Kite Giedraitis's Categorizations of 41edo Scales

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

A review of 12-equal scales

There are three broad categories of 12-equal 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-equal 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-equal accurately represents only primes 2, 3 and 5 (as well as 17 and 19, and various other higher primes). 41-equal 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-equal doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.

41-equal scales

41-equal 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-equal 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-equal scales: pentatonic, diatonic, semitonal, trientonal and microtonal. The three latter ones fall under the general category of chromatic.

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 an off-5th (either an ^5 or a v5), and would often become fuzzy to avoid the wolf.

In addition to these broad categories, every 41-equal 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. Rotating these scales makes the minor (wa), upminor (ya), upmajor (za) and double-downmajor (ila) pentatonic scales.

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)

Chromaticism: semitonal, trientonal and microtonal scales

Most 41-equal intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. Traditional 12-equal chromaticism, which translates to runs played on every other fret, is called semitonal, a conventional term referring to the 12-equal semitone. Playing a run of notes one fret apart is called trientonal, which means "by third-tones". In a guitar context, it can be called fretwise. 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. Chromatic is an umbrella term that includes semitonal, trientonal/fretwise and microtonal.

scale type --> semitonal trientonal or fretwise 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 trientonal/fretwise 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/trientonal. Scales of 1, 2 and 3 edosteps are trientonal/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-equal yaza harmonic scales are usually semitonal or trientonal.

In 12-equal, a song is generally in a major or minor key, and uses a major or minor scale. A ya piece in 41-equal often is as well. But unlike 12-equal, 41-equal 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 12-equal ya 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 41-equal 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-equal scale creates double-up and double-down intervals, i.e. mid intervals. This increases the odd limit and/or the prime limit, so yaza 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-equal MOS scales

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. Moves are explained in Kite Guitar Scales.

We can find all non-awkward MOS scales by requiring that one step size be an odd number of edosteps and the other be even, and further requiring that there are exactly 3 of the first step size. Then we simply make a table with odd step sizes on the top and even ones on the side. Not all odd/even combinations make a MOS scale, because 41 minus the 3 odd steps isn't always a multiple of the even step size. In those cases a 3rd step size is used once. It's named either XL or xs or m. Often there is more than one 3rd step possible. Alternatively, we can avoid the 3rd step size by allowing non-octave scales, as in the Bohlen-Pierce scale in the bottom row.

Each column header is a string-hopping move. The first column heading is "-5 = 3\41 = m2", which means that you go back 5 frets when hopping, which equals 3 edosteps, which equals a plain minor 2nd. Each row header is a string-sliding move in a similar format. "+1 = 2\41 = vm2" means go up 1 fret = 2\41 = a downmiinor 2nd.

Geometrically, each column header defines a diagonal line, except the vM3 column which defines a line parallel to the frets. Each row header says how many frets apart these lines are. These geometrical patterns make (possibly non-octave) MOS scales. Often the first line of each table entry describes this geometry. Once one masters these geometrical patterns, one can flit about the fretboard and use MOS scales to quickly span large intervals.

TO DO: make fretboard diagrams of these geometrical patterns.

The Laquinyo scales use +1 moves and make a solid block on the fretboard. The Laquinyo generator is a vM3.

The Sasa-tritribizo scales use +2 moves and -1 / -3 / -5 moves and make a "checkerboard" pattern on the fretboard. The Sasa-tritribizo generator is an ^M3.

-5 = 3\41 = m2 -4 = 5\41 = ~2 -3 = 7\41 = M2 -2 = 9\41 = vm3 -1 = 11\41 = ^m3 -0 = 13\41 = vM3 --1 = 15\41 = ^M3
+1 = 2\41 = vm2

Laquinyo

(P8, P12/5)

solid block

gen = vM3 3L 16s = 19

L=3, s=2

solid block

gen = vM3 3L 13s = 16

L=5, s=2

solid block

gen = vM3 3L 10s = 13

L=7, s=2

solid block

gen = vM3 3L 7s = 10

L=9, s=2

solid block

gen = vM3 3L 4s = 7

L=11, s=2

double harmonic vminor

2L 1m 4s = 7

L=13, m=7, s=2

vminor Sakura

2L 1m 2s = 5

L=15, m=7, s=2

+2 = 4\41 = ^m2

Sasa-tritribizo

checkerboard

gen = ^M3 8L 3s = 11

L=4, s=3

alternate frets

2L 7s 1xs = 10

L=5, s=4, xs=3

twin vminor pentatonic

1XL 3L 5s = 9

XL=6, L=5, s=4

65445-4454

checkerboard

gen = ^M3 3L 5s = 8

L=7, s=4

alternate frets

2L 1m 4s = 7

L=9, m=7, s=4

double harmonic:

4947-494

checkerboard

gen = ^M3 3L 2s = 5

L=11, s=4

90-degree zigzag

2L 1m 2s = 5

L=13, m=7, s=4

^minor Sakura scale:

7,4,13,4,13

Indochinese scale:

13,4,7,13,4

+3 = 6\41 = vM2

Saquadyo

(P8, P5/4)

1XL 4L 3s = 8

XL=8, L=6, s=3

ya equi-hepta

1XL 4L 2s = 7

XL=7, L=6, s=5

every 3rd fret

whole-tone

1XL 3L 2s = 6

XL=8, L=7, s=6

diagonal lines

3x9 + 8 + 6 = 5

3L 1m 1s = 4L 1s

ya pentatonic

2L 1m 2s = 5

L=11, m=7, s=6

+4 = 8\41 = ^M2

Latrizo

(P8, P5/3)

diagonal lines

gen = ^m3 4L 3s = 7

L=8, s=3

every 4th fret

dots only

1XL 2L 3s = 6

XL=10, L=8, s=5

2L 1m 3s 1xs = 7

L=8, m=6, s=5, xs=4

6585-854

2L 3s 1xs = 6

L=8, s=7, xs=4

787-874

1XL 2L 2s = 5

XL=11, L=8, s=7

787-8,11

za pentatonic

2L 2s 1xs = 5

L=9, s=8, xs=7

+5 = 10\41 = m3 every 5th fret

3L 1m 2s = 6

L=10, m=5, s=3

Bohlen-Pierce

P12 = 4L 5s = 9

L=10, s=5

wa pentatonic

gen = P5 2L 3s = 5

L=10, s=7

(Notes to myself)

Changing awkward very-near-equal scales to non-awkward less-equal scales:

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

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

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

octatonic: 5 5 5 5 5 5 5 6 --> 5 5 5 6 6 6 4 4

nonatonic: 5 5 5 5 5 4 4 4 4 --> 5 5 5 4 4 4 4 4 6

decatonic: 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 -- not awkward!!!

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


The double harmonic scale C ^Db vE F G ^Ab vB C has these chords:

Cv CvM7

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

vE^m, vE^m6,

F^m, F^m,vM7

Mathematically, 41-equal 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 there's only one L
Table of 41-equal 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