Kite Guitar Scales

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Revision as of 05:25, 27 July 2020 by TallKite (talk | contribs) (adding equiheptatonic and decatonic, a work in progress)
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Printable charts for the downmajor tuning of the Kite Guitar. One is of scale degrees, the other is of the three main heptatonic scales. In the latter, some scale degrees appear more than once. In general, use the one that agrees with the current chord.


Overview

There are many possible scales. Those listed here are select ones with a low prime limit and/or a low odd limit.

Every scale can be thought of as a chord, e.g. the 12edo major pentatonic scale is a 6add9 pentad. Many pentads and heptads have an innate comma which 41edo does not temper out. Thus many Kite Guitar scales are "fuzzy", meaning a scale degree may vary by 1 edostep. In the tables below, a note that may be either a M2 or a vM2 is indicated by (v)M2. In general, major scales have a fuzzy 2nd and minor scales have a fuzzy 4th. But the chord progression may make other degrees fuzzy. For example, Iv - IVv - Vv7 - Iv requires a fuzzy 4th.

The modes of a scale are grouped together. Not every mode is shown. Two modes of a scale will use the same prime subgroup, so modes are grouped by subgroup.

Each scale has steps of various sizes, shown in the far right columns as both intervals and edosteps. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio can be calculated directly from the edosteps. For example, the downminor heptatonic scale has a very large L/s ratio of 8/2 = 4, giving it a lopsided feel. But the downminor pentatonic scale has a very small L/s ratio of only 9/7 = 1.29, giving it an equipentatonic feel.

Harmonic and subharmonic scales are segments of the harmonic and subharmonic series. They are not fuzzy. Harmonic and subharmonic may be abbreviated as har- and subhar-, e.g. harmajor pentatonic. Pentatonic scales use (sub)harmonics 5-10, and heptatonic scales use (sub)harmonics 7-14.

Pentatonic Scales

Every pentatonic scale has 5 modes, but only those modes with a non-fuzzy 5th are listed.

Major and minor scales

The za scales are nearly equipentatonic, dividing the P4 into two nearly equal steps of ^M2 and vm3 (8 and 9).

subgroup name scale as a chord step sizes
ya

(2.3.5)

downmajor P1 (v)M2 vM3 P5 vM6 P8 v6,(v)9 chord vM2, M2, ^m3 6 7 11
upminor P1 ^m3 (^)4 P5 ^m7 P8 ^m7,(^)11 chord
za

(2.3.7)

downminor P1 vm3 (v)4 P5 vm7 P8 vm7,(v)11 chord M2, ^M2, vm3 7 8 9
upmajor P1 (^)M2 ^M3 P5 ^M6 P8 ^6,(^)9 chord

Harmonic and subharmonic scales

These are named after the triad implied by the 3rd and 5th, minus the up or down. Note that the harmonic major scale contains a minor 7th, and the harmonic minor scale contains a major 6th. Likewise with the subharmajor and subharminor scales. A harmonic diminished pentatonic scale would be P1 ^m3 d5 ^m6 ^m7 P8 = 5:6:7:8:9. But it's not very plausible, and would be heard as one of the other modes.

subgroup name scale as a chord step sizes
yaza

(2.3.5.7)

harmonic major P1 M2 vM3 P5 vm7 P8 v9 = 8:9:10:12:14 vM2, M2, ^M2,

vm3, ^m3

6 7 8 9 11
harmonic minor P1 vm3 P4 P5 vM6 P8 vm6,11 = 6:7:8:9:10
" subharmonic major P1 M2 ^M3 P5 ^m7 P8 ^9 = 9/(9:8:7:6:5) " "
subharmonic minor P1 ^m3 P4 P5 ^M6 P8 ^m6,11 = 12/(12:10:9:8:7)
subharmonic diminished P1 vm3 d5 vm6 vm7 P8 vm7(b5),vm6 = 14/(14:12:10:9:8)

Heptatonic Scales

Major and minor scales

As with chords, adding up or down to a scale name affects the 3rd, 6th and 7th. However, there are fuzzy notes not implied by the name. Without these fuzzy notes, downmajor and upminor would not be modes of each other.

subgroup name scale step sizes
ya

(2.3.5)

downmajor P1 (v)M2 vM3 P4 P5 vM6 vM7 P8 ^m2, vM2, M2 4 6 7
upminor P1 M2 ^m3 (^)4 P5 ^m6 ^m7 P8
za

(2.3.7)

upmajor P1 (^)M2 ^M3 P4 P5 ^M6 ^M7 P8 vm2, M2, ^M2 2 7 8
downminor P1 M2 vm3 (v)4 P5 vm6 vm7 P8

Harmonic and subharmonic scales

These all have the same prime subgroup, yazalatha (2.3.5.7.11.13). Adding the 15th harmonic (the bolded note) makes an octotonic scale that uses harmonics 8-16. Again, the scales are named after the triad implied by the 3rd and 5th, minus the up or down. If there are two 3rds, the unbolded one is used. Each scale contains the similarly-named pentatonic scale, e.g. the harmajor scale contains the harmajor pentatonic scale.

scale as a chord step sizes
harmonic major P1 M2 vM3 ~4 P5 ~6 vm7 vM7 P8 8:9:10:11:12:13:14:15 ^m2, ~2, vM2, M2, ^M2 4 5 6 7 8
harmonic minor P1 ~2 vm3 vM3 P4 P5 vM6 ~7 P8 12:13:14:15:16:18:20:22
subharmonic major P1 M2 ^m3 ^M3 ~4 P5 ~6 ^m7 P8 18/(18:16:15:14:13:12:11:10) " "
subharmonic minor P1 ~2 ^m3 P4 P5 ^m6 ^M6 ~7 P8 24/(24:22:20:18:16:15:14:13)

One of the hallmarks of harmonic and subharmonic scales is that each step has a unique size. Unfortunately, in 41edo, these scales do not have unique step sizes. The heptatonic scales run 8 7 6 6 5 5 4. The octotonic step sizes are worse, 7 6 6 5 5 4 4 4. Only the pentatonic scales have unique step sizes.

The seven modes

Generalizing the seven modes to 41edo is tricky. Five of the seven ya modes are formed from this collection of notes: 

  D ----- A ----- E ----- B
   \     / \     / \     / \
    \   /   \   /   \   /   \
     \ /     \ /     \ /     \
     ^F ---- ^C ---- ^G ---- ^D

Five of the seven za modes are formed from this collection: 

   ------- ------- -------
   \     / \     / \     / \
    \   /   \   /   \   /   \
 vF  \ / vC  \ / vG  \ / vD  \
      D ----- A ----- E ----- B

In both cases, the D is fuzzy. But the two dorian scales and the two locrian scales are not from these lattices, and are not actually modes of the other scales.

To be consistent, the two dorian scales should have a fuzzy tonic. To avoid this, and to provide all six triads, there are two fuzzy notes. Note that the 6th of the updorian scale can be downed.

To be consistent, the uplocrian or downlocrian scale should have an upflat or downflat 5th. To get a plain flat 5th, and thus a more consonant 5:6:7 or 7/(7:6:5) tonic triad, the 5th is fuzzy as well as the 3rd.

subgroup name scale step sizes
ya

(2.3.5)

downlydian P1 M2 vM3 vA4 P5 (v)M6 vM7 P8 ^m2, vM2, M2 4 6 7
downmajor P1 (v)M2 vM3 P4 P5 vM6 vM7 P8
downmixolydian P1 vM2 vM3 P4 (v)5 vM6 m7 P8
upminor P1 M2 ^m3 (^)4 P5 ^m6 ^m7 P8
upphrygian P1 ^m2 ^m3 P4 P5 ^m6 (^)m7 P8
" updorian P1 M2 ^m3 (^)4 P5 (v)M6 ^m7 P8 ^m2, ~2, vM2, M2 4 5 6 7
" uplocrian P1 ^m2 (^)m3 P4 (^)d5 ^m6 m7 P8 m2, ^m2, vM2, M2, ^M2 3 4 6 7 8
za

(2.3.7)

uplydian P1 M2 ^M3 ^A4 P5 (^)M6 ^M7 P8 vm2, M2, ^M2 2 7 8
upmajor P1 (^)M2 ^M3 P4 P5 ^M6 ^M7 P8
upmixolydian P1 ^M2 ^M3 P4 (^)5 ^M6 m7 P8
downminor P1 M2 vm3 (v)4 P5 vm6 vm7 P8
downphrygian P1 vm2 vm3 P4 P5 vm6 (v)m7 P8
yaza downdorian P1 M2 vm3 (v)4 P5 (v)M6 vm7 P8 vm2, ~2, M2, ^M2 2 5 7 8
" downlocrian P1 vm2 (v)m3 P4 (v)d5 vm6 m7 P8 vm2, m2, vM2, M2, ^M2 2 3 6 7 8

Near-equiheptatonic scales

These are a cross between the usual modes and the harmonic or subharmonic scales. Obviously they are reminiscent of 7-edo. The 4th is divided into three nearly equal steps of two vM2's and a ~2 (6 6 5), thus it's also reminiscent of the third-4th pergen and the Triyo temperament.

The smallest step of the upminor or downmajor scale is widened by 1 edostep to a mid-2nd.

As can be seen from this picture, the upminor scale falls on two arms of the 41edo spiral of 5ths. Only 1 fuzzy note is needed to avoid wolf fifths. But the

mid-downmajor - 7 6 6 5 - 6 6 5 --> 6 5 7 6 - 6 5 6 = vM2 ^m3

mid-upminor - 5667-566 --> mid = 6675-665

6657 = P1 vM2 ~3 P4 P5

6567 = P1 vM2 ^m3 P4 P5

5667 = P1 ~2 ^m3 P4 P5

7665 = P1 M2 ^m3 ~4 P5

6765 = P1 vM2 ^m3 ~4 P5

6675 = P1 vM2 ~3 ~4 P5

665 = P5 vM6 ~7 P8

656 = P5 vM6 ^m7 P8

566 = P5 ~6 ^m7 P8

subgroup name scale as edosteps as (sub)harmonic series fragments as chain of 5ths step sizes
yala

(2.3.5.11)

mid-major P1 M2 vM3 ~4 P5 vM6 ~7 P8 7665-665 (8:9:10:11:12)/8 + (9:10:11:12)/6 P152 vM63 ~74 ~2, vM2, M2 5 6 7
mid? P1 vM2 ~3 P4 P5 vM6 ~7 P8 6657-665 (9:10:11:12)/9 + (8:9:10:11:12)/6 P415 vM26 ~37
" mid-minor P1 ~2 ^m3 P4 P5 ~6 ^m7 P8 5667-566 12/(12:11:10:9:8) + 18/(12:11:10:9) ~26 ^m37 P415 " "
? P1 vM2 ~3 ~4 P5 vM6 ~7 P8 6675-665 P15 vM26 ~374
" ? P1 vM2 vM3 ~4 P5 vM6 ~7 P8 6765-665 P15 vM263 ~74 " "
? P1 vM2 ^m3 P4 P5 vM6 ^m7 P8 6567-656 ^m37 P415 vM26

Decatonic - Ten is the New Twelve

"The Flight of the Bumblebee" has simple 5-limit triads, but a scale that is clearly dodecatonic. The evenly-spaced 12edo scale is quite fitting for this piece. How would this piece translate to the Kite guitar? Poorly, because the scale would be either very uneven (steps of 2, 3 and 4, L/s ratio of 2), or very awkward to play (all plain notes, lots of jumping between strings).

Is there an easily playable chromatic-sounding scale with nearly even steps? We need three odd numbers and the rest even. If the even number is 6 or 8, we get the equiheptatonic (41/6 is about 7) or equipentatonic (41/8 is about 5) scales. The obvious answer is 4, which makes a decatonic scale.

subgroup name scale as edosteps as a chord step sizes
yalaza

(2.3.5.7.11)

twin downminor pentatonic #1 P1 ~2 vm3 vM3 (v)4 d5 P5 ~6 vm7 vM7 P8 5444-34-5444 12:13:14:15:16 m2, ^m2, ~2 3 4 5
twin downminor pentatonic #2 P1 ^m2 vm3 vM3 (v)4 A4 P5 ^m6 vm7 vM7 P8 4544-43-4544
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