math>U\|D</math>.		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 <math>U\|D(P)</math>. If <math>P = 1</math> we may omit it and just write <math>U\|D</math>.

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

	== Rationale ==		== Rationale ==

ArrowHead294 at 12:59, 12 March 2025

2025-03-12T12:59:10Z

Show changes

ArrowHead294 at 18:29, 11 February 2025

2025-02-11T18:29:04Z

← Older revision		Revision as of 18:29, 11 February 2025
Line 161:		Line 161:
	* [[Jake Freivald's mode numbering system]]		* [[Jake Freivald's mode numbering system]]

			{{Navbox notation}}
	{{~~Notation navbox~~}}

	[[Category:Mode]]		[[Category:Mode]]
	~~[[Category:Notation]]~~

BudjarnLambeth: {{Notation navbox}}

2024-12-03T00:04:04Z

← Older revision		Revision as of 00:04, 3 December 2024
Line 160:		Line 160:
	* [[Genchain mode numbering]]		* [[Genchain mode numbering]]
	* [[Jake Freivald's mode numbering system]]		* [[Jake Freivald's mode numbering system]]


			{{Notation navbox}}

	[[Category:Mode]]		[[Category:Mode]]
	[[Category:Notation]]		[[Category:Notation]]

TallKite: /* Things to watch out for */ made the statement about UDP's issues being avoidable less specific.

2024-11-10T06:28:44Z

Things to watch out for: made the statement about UDP's issues being avoidable less specific.

← Older revision		Revision as of 06:28, 10 November 2024
Line 154:		Line 154:
	** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700¢ for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700¢), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700¢), meaning that the bright generator is the fifth. As a result, Dominant[12] 7\|4 is ambiguous.		** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700¢ for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700¢), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700¢), meaning that the bright generator is the fifth. As a result, Dominant[12] 7\|4 is ambiguous.
	* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5\|2 is ambiguous.		* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5\|2 is ambiguous.
	~~Genchain mode numbering solves~~ all these issues (see below).		Not all notations have these issues (see below).

	== See also ==		== See also ==

TallKite: /* Things to watch out for */

2024-11-09T02:11:25Z

Things to watch out for

← Older revision		Revision as of 02:11, 9 November 2024
Line 154:		Line 154:
	** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700¢ for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700¢), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700¢), meaning that the bright generator is the fifth. As a result, Dominant[12] 7\|4 is ambiguous.		** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700¢ for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700¢), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700¢), meaning that the bright generator is the fifth. As a result, Dominant[12] 7\|4 is ambiguous.
	* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5\|2 is ambiguous.		* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5\|2 is ambiguous.
			Genchain mode numbering solves all these issues (see below).

	== See also ==		== See also ==

← Older revision		Revision as of 13:10, 12 March 2025
Line 105:		Line 105:
	A [[periodic scale]] ''S'' associates an interval ''S''(''i'') to every integer ''i'', such that there is a period (strictly, a quasiperiod) {{nowrap\|Q > 0}} and an interval of repetition ''R'' such that {{nowrap\|''S''(''i'' + ''Q'') {{=}} ''S''(''i'') + ''R''}}. ''Q'' is chosen so as to be minimal; there is no smaller period. ''S'' is monotone if {{nowrap\|''i'' < ''j''}} implies that {{nowrap\|''S''(''i'') < ''S''(''j'')}}.		A [[periodic scale]] ''S'' associates an interval ''S''(''i'') to every integer ''i'', such that there is a period (strictly, a quasiperiod) {{nowrap\|Q > 0}} and an interval of repetition ''R'' such that {{nowrap\|''S''(''i'' + ''Q'') {{=}} ''S''(''i'') + ''R''}}. ''Q'' is chosen so as to be minimal; there is no smaller period. ''S'' is monotone if {{nowrap\|''i'' < ''j''}} implies that {{nowrap\|''S''(''i'') < ''S''(''j'')}}.

	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 ~~<math>~~U\|D(P)~~</math>~~. If ~~<math>~~P = 1~~</math>~~ we may omit it and just write ~~<math>~~U\|D~~</math>~~.		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;{{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.