Kite Giedraitis's Categorizations of 41edo Scales

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Kite's personal thoughts on 41-edo scales as they relate to the Kite Guitar. See also Kite Guitar Scales.

A review of 12-edo scales

There are three broad categories of 12-edo scales: pentatonic, diatonic and chromatic:

scale type --> pentatonic diatonic chromatic
scale steps M2 m3 m2 M2 (A2) A1 or m2 (M2)
semitones per scale step 2 3 1 2 (3) 1 (2)
example scale C D E G A C C D E F G A B C C Db D Eb E F F# G Ab A Bb B C
scale steps in semitones 2 2 3 2 3 2 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1

Strictly speaking, "diatonic" means a maximally even 5L2s scale, but here it's used more loosely. The maximally even requirement is relaxed. Also, occasional augmented 2nds are allowed, and the harmonic minor scale is considered to be diatonic. But in fact, common diatonic scales tend to avoid adjacent minor 2nds. Likewise, pentatonic scales tend to avoid adjacent minor 3rds. Hemitonic pentatonic scales such as C D Eb G Ab C and C E F G B C are for our purposes considered to be diatonic scales with missing notes, as are most hexatonic scales.

Some music falls entirely within one category. Much world music is entirely pentatonic, much folk music is entirely diatonic, and serial music is entirely chromatic. But often a piece of music falls into two categories. The melody of "Ash Grove" is mostly diatonic, but the occasional augmented 4th makes it somewhat chromatic. Likewise, "Greensleeves" has an occasional major 7th. The melody of "Let It Be" is mostly pentatonic, but the occasional perfect 4th makes it somewhat diatonic. The blues scale 1 b3 4 b5 5 b7 8 is likewise somewhat pentatonic and somewhat chromatic, as are most blues melodies. This overlap in categories is why diatonic scales avoid adjacent minor 2nds, because otherwise it would become a diatonic/chromatic scale.

In addition, there are unconventional scales like the whole tone, diminished and Tcherepnin scales. But these are much less common, because they don't map to a compact shape in the JI lattice. (See "Convexity and the well-formedness of musical objects", Aline Honingh and Rens Bod, Journal of New Music Research, 2005.)

Prime subgroups

Imperfect degrees in 12-edo have two qualities, major and minor, and each one implies two colors.

quality minor major
color 4thwd wa gu yo 5thwd wa
prime 3-under 5-under 5-over 3-over

12-edo accurately represents only primes 2, 3 and 5 (as well as 17 and 19, and various other higher primes). 41-edo accurately represents primes 7, 11 and 13 as well. There are 7 qualities:

quality downminor minor upminor mid downmajor major upmajor
color zo 4thwd wa gu lo/lu/tho/thu yo 5thwd wa ru
prime 7-over 3-under 5-under 11-over/under, 13-over/under 5-over 3-over 7-under

In color notation, these subgroups are named wa = 2.3 = 3-limit, ya = 2.3.5 = 5-limit, za = 2.3.7, and ila = 2.3.11. The subgroups can combine, e.g. yaza = 2.3.5.7. Note that 7-limit includes both za and yaza. 41-edo doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.

41-edo scales

41-edo has an enormous variety of scales. There are many thousands of unconventional scales, but we will focus on the ones that map compactly to the JI lattice. These are scales that contain numerous perfect 5ths. Two notes a perfect fifth apart generally have the same quality. So compact scales use only a few qualities, and thus a small prime subgroup.

In practice, 41-edo scales tend to be "fuzzy", meaning that one or two scale notes may sometimes shift by an edostep. For example, a major scale may contain both a M2 and a vM2, and use whichever one is required by the harmony at the moment.

There are five broad categories of 41-edo scales: pentatonic, diatonic, semitonal, chromatic and microtonal.

Pentatonic scales

There are four basic categories of pentatonic scales, one for each of the prime subgroups:

scale type --> wa pentatonic ya pentatonic za pentatonic ila pentatonic
scale steps M2 m3 vM2 M2 m3 ^m3 M2 ^M2 vm3 m3 ~2 M2 m3 ~3
edosteps per scale step 7 10 6 7 10 11 7 8 9 10 5 7 10 12
example scale C D E G A C C D vE G vA C C vEb F G vBb C C vvE F G vvB C
scale steps in edosteps 7 7 10 7 10 7 6 11 6 11 9 8 7 9 8 12 5 7 12 5

The za scale is the most equally distributed, thus arguably the most pentatonic-friendly of the subgroups. The za pentatonic example has an L/s ratio of only 1.29, whereas the ya example has a 1.83 ratio, and the ila example has a 2.4 ratio.

A scale needn't have every single step size on the list in order to be in the category, just most of them. In practice, a non-wa pentatonic scale will often lack a m3 step, as in the examples. But a fuzzy pentatonic scale often will have a m3, e.g. C D vE G vA/A C. Ya and za scales generally contain a wolf 5th (either an ^5 or a v5), and would often become fuzzy to avoid the wolf.

In addition to these broad categories, every 41-edo scale has a unique name that uses ups and downs. The 4 pentatonic examples above are major pentatonic, downmajor pentatonic, downminor pentatonic and double-upminor pentatonic.

These subgroups can be combined to make another four subgroups. Yala pentatonic scales tend to have wolf 5ths, and thus may be fuzzy. A yazala pentatonic scale must be fuzzy, in order to contain so many different step sizes.

scale type --> yaza pentatonic yala pentatonic zala pentatonic yazala pentatonic
scale steps vM2 M2 ^M2 vm3 m3 ^m3 ~2 vM2 M2 m3 ^m3 ~3 ~2 M2 ^M2 vm3 m3 ~3 ~2 vM2 M2 ^M2 vm3 m3 ^m3 ~3
edosteps 6 7 8 9 10 11 5 6 7 10 11 12 5 7 8 9 10 12 5 6 7 8 9 10 11 12
example C D vE G vBb C C D vE G vvB C C vEb F G vvB C C ^Eb/vvE F G vBb C
edosteps 7 6 11 9 8 (harmonics 6-10) 7 6 11 12 5 9 8 7 12 5 11/12 6/5 7 9 8

Diatonic scales

There are four basic categories of diatonic scales. In practice, a non-wa scale will often lack a m2 step, unless it's fuzzy. The ya and za diatonic scales have wolf 5ths, and thus tend to be fuzzy. The ila scale is the most equally distributed, thus arguably the most diatonic-friendly. Ya is also fairly equal. Za scales tend to have a very lopsided L/s ratio.

scale type --> wa diatonic ya diatonic za diatonic ila diatonic
scale steps m2 M2 m2 ^m2 vM2 M2 vm2 m2 M2 ^M2 m2 ~2 M2
edosteps 3 7 3 4 6 7 2 3 7 8 3 5 7
example C D E F G A B C C D vE F G vA vB C C D vEb F G vAb vBb C C vvD Eb F G vvA Bb C
edosteps 7 7 3 7 7 7 3 7 6 4 7 6 7 4 7 2 8 7 2 7 8 5 5 7 7 5 5 7

There are four additional subgroups for diatonic scales:

scale type --> yaza diatonic yala diatonic zala diatonic yazala diatonic
scale steps vm2 m2 ^m2 vM2 M2 ^M2 m2 ^m2 ~2 vM2 M2 vm2 m2 ~2 M2 ^M2 vm2 m2 ^m2 ~2 vM2 M2 ^M2
edosteps 2 3 4 6 7 8 3 4 5 6 7 2 3 5 7 8 2 3 4 5 6 7 8
example C vD vEb F G vAb vBb C C vvD ^Eb F G ^Ab Bb C C vvD Eb F G vAb vBb C C D vE ^^F G vvA vBb C
edosteps 6 3 8 7 2 7 8 5 6 6 7 4 6 7 5 5 7 7 2 7 8 7 6 6 5 5 4 8 (harmonics 8-14)

Semitonal, chromatic and microtonal scales

Most 41-edo intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. On the Kite guitar, playing a run of notes one fret apart inherits the term "chromatic" from 12-edo. This term is especially appropriate if using color notation to describe 41-edo intervals, since chromatic means "colorful". Traditional 12-edo chromaticism, which translates to runs played on every other fret, is called semitonal, a conventional term referring to the 12-edo semitone. Microtonal scales differ from fuzzy scales in having many sequential ^1 intervals, and no steps larger than a vm2. Thus fuzzy means partly but not fully microtonal, and a fuzzy diatonic scale could be called a diatonic/microtonal scale.

scale type --> semitonal chromatic microtonal
scale steps (vm2) m2 A1 (~2) vm2 m2 ^1 vm2
edosteps (2) 3 4 (5) 2 3 1 2
example C vDb vD vEb vE F Gb G... C vDb ^Db vD vEb ^Eb vE ^E... C vDb ^Db vD D ^D vEb ^Eb...
edosteps 2 4 3 4 4 3 4... 2 2 2 3 2 2 2... 2 2 2 1 1 1 2...

On the Kite guitar, going up an "even" interval (one that has an even number of edosteps) keeps one on the same string, and an "odd" one takes you to the next string. An octave spans 3 strings, thus a scale often has only 3 odd intervals. The exceptions are generally either fuzzy or awkward to play. The latter include wa, ila and zala diatonic, and microtonal scales with many ^1 steps.

From this we can deduce that chromatic scales are often 19 tones, and microtonal ones are often 22. We can also deduce that a semitonal scale of 12 notes usually has two vm2's. If there are more vm2's, the scale is semitonal/chromatic. Scales of 1, 2 and 3 edosteps are chromatic/microtonal.

Harmonic scales

In Western music, harmonies often require notes that the melody doesn't. For example, "Auld Lang Syne" has a pentatonic melody but diatonic harmonies. Often the melody is diatonic but the harmonies are at least somewhat chromatic. The score will have accidentals in the piano part but not the vocal part. The scale used by the melody is the melodic scale, and the one used to construct chords is the harmonic scale. 41-edo yaza harmonic scales are usually semitonal or chromatic.

In 12-edo, a song is generally in a major or minor key, and uses a major or minor scale. A ya piece in 41-edo often is as well. But unlike 12-edo, 41-edo allows the use of yaza chords such as 4:5:6:7. If this is one's tonic chord, both major and minor are used simultaneously. A simple Iv7 - IVv7 progression has both a downmajor 3rd and a downminor 3rd. Clearly the major/minor duality no longer applies. Instead, there is an up/down duality.

For ya or za scales, one chooses a 7-note subset of the 12 notes, and lets the imperfect degrees be either major or minor, or some combination. For yaza scales, choose a 12-note subset, and let all but the tonic, 4th and 5th be either upped or downed. (The M2 and m7 may also be plain.) Up is utonal and down is otonal. Combining upped and downed intervals in a 41-edo scale creates double-up and double-down intervals, i.e. mid intervals. This increases the odd limit and/or the prime limit, so scales tend not to mix up and down.

Harmonic scales aren't played sequentially to create melodies, and having more than 3 odd intervals isn't awkward. Often a harmonic scale is fuzzy, and uses pitch shifts of one edostep. Such a scale could be classified as diatonic/microtonal or semitonal/microtonal.

ya: 5-over maps to major, so major is more otonal than minor, and for a scale using A B C D E F G, C is the obvious tonic.

za: 7-over maps to minor, so minor is otonal, and for a C D E G A scale, A is the obvious tonic.

ila: 11-over and 11-under both map to neutral, so no obvious tonic.

yaza: D is the obvious tonic for either scale

41-edo MOS scales

Mathematically, 41edo has 20 edo-distinct temperaments, and each one has infinite MOS scales. This table only lists musically useful ones. MOS scales listed are those with:

  • 5-13 notes
  • s >= 2
  • L/s <= 4.5
  • L/s <= 2 if 1L
Table of Temperaments by generator
edosteps Cents Temperament(s) Pergen MOS Scales L s moves
1 = ^1 29.27 Sepla-sezo = [-100 33 0 17 (P8, P4/17) lopsided, s=1
2 = vm2 58.54 Hemimiracle (P8, P5/12) 20 notes
3 = m2 87.80 88cET (approx), Octacot (P8, P5/8) 13 = 1L 12s 5 3 -5, -4
4 = ^m2 117.07 Miracle (P8, P5/6) 10 = 1L 9s 5 4 +2, -4
5 = ~2 146.34 Bohlen-Pierce, Bohpier (P8, P12/13) 8 = 1L 7s

9 lopsided

6 5 +3, -4
6 = vM2 175.61 Tetracot, Bunya, Monkey (P8, P5/4) 7 = 6L 1s

13 lopsided

6 5 +3, -4
7 = M2 204.88 Baldy (P8, c3P4/20) 6 = 5L 1s

11 lopsided

7 6 +3, -3
8 = ^M2 234.15 Rodan, Guiron (P8, P5/3) 5 = 1L 4s

6, 11 lopsided

9 8 +4, -2
9 = vm3 263.41 Septimin (P8, ccP4/11) 5 = 4L 1s

9 = 5L 4s

9 5

5 4

-2, -4

+2, -4

10 = m3 292.68 Quasitemp (P8, c3P4/14) lopsided, s=1
11 = ^m3 321.95 Superkleismic (P8, ccP4/9) 7 = 4L 3s

11 = 4L 7s

8 3

5 3

+4, -5

-5, -4

12 = ~3 351.22 Hemififths, Karadeniz (P8, P5/2) 7 = 3L 4s

10 = 7L 3s

7 5

5 2

-4, -3

+1, -3

13 = vM3 380.49 Magic (Latrizo), Witchcraft (P8, P12/5) 10 = 3L 7s

13 = 3L 10s

9 2

7 2

+1, -2

+1, -3

14 = M3 409.76 Hocus (P8, c3P4/10) lopsided, s=1
15 = ^M3 439.02 Sasa-tritribizo = [5 -35 0 18 (P8, c6P5/18) 5 = 3L 2s

8 = 3L 5s

11 = 8L 3s

11 4

7 4

4 3

+2, -1

+2, -3

+2, -5

16 = v4 468.29 Barbad (P8, c7P4/19) 5 = 3L 2s

8 = 5L 3s

13 = 5L 8s

9 7

7 2

5 2

-2, -3

+1, -3

+1, -4

17 = P4 497.56 Schismatic (Helmholtz,

Garibaldi, Cassandra)

(P8, P5) 5 = 2L 3s

7 = 5L 2s

12 = 5L 7s

10 7

7 3

4 3

+5, -3

-3, -5

+2, -5

18 = ^4 526.83 Trismegistus (P8, c6P5/15) 5 = 2L 3s

7 = 2L 5s

9 = 7L 2s

13 5

8 5

5 3

-4, -0

+4, -4

-4, -5

19 = ~4 556.10 Sasa-quadquadlu = [57 -1 0 0 -16 (P8, c7P4/16) 7 = 2L 5s

9 = 2L 7s

11 = 2L 9s

13 = 2L 11s

13 3

10 3

7 3

4 3

-5, -0

+5, -5

-5, -3

+2, -3

20 = d5 585.37 Pluto (P8, c3P4/7) lopsided, s=1

The moves column is explained Most MOS scales either lack a perfect 5th or are awkward to play on the Kite Guitar. Awkward scales require more than 3 string hops per octave, or moves by more than 4 frets.

The Laquinyo scales use +1 and make a solid block.

The Sasa-tritribizo scales use +2 and -1 / -3 / -5 and make a "checkerboard" pattern on the fretboard.

--1 = 15 = ^M3 -0 = 13 = vM3 -1 = 11 = ^m3 -2 = 9 = vm3 -3 = 7 = M2 -4 = 5 = ~2 -5 = 3 = m2
+1 = 2 = vm2 vminor Sakura

5 = 2x15 + 7 + 2x2

double harmonic vminor

7 = 2x13 + 7 + 4x2

solid block

7 = 3L 4s

solid block

10 = 3L 7s

solid block

13 = 3L 10s

solid block

16 = 3L 13s

solid block

19 = 3L 16s

+2 = 4 = ^m2 90-degree zigzag

Sakura scale: 7,4,13-4,13

Indochinese scale

5 = 2L 1m 2s

checkerboard

5 = 3L 2s

alternate frets

double harmonic

7 = 2L 1m 4s

m = 7 = -3

checkerboard

8 = 3L 5s

alternate frets

twin vminor

10 = 2x5 + 7x4 + 3

9 = 6 + 3x5 + 5x4

65445-4454

checkerboard

11 = 8L 3s

+3 = 6 = vM2 ya pentatonic

5 = 2L 1m 2s

m = 7 = -3

diagonal lines

5 = 3x9 + 8 + 6

3L 1m 1s = 4L 1s

every 3rd fret

whole-tone

6 = 1XL 3L 2s

XL = 8 = +4

ya equi-hepta

7 = 1XL 4L 2s

XL = 7 = -3

+4 = 8 = ^M2 za pentatonic

5 = 2L 2s 1xs

xs = 7 = -3

??????

6 = 2x8 + 3x7 + 4

787-874

5 = 787-8,11

every 4th fret

dots only

6 = 10 + 2x8 + 3x5

7 = 2x8 + 3x5 + 6 + 4

6585-854

7 = 4L 3s
+5 = 10 = m3 5 = 2L 3s Bohlen-Pierce

9 = 4L 5s = P12

penta: 8 8 8 8 9 --> 9 9 7 8 8, 9 9 9 8 6, 7 7 7 10 10

hexa: 7 7 7 7 7 6 --> 7 7 7 6 6 8, 7 7 9 6 6 6, 9 9 7 6 6 4, ...

hepta: 6 6 6 6 6 6 5 --> 5 5 7 6 6 6 6, 7 7 7 6 6 4 4

octa: 5 5 5 5 5 5 5 6 --> 5 5 5 6 6 6 4 4

nona: 5 5 5 5 5 4 4 4 4 --> 5 5 5 4 4 4 4 4 6

deca: 4 4 4 4 4 4 4 4 4 5 --> 5 5 3 4 4 4 4 4 4 4

eleven: 4 4 4 4 4 4 4 4 3 3 3

twelve: 4 4 4 4 4 3 3 3 3 3 3 3 --> 3 3 3 4 4 4 4 4 4 4 2 2

Double harmonic: 4947-494

C ^Db vE F G ^Ab vB C

Cv CvM7

^Dbv, ^Dbv7, ^DbvM7, ^Dv7(v5), ^Dbvm, ^Dbvm7, ^Dbvm7(b5)

vE^m, vE^m6,

F^m, F^m,vM7