Extended-diatonic interval names: Difference between revisions

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In Hunt's system when used in 41-tET or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings. [[User:PiotrGrochowski/Extra-Diatonic Intervals|Other]] size-based systems also exist, but are less thoroughly described and less well known. In all these systems, interval arithmetic is not conserved across all tunings.

==Ups and Downs==

One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from ~~the diatonic scale~~. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). P1, P4, P5 and P8 are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as ~~it's~~ conservation of interval arithmetic. Rank-2 temperaments may also be described, with the addition of an additional pair of qualifiers - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.

One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from pythagorean intervals. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'.

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31-tET in Ups and Downs:

1 ^1 vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 4 ^4 A4 d5 v5 5 ^5 vm6 m6 ~6 M6 ^M6 vm7 m7 ~7 M7 ^M7 v8 8

P1 ^1 A1/vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 P4 ^4 A4 d5 v5 P5 ^5 A5/vm6 m6 ~6 M6 ^M6 vm7 m7 ~7 M7 ^M7 v8 P8

In Jones' relabeling 31edo appears as in Fokker/Keenan Extended-diatonic Interval-names.

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 In Hunt's system when used in 41-tET or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings. [[User:PiotrGrochowski/Extra-Diatonic Intervals|Other]] size-based systems also exist, but are less thoroughly described and less well known. In all these systems, interval arithmetic is not conserved across all tunings.
 ==Ups and Downs==
-One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). P1, P4, P5 and P8 are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as it's conservation of interval arithmetic. Rank-2 temperaments may also be described, with the addition of an additional pair of qualifiers - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.
+One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from pythagorean intervals. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.
 [[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'.
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 -tET in Ups and Downs:
-^1 vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 4 ^4 A4 d5 v5 5 ^5 vm6 m6 ~6 M6 ^M6 vm7 m7 ~7 M7 ^M7 v8 8
+P1 ^1 A1/vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 P4 ^4 A4 d5 v5 P5 ^5 A5/vm6 m6 ~6 M6 ^M6 vm7 m7 ~7 M7 ^M7 v8 P8
 In Jones' relabeling 31edo appears as in Fokker/Keenan Extended-diatonic Interval-names.