Kite Giedraitis's Categorizations of 41edo Scales: Difference between revisions

!scale steps

|M2

|m3

|m2

|M2

|(A2)

|A1 or m2

|(M2)

!semitones per scale step

|2

|3

|1

|2

|(3)

|1

|(2)

!example scale

| colspan="2" |C D E G A C

| colspan="3" |C D E F G A B C

| colspan="2" |C Db D Eb E F F# G Ab A Bb B C

!scale steps in semitones

| colspan="2" |2 2 3 2 3

| colspan="3" |2 2 1 2 2 2 1

| colspan="2" |1 1 1 1 1 1 1 1 1 1 1 1

Imperfect degrees in 12-equal have two qualities, major and minor, and each one implies two [[Color notation|colors]].  

!color

|4thwd wa

|gu

|yo

|5thwd wa

!prime

|3-under

|5-under

|5-over

|3-over

!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

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.

!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

| colspan="2" |C D E G A C

| colspan="4" |C D vE G vA C

| colspan="4" |C vEb F G vBb C

| colspan="4" |C vvE F G vvB C

!scale steps in edosteps

| colspan="2" |7 7 10 7 10

| colspan="4" |7 6 11 6 11

| colspan="4" |9 8 7 9 8

| colspan="4" |12 5 7 12 5

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. [[Kite Guitar Exercises and Techniques by Kite Giedraitis|Rotating]] these scales makes the minor (wa), upminor (ya), upmajor (za) and double-downmajor (ila) pentatonic scales.  

!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

| colspan="6" |C D vE G vBb C

| colspan="6" |C D vE G vvB C

| colspan="6" |C vEb F G vvB C

| colspan="8" |C ^Eb/vvE F G vBb C

!edosteps

| colspan="6" |7 6 11 9 8  (harmonics 6-10)

| colspan="6" |7 6 11 12 5

| colspan="6" |9 8 7 12 5

| colspan="8" |11/12 6/5 7 9 8

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 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

| colspan="2" |C D E F G A B C

| colspan="4" |C D vE F G vA vB C

| colspan="4" |C D vEb F G vAb vBb C

| colspan="3" |C vvD Eb F G vvA Bb C

!edosteps

| colspan="2" |7 7 3 7 7 7 3

| colspan="4" |7 6 4 7 6 7 4

| colspan="4" |7 2 8 7 2 7 8

| colspan="3" |5 5 7 7 5 5 7

!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

| colspan="6" |C vD vEb F G vAb vBb C

| colspan="5" |C vvD ^Eb F G ^Ab Bb C

| colspan="5" |C vvD Eb F G vAb vBb C

| colspan="7" |C D vE ^^F G vvA vBb C

!edosteps

| colspan="6" |6 3 8 7 2 7 8

| colspan="5" |5 6 6 7 4 6 7

| colspan="5" |5 5 7 7 2 7 8

| colspan="7" |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 steps

|(vm2)

|m2

|A1

|(~2)

|vm2

|m2

|^1

|vm2

!edosteps

|(2)

|3

|4

|(5)

|2

|3

|1

|2

!example

| colspan="4" |C vDb vD vEb vE F Gb G...

| colspan="2" |C vDb ^Db vD vEb ^Eb vE ^E...

| colspan="2" |C vDb ^Db vD D ^D vEb ^Eb...

!edosteps

| colspan="4" |2 4 3 4 4 3 4...

| colspan="2" |2 2 2 3 2 2 2...

| colspan="2" |2 2 2 1 1 1 2...

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.   

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.   

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.

==41-equal MOS 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.  

|+

!

!-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

|solid block

|solid block

|solid block

|solid block

|solid block

|double harmonic vminor

|vminor Sakura

|checkerboard

|alternate frets

|checkerboard

|alternate frets

|checkerboard

|90-degree zigzag

|

|1XL 4L 3s = 8

|ya equi-hepta

|every 3rd fret

|diagonal lines

|ya pentatonic

|

|

|diagonal lines

|every 4th fret

|2L 3s 1xs = 6

|za pentatonic

|

|

|

|every 5th fret

|Bohlen-Pierce

|wa pentatonic

|

|

|

|

=== (Notes to myself) ===

|+Table of 41-equal Temperaments by generator

!edosteps

!Cents

!Temperament(s)

![[Pergen]]

!MOS Scales

!L s

!moves

|1 = ^1

|29.27

|Sepla-sezo = {{monzo| -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

|6 5

| +3, -4

|6 = vM2

|175.61

|[[Tetracot]], [[Bunya]], [[Monkey]]

|(P8, P5/4)

|7 = 6L 1s

|6 5

| +3, -4

|7 = M2

|204.88

|[[Baldy]]

|(P8, c³P4/20)

|6 = 5L 1s

|7 6

| +3, -3

|8 = ^M2

|234.15

|[[Rodan]], [[Guiron]]

|(P8, P5/3)

|5 = 1L 4s

|9 8

| +4, -2

|9 = vm3

|263.41

|[[Septimin]]

|(P8, ccP4/11)

|5 = 4L 1s

|9 5

| -2, -4

|10 = m3

|292.68

|[[Quasitemp]]

|(P8, c³P4/14)

|lopsided, s=1

|

|

|11 = ^m3

|321.95

|[[Superkleismic]]

|(P8, ccP4/9)

|'''7 = 4L 3s'''

|'''8 3'''

|12 = ~3

|351.22

|[[Hemififths]], [[Karadeniz]]

|(P8, P5/2)

|7 = 3L 4s

|7 5

| -4, -3

|13 = vM3

|380.49

|[[Magic|Magic (Latrizo)]], [[Witchcraft]]

|(P8, P12/5)

|'''10 = 3L 7s'''

|'''9 2'''

|'''+1, -2'''

|14 = M3

|409.76

|[[Hocus]]

|(P8, c³P4/10)

|lopsided, s=1

|

|

|15 = ^M3

|439.02

|Sasa-tritribizo = {{monzo| 5 -35 0 18 }}

|(P8, c⁶P5/18)

|'''5 = 3L 2s'''

|'''11 4'''

|'''+2, -1'''

|16 = v4

|468.29

|[[Barbad]]

|(P8, c⁷P4/19)

|5 = 3L 2s

|9 7

| -2, -3

|17 = P4

|497.56

|[[Schismatic]] ([[Helmholtz]],

|(P8, P5)

|5 = 2L 3s

|10 7

| +5, -3

|18 = ^4

|526.83

|[[Trismegistus]]

|(P8, c⁶P5/15)

|5 = 2L 3s

|13 5

| -4, -0

|19 = ~4

|556.10

|Sasa-quadquadlu = {{monzo| 57 -1 0 0 -16 }}

|(P8, c⁷P4/16)

|7 = 2L 5s

|13 3

| -5, -0

|20 = d5

|585.37

|[[Pluto]]

|(P8, c³P4/7)

|lopsided, s=1

|

|