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Kite Guitar Scales: Difference between revisions

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→Decatonic - the semitonal scale (2L 7s 1xs): finished this section. also added a paragraph about alternating generators to the overview.
TallKite (talk | contribs)
autopatrolled, Sysops
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added a section on the 11-tone checkerboard scale, other minor changes too
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Certain Asian music uses very "lopsided" scales such as P1 M3 P4 P5 M7 P8 (SE Asia) and P1 M2 m3 P5 m6 P8 (Japan). While there is a certain charm to these, scales with equal or roughly equal sizes are also attractive. The only such 12edo scales are the whole tone scale and the full 12-note gamut. Since 41 is a prime number, it has no strictly equal scales. But there are many nearly-equal scales, or near-edos.   
Certain Asian music uses very "lopsided" scales such as P1 M3 P4 P5 M7 P8 (SE Asia) and P1 M2 m3 P5 m6 P8 (Japan). While there is a certain charm to these, scales with equal or roughly equal sizes are also attractive. The only such 12edo scales are the whole tone scale and the full 12-note gamut. Since 41 is a prime number, it has no strictly equal scales. But there are many nearly-equal scales, or near-edos.   


If N goes into 41 X times with a remainder of Y, then the near-N-edo scale has steps YL and (N-Y)s, where L=X+1 and s=X. This near-N-edo scale is altered slightly so that there are only 3 odd numbers, and the rest are even. This avoids an awkward scale and also tends to make the intervals well tuned. For example, the unaltered whole-tone scale would have thirds of mostly 14/11 with some 5/4, but the the altered one has thirds of mostly 5/4 with some 9/7.   
If N goes into 41 X times with a remainder of Y, then the near-N-edo scale has steps YL and (N-Y)s, where L=X+1 and s=X. This near-N-edo scale is altered slightly so that there are only 3 odd numbers, and the rest are even. This avoids an awkward scale and also tends to make the intervals well tuned. For example, the unaltered whole-tone scale would have thirds of mostly 14/11 (plain M3) with some 5/4, but the the altered one has thirds of mostly 5/4 with some 9/7.   


The alteration is done so that it produces only 1 additional step size which is either 1 edostep larger than L or else 1 edostep smaller than s. If possible (and it often is), the alteration is done so that this new step size occurs only once. This is ideal because almost all steps are within the original L-to-s range, and the original (small) L/s ratio still describes the overall sound of the scale. If the new step size occurs more than once, the 3 step sizes are named L, m and s. If it only occurs once, the new step size is named either XL or xs, for extra large/small. The new step is '''bolded''' in the table below. It occurs more than once for near-edos 8 and 12-17, and not at all for near-edos 11 and 19.   
The alteration is done so that it produces only 1 additional step size which is either 1 edostep larger than L or else 1 edostep smaller than s. If possible (and it often is), the alteration is done so that this new step size occurs only once. This is ideal because almost all steps are within the original L-to-s range, and the original (small) L/s ratio still describes the overall sound of the scale. If the new step size occurs more than once, the 3 step sizes are named L, m and s. If it only occurs once, the new step size is named either XL or xs, for extra large/small. The new step is '''bolded''' in the table below. It occurs more than once for near-edos 8 and 12-17, and not at all for near-edos 11 and 19.   
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|3L 3m '''2s'''
|3L 3m '''2s'''
|1.5
|1.5
|0
|1
| +3, +2, -4
| +3, +2, -4
|2 dim6/dim7 tetrads
|2 dim6/dim7 tetrads
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|8L 3s
|8L 3s
|1.33
|1.33
|
|0
| +2, -5
| +2, -5
|
|
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|
|
|}
|}
These near-equal scales can be used to translate music from a small edo to the Kite guitar.


=== Tritonic and Tetratonic ===
=== Tritonic and Tetratonic ===
Line 1,030: Line 1,031:
Because of the prominence of the "octa-5th" (i.e. tritone) in octatonic chords, this interval plays a role analogous to the perfect 5th in other scales. Every octotonic scale contains eight tritones. The most consonant tritone is the dim 5th = 7/5. Of course all eight tritones can't be dim 5ths without dual-ness, but half of them can be. In particular, the tonic chord can be a dim7 chord that contains two dim 5ths. The only two such chords that are playable are the ^dim7 and vdim7 chords. If we require that the remaining four notes of the scale make another such chord, there are only three near-equal octotonic scales. Each has two main modes, depending on which of the dim7 chords is considered to be the tonic chord.     
Because of the prominence of the "octa-5th" (i.e. tritone) in octatonic chords, this interval plays a role analogous to the perfect 5th in other scales. Every octotonic scale contains eight tritones. The most consonant tritone is the dim 5th = 7/5. Of course all eight tritones can't be dim 5ths without dual-ness, but half of them can be. In particular, the tonic chord can be a dim7 chord that contains two dim 5ths. The only two such chords that are playable are the ^dim7 and vdim7 chords. If we require that the remaining four notes of the scale make another such chord, there are only three near-equal octotonic scales. Each has two main modes, depending on which of the dim7 chords is considered to be the tonic chord.     


The scales are named after the root of the non-tonic dim7 chord. This chord is always upped or downed (^d7 vs. vd7) to match the root. If the tonic chord is upped or downed the opposite way, the two dim7 chords, and hence the entire scale, can easily be deduced from the name: the <u>up</u>flat-2 octotonic scale has an <u>up</u>dim7 chord on the ^bII and a <u>down</u>dim7 chord on the I. The octave inverse of ^b2 is vM7, thus the other main mode of the upflat-2 scale is the down-7 scale. If the tonic chord is upped or downed the same way, we must add that direction to the name: the up-3 up scale has an updim7 chord on ^III and an updim7 chord on I.   
The scales are named after the root of the non-tonic dim7 chord. This chord is always upped or downed (^d7 vs. vd7) to match the root. If the tonic chord is upped or downed the opposite way, the two dim7 chords, and hence the entire scale, can easily be deduced from the name: the <u>up</u>flat-2 octotonic scale has an <u>up</u>dim7 chord on the <u>up</u>flat II and a <u>down</u>dim7 chord on the I. The octave inverse of ^b2 is vM7, thus the other main mode of the upflat-2 scale is the down-7 scale. If the tonic chord is upped or downed the same way, we must add that direction to the name: the up-3 up scale has an updim7 chord on up-III and an updim7 chord on I.   


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 1,172: Line 1,173:
34343
34343
|4 3
|4 3
L/s = 1.5
L/s = 1.33
|5L 7s
|5L 7s
or 12L
or 12L
Line 1,218: Line 1,219:


The twin downminor scale consists of two downminor pentatonic scales, offset from each other by two frets. Likewise with the twin upmajor scale. The twin downminor's down-7 mode is (12:13:14:15:16:17:18)/12 plus (12:13:14:15:16)/8, except that prime 17 isn't well tuned.
The twin downminor scale consists of two downminor pentatonic scales, offset from each other by two frets. Likewise with the twin upmajor scale. The twin downminor's down-7 mode is (12:13:14:15:16:17:18)/12 plus (12:13:14:15:16)/8, except that prime 17 isn't well tuned.
The "mixed" scale that uses both upmajor and downminor is distributionally even, and each string has 3 or 4 notes. The two unmixed "twin" scales are less even, and one string holds only 2 notes.
{| class="wikitable center-all"
{| class="wikitable center-all"
!subgroup
!subgroup
Line 1,389: Line 1,392:
|345-444-5444
|345-444-5444
|}
|}
=== Eleven-tone - The checkerboard scale (8L 3s) ===
This scale is notable for not needing a 3rd step size or a 3rd move. It gets its name from the fact that it uses every other fret of each string, and each string's notes are offset by one fret from the neighboring strings. Thus the scale chart looks like an actual checkerboard. It's a [[MOS scale]] generated by the ^M3, which is 15\41. The complete genchain runs from -10 generators to +10:
M3 ~6 m2 ^4 vm7 M2 ~5 vM7 ^m3 vm6 '''<u>P1</u>''' ^M3 vM6 ^m2 ~4 m7 ^M2 v5 M7 ~3 m6
The official [[Color notation/Temperament Names|name]] for this temperament is Sasa-tritribizo, with an extremely complex [[pergen]] (P8, c⁶P5/18). Better to call it checkerboard! Checkerboard[5] and Checkerboard[8] are also MOS scales, with L/s ratios of 2.75 and 1.75 respectively.
11 generators add up to an ^1, showing how very near 11-edo this scale is. 7 generators add up to a down-5th, thus all modes contain 4 down-5ths. The 11 modes of Checkerboard[11] are listed here in order from sharpest to flattest. The notes that differ from the neighboring modes are '''bolded'''.
{| class="wikitable center-all"
!subgroup
!name
! colspan="12" |scale
!genchain
!edosteps
!step sizes
!step count
!moves
|-
! rowspan="11" |yazala
(2.3.5.7.11)
!#1
|P1
|^m2
|^M2
|~3
|^M3
|~4
|v5
|'''m6'''
|vM6
|m7
|M7
|P8
|P1 ... m6
|444-3444-344-3
| rowspan="11" |4 3
L/s = 1.33
| rowspan="11" |8L 3s
| rowspan="11" | +2, -5
|-
!#2
|P1
|^m2
|^M2
|'''~3'''
|^M3
|~4
|v5
|'''vm6'''
|vM6
|m7
|M7
|P8
|vm6 ... ~3
|444-344-3444-3
|-
!#3
|P1
|^m2
|^M2
|'''^m3'''
|^M3
|~4
|v5
|vm6
|vM6
|m7
|'''M7'''
|P8
|^m3 ... M7
|44-3444-3444-3
|-
!#4
|P1
|^m2
|^M2
|^m3
|^M3
|~4
|'''v5'''
|vm6
|vM6
|m7
|'''vM7'''
|P8
|vM7 ... v5
|44-3444-344-34
|-
!#5
|P1
|^m2
|'''^M2'''
|^m3
|^M3
|~4
|'''~5'''
|vm6
|vM6
|m7
|vM7
|P8
|~5 ... ^M2
|44-344-3444-34
|-
!#6
|P1
|^m2
|'''M2'''
|^m3
|^M3
|~4
|~5
|vm6
|vM6
|'''m7'''
|vM7
|P8
|M2 ... m7
|4-3444-3444-34
|-
!#7
|P1
|^m2
|M2
|^m3
|^M3
|'''~4'''
|~5
|vm6
|vM6
|'''vm7'''
|vM7
|P8
|vm7 ... ~4
|4-3444-344-344
|-
!#8
|P1
|'''^m2'''
|M2
|^m3
|^M3
|'''^4'''
|~5
|vm6
|vM6
|vm7
|vM7
|P8
|^4 ... ^m2
|4-344-3444-344
|-
!#9
|P1
|'''m2'''
|M2
|^m3
|^M3
|^4
|~5
|vm6
|'''vM6'''
|vm7
|vM7
|P8
|m2 ... vM6
|3444-3444-344
|-
!#10
|P1
|m2
|M2
|^m3
|'''^M3'''
|^4
|~5
|vm6
|'''~6'''
|vm7
|vM7
|P8
|~6 ... ^M3
|3444-344-3444
|-
!#11
|P1
|m2
|M2
|^m3
|'''M3'''
|^4
|~5
|vm6
|~6
|vm7
|vM7
|P8
|M3 ... P1
|344-3444-3444
|}
The scale completely lacks perfect 4ths and 5ths, giving it a very unusual sound. But it's full of low-odd-limit ratios, so it sounds both alien and harmonious. There are many augmented and diminished chords. Any three adjacent notes on the genchain form Bohlen-Pierce's up-sesquiaug chord P1 ^M3 ^^#5 (aka up-down6-no5 P1 ^M3 vM6), thus every scale has nine such chords. Likewise nine each of that chord's homonyms, up-halfaug (P1 ^M3 ^^5=vm6) and upminor-halfaug (P1 - ^m3 - ^^5=vm6). There are also seven upminor-mid5 chords (P1 ^m3 ~5) and five upminor-down7-mid5 chords (P1 ^m3 ~5 vm7).
Because it's a MOS scale, each column between P1 and P8 in the table above contains only two types of interval. Melodically, the smaller intervals sound vaguely like 12-edo:
* 1 step = minorish 2nd = 88¢ or 117¢
* 2 steps = majorish 2nd = 205¢ or 234¢
* 3 steps = minorish 3rd = 322¢ or 351¢
* 4 steps = majorish 3rd = 410¢ or 439¢
* 5 steps = middish 4th = 527¢ or 566¢
* 6 steps = middish 5th = 644¢ or 673¢


=== Nineteen-tone - The chromatic scale (3L 16s) ===
=== Nineteen-tone - The chromatic scale (3L 16s) ===
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