Kite Guitar Scales

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 41edo scales. Those discussed here are those with at least 5 notes, and which contain a plain perfect 5th. Scales that are awkward to play on the Kite guitar are avoided. An awkward scale has a step which requires a jump of more than four frets. Thus plain minor 2nds and 3rds are avoided. A scale naturally hops from one string to the next as it goes up or down. Unlike other guitars, the Kite guitar doesn't let one hop freely. For example, the 3-limit scale fragment P1 M2 M3 P4 requires 3 hops, 2 upward and 1 downward. Any scale which doesn't have exactly three upward hops per octave is awkward, because the downward hop will always be at least 6 frets, and usually 7 or more. Almost every scale with a low prime limit and/or a low odd limit is not awkward.

Most MOS scales are awkward. The only non-awkward MOS scales that contain a perfect 5th are those from the Laquinyo temperament, which are generated by the downmajor 3rd. These have a small step of one fret. They have either a very lopsided L/s ratio or more than 12 notes. Besides these, the least awkward MOS scales with a 5th are the plain pentatonics: P1 M2 M3 P5 M6 P8 (major), or P1 m3 P4 P5 m7 P8 (minor), or the two thirdless modes.

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 order to avoid a wolf 5th. 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 it depends on the chord progression. For example, Iv - IVv - Vv7 - Iv requires a major scale with a fuzzy 4th.

Intervals can also be thought of as fuzzy. For example, a fuzzy major 2nd can be either a M2 or a vM2. Thus the downmajor scale 7647-674 is a fuzzy 5L2s MOS scale.

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. It can also be thought of as a fuzzy 2L3s MOS scale.

Harmonic and subharmonic scales are contiguous segments of the harmonic and subharmonic series respectively. 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. In harmonic scales, the step sizes get smaller as you ascend. In subharmonic scales, they get larger. In general, given a choice between an Ls sequence and an sL sequence, the first is often more otonal, and more consonant. For example, P1-M2-vM3 vs. P1-vM2-vM3, or P1-vm3-P4 vs. P1-^M2-P4, or even P1-vM3-P5 vs. P1-^m3-P5. Likewise for the choice between LLs and LsL and sLL, or between Lss and sLs and ss, the first is generally more consonant.

See also Kite Giedraitis's Categorizations of 41edo Scales.

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 downminor 7th, and the harmonic minor scale contains a downmajor 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	nickname	scale						as a chord	step sizes
yaza (2.3.5.7)	harmonic major	harmajor	P1	M2	vM3	P5	vm7	P8	v9 = 8:9:10:12:14	vM2, M2, ^M2, vm3, ^m3	6 7 8 9 11
	harmonic minor	harminor	P1	vm3	P4	P5	vM6	P8	vm6,11 = 6:7:8:9:10
"	subharmonic major	subharmajor	P1	M2	^M3	P5	^m7	P8	^9 = 9/(9:8:7:6:5)	"	"
	subharmonic minor	subharminor	P1	^m3	P4	P5	^M6	P8	^m6,11 = 12/(12:10:9:8:7)
	subharmonic diminished	subhardim	P1	vm3	d5	vm6	vm7	P8	vm7(b5),vm6 = 14/(14:12:10:9:8)

All five of these scales are "anti-MOS" in the sense that each scale step has a unique size.

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). They use harmonics 7-14. 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. Subhardim = 14/(14:13:12:11:10:9:8) is a theoretical possibility.

name	nickname	scale									as a chord	step sizes
harmonic major	harmajor	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	harminor	P1	~2	vm3	vM3	P4	P5	vM6	~7	P8	12:13:14:15:16:18:20:22
subharmonic major	subharmajor	P1	M2	^m3	^M3	~4	P5	~6	^m7	P8	18/(18:16:15:14:13:12:11:10)	"	"
subharmonic minor	subharminor	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 "anti-MOS" pentatonic scales have unique step sizes.

The seven diatonic modes

Generalizing major and minor to 41edo is fairly straightforward. Some of the other modes are 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								as a chain of 5ths	step sizes
ya (2.3.5)	downlydian	P1	M2	vM3	vA4	P5	(v)M6	vM7	P8	P15M26 vM637vA4	^m2, vM2, M2	4 6 7
	downmajor	P1	(v)M2	vM3	P4	P5	vM6	vM7	P8	P415M2 vM2637
	downmixolydian	P1	vM2	vM3	P4	(v)5	vM6	m7	P8	m7P415 v5vM263
	upminor	P1	M2	^m3	(^)4	P5	^m6	^m7	P8	^m637^4 P415M2
	upphrygian	P1	^m2	^m3	P4	P5	^m6	(^)m7	P8	^m2637 m7P415
"	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

It would also be possible to define the modes based on the harmonic and subharmonic scales. For example, the downmixolydian scale could be P1 M2 vM3 P4 P5 vM6 vm7 P8, which contains a 4:5:6:7:9 chord. But this scale has two wolf 5ths.

Near-equidistant Scales

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. We've seen how the upmajor and downminor pentatonic scales are nearly equi-pentatonic.

Heptatonic

These 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 two main ones are equi-major and equi-minor. Equi-minor is somewhat like maqam Bayati. Equi-major is equi-minor octave-inverted.

These scales can be derived from the seven modes by widening the two smallest steps by 1 edostep, from an upminor 2nd to a mid 2nd. The step sizes are 1L4m2s (L=7, m=6, s=5). Treating the sole large step as a fuzzy medium step, they are fuzzy 5L2s MOS scales.

As can be seen from the 41-edo spiral of 5ths, the upminor scale occupies two arms of the 41edo spiral of 5ths. Only one fuzzy note is needed to avoid wolf fifths. But these scales occupy three arms, and would need two fuzzy notes.

subgroup	name	scale								as edosteps	as (sub)harmonic series fragments	as a chain of 5ths	step sizes
yala (2.3.5.11)	equi-major	P1	(v)M2	vM3	~4	P5	vM6	~7	P8	7665-665	(8:9:10:11:12)/8 + (9:10:11:12)/6	P152 vM263 ~74	~2, vM2, M2	5 6 7
	equi-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
	equi-dorian	P1	vM2	^m3	(^)4	P5	vM6	^m7	P8	6567-656		^m37^4 P415 vM26
"	equi-minor	P1	~2	^m3	(^)4	P5	~6	^m7	P8	5667-566	12/(12:11:10:9:8) + 18/(12:11:10:9)	~26 ^m37^4 P415	"	"
	?	P1	vM2	~3	~4	P5	vM6	~7	P8	6675-665		P15 vM26 ~374

Dodecatonic

"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 awkward to play (all plain notes, lots of hopping between strings), or very uneven (steps of 2, 3 and 4, L/s ratio of 2). The obvious uneven scale is the harmonic duodene, with 3 fuzzy notes to avoid wolf 5ths.

subgroup	name	scale													as a chain of 5ths	step sizes
ya (2.3.5)	harmonic duodene	P1	^m2	(v)M2	^m3	vM3	P4	(v)A4	P5	^m6	vM6	(^)m7	vM7	P8	A4^m2637 m7P415M2 vM2637vA4	vvA1, m2, ^m2, (~2)	2 3 4 (5)

Is there an easily playable chromatic-sounding scale with nearly equal steps? Imagine such a scale expressed in edosteps. To avoid awkward string-hopping, we need three odd numbers and the rest even. If the even number is 8, we get the equipentatonic scales, because one-eighth of 41 is about 5. If the even number is 6, we get the equiheptatonic scales, because one-sixth of 41 is about 7. The next even number is 4, which makes a decatonic scale.

Decatonic - ten is the new twelve

The twin downminor scale consists of two downminor pentatonic scales, offset from each other by two frets. Mode #1 is (12:13:14:15:16:17:18)/12 plus (12:13:14:15:16)/8, except that prime 17 isn't well tuned.

subgroup	name	scale											as edosteps	as a chord	step sizes
yalaza (2.3.5.7.11)	twin downminor #1	P1	~2	vm3	vM3	(v)4	d5	P5	~6	vm7	vM7	P8	544-434-5444	12:13:14:15:16	m2, ^m2, ~2	3 4 5
	twin downminor #2	P1	^m2	vm3	vM3	(v)4	A4	P5	^m6	vm7	vM7	P8	454-443-4544
"	twin upmajor	P1	m2	(^)M2	^m3	^M3	d5	P5	^m6	^M6	vM7	P8	344-454-4454		"	"
"															"	"