Modal union - Revision history

Inthar: /* Properties */

2024-01-22T03:07:11Z

Properties

← Older revision		Revision as of 03:07, 22 January 2024
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	== Properties ==		== Properties ==
	For any [[MOS scale]] with a ''n''L 1s step pattern~~, no matter the~~ tuning, the modal union is a child MOS scale with either an (''n''+1)L ''n''s or an ''n''L (''n''+1)s step pattern. The archeotonic example above shows that the modal union of [[6L 1s]] is [[6L 7s]] for a hard tuning, while it would be [[7L 6s]] for a soft tuning.		For any [[MOS scale]] with a ''n''L 1s step pattern in any non-[[basic]] tuning, the modal union is a child MOS scale with either an (''n''+1)L ''n''s or an ''n''L (''n''+1)s step pattern. The archeotonic example above shows that the modal union of [[6L 1s]] is [[6L 7s]] for a hard tuning, while it would be [[7L 6s]] for a soft tuning.

	For any ''n''-tone MOS scale tuned as a subset of an [[equal tuning]] with fewer than 2''n''−1 tones per period, the modal union is the tuning itself. The case of the 12edo diatonic scale mentioned above is an example of that property. It could also be generalized to non-MOS scales with a different restriction on the number of tones for the tuning.		For any ''n''-tone MOS scale tuned as a subset of an [[equal tuning]] with fewer than 2''n''−1 tones per period, the modal union is the tuning itself. The case of the 12edo diatonic scale mentioned above is an example of that property. It could also be generalized to non-MOS scales with a different restriction on the number of tones for the tuning.

Inthar at 15:23, 3 August 2022

2022-08-03T15:23:05Z

← Older revision		Revision as of 15:23, 3 August 2022
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	{{About\|a type of scale\|a collection of scales\|Dome}}		{{About\|a type of scale\|a collection of scales\|Dome}}

	The '''modal union''' of a [[periodic scale]] is the scale obtained by superimposing all its [[mode]]s on the same tonal center. The resulting scale contains every mode of the original scale on ~~every note~~.		The '''modal union''' of a [[periodic scale]] is the scale obtained by superimposing all its [[mode]]s on the same tonal center. The resulting scale contains every mode of the original scale on the tonic.

	The modal union of an ''n''-tone scale has at most 2''n''−1 distinct tones; it has fewer tones if there are enharmonic equivalents. For example, the basic [[5L 2s\|diatonic]] scale of [[12edo]] has 7 tones, but its modal union only has 12 tones instead of 13, because the augmented fourth of the lydian mode is enharmonically equivalent to the diminished fifth of the the locrian mode. However, in other tunings of the diatonic scale, such as [[17edo]] (hard diatonic) or [[19edo]] (soft diatonic), the modal union has 13 tones, and the enharmonic notes differ by a [[diesis]].		The modal union of an ''n''-tone scale has at most 2''n''−1 distinct tones; it has fewer tones if there are enharmonic equivalents. For example, the basic [[5L 2s\|diatonic]] scale of [[12edo]] has 7 tones, but its modal union only has 12 tones instead of 13, because the augmented fourth of the lydian mode is enharmonically equivalent to the diminished fifth of the the locrian mode. However, in other tunings of the diatonic scale, such as [[17edo]] (hard diatonic) or [[19edo]] (soft diatonic), the modal union has 13 tones, and the enharmonic notes differ by a [[diesis]].

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2022-08-03T05:40:47Z

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{{About|a type of scale|a collection of scales|Dome}}

The '''modal union''' of a [[periodic scale]] is the scale obtained by superimposing all its [[mode]]s on the same tonal center. The resulting scale contains every mode of the original scale on every note.

The modal union of an ''n''-tone scale has at most 2''n''−1 distinct tones; it has fewer tones if there are enharmonic equivalents. For example, the basic [[5L 2s|diatonic]] scale of [[12edo]] has 7 tones, but its modal union only has 12 tones instead of 13, because the augmented fourth of the lydian mode is enharmonically equivalent to the diminished fifth of the the locrian mode. However, in other tunings of the diatonic scale, such as [[17edo]] (hard diatonic) or [[19edo]] (soft diatonic), the modal union has 13 tones, and the enharmonic notes differ by a [[diesis]].

== Examples ==
'''Soft diatonic scale (3:2 5L 2s, 19edo)'''
<pre>
Lydian: D - - E - - F# - - G# - A - - B - - C# - (D)
Ionian: D - - E - - F# - G - - A - - B - - C# - (D)
Mixolydian: D - - E - - F# - G - - A - - B - C - - (D)
Dorian: D - - E - F - - G - - A - - B - C - - (D)
Aeolian: D - - E - F - - G - - A - Bb - - C - - (D)
Phrygian: D - Eb - - F - - G - - A - Bb - - C - - (D)
Locrian: D - Eb - - F - Gb - - - A - Bb - - C - - (D)

Modal union: D - Eb E - F F# Gb G G# - A - Bb B - C C# - (D)
</pre>

'''Hard archeotonic scale (3:1 6L 1s, 19edo)'''
<pre>
Ryonian: J - - K - - L - - M - - N& - - O& - - P&(J)
Karakalian: J - - K - - L - - M - - N& - - O& P - - (J)
Lobonian: J - - K - - L - - M - - N& O - - P - - (J)
Horthathian: J - - K - - L - - M N - - O - - P - - (J)
Oukranian: J - - K - - L M@ - - N - - O - - P - - (J)
Tamashian: J - - K L@ - - M@ - - N - - O - - P - - (J)
Zo-Kalarian: J K@ - - L@ - - M@ - - N - - O - - P - - (J)

Modal union: J K@ - K L@ - L M@ - M N - N& O - O& P - P&(J)
</pre>

'''Basic left-handed diasem scale (3:2:1 5L 2M 2s, 21edo)'''
<pre>
LH Lydian: J - - K - - L& M - - N^ - O - - P& Q - - R^ - (J)
LH Ionian: J - - K L - - M - N - - O - - P& Q - - R^ - (J)
LH Mixo: J - - K L - - M - N - - O P - - Q - R - - (J)
LH Bright Dorian: J K@ - - L - Mv - - N - - O P - - Q - R - - (J)
LH Bright Aeolian: J K@ - - L - Mv - - N O@ - - P - Qv - - R - - (J)

LH Dark Dorian: J - - K - L^ - - M^ - - N& O - - P& - Q^ - - R&(J)
LH Dark Aeolian: J - - K - L^ - - M^ N - - O - P^ - - Q^ - - R&(J)
LH Phrygian: J - Kv - - L^ - - M^ N - - O - P^ - - Q^ R - - (J)
LH Locrian: J - Kv - - L^ Mv - - N - Ov - - P^ - - Q^ R - - (J)

Modal union: J K@ Kv K L L^ Mv M M^ N N^ Ov O P P^ Qv Q Q^ R R^ R&(J)

& = +2\21 | ^ = +1\21 | v = −1\21 | @ = −2\21
</pre>

== Properties ==
For any [[MOS scale]] with a ''n''L 1s step pattern, no matter the tuning, the modal union is a child MOS scale with either an (''n''+1)L ''n''s or an ''n''L (''n''+1)s step pattern. The archeotonic example above shows that the modal union of [[6L 1s]] is [[6L 7s]] for a hard tuning, while it would be [[7L 6s]] for a soft tuning.

For any ''n''-tone MOS scale tuned as a subset of an [[equal tuning]] with fewer than 2''n''−1 tones per period, the modal union is the tuning itself. The case of the 12edo diatonic scale mentioned above is an example of that property. It could also be generalized to non-MOS scales with a different restriction on the number of tones for the tuning.

[[Category:Mode]]
[[Category:Scale]]