13edo scales

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Since 13 is prime, 13edo has more MOS scale types than 12edo; somewhat amazingly, all of them have a good number of consonant chords if you know where to look.

Overview

The intervals themselves are not very alien except near the middle of the octave, just a bit darker compared to their 12edo counterparts. They are familiar enough that they can be given pseudo-diatonic names:

Degree Cents Pseudo-Diatonic Category
0 0.00 Unison (P1)
1 92.31 Minor second (m2)
2 184.62 Major second (M2)
3 276.92 Minor third (m3)
4 369.23 Major third (M3), Diminished fourth (d4)
5 461.54 Minor fourth (m4)
6 553.85 Major fourth (M4), Minor tritone (mᴛ)
7 646.15 Minor fifth (m5), Major tritone (Mᴛ)
8 738.46 Major fifth (M5)
9 830.77 Minor sixth (m6), Augmented fifth (A5)
10 923.08 Major sixth (M6)
11 1015.38 Minor seventh (m7)
12 1107.69 Major seventh (M7)
13 1200.00 Octave (P8)

My (Inthar lus Lăneaf's) subjective perception of the relative consonance of different intervals from the most consonant to the most dissonant (octave equivalents are not taken into account):

  • Basals (the most consonant): major second, major and minor thirds
  • Glitterers (intermediate, buzzy consonance): major and minor fourths, major and minor sixths, major and minor sevenths, minor ninth
  • Flarers (the most dissonant): minor and major fifths, the most dissonant and categorically ambiguous intervals.

Cheat sheet of important MOS scale types with L = major second, s = minor second:

MOS type Generator Most common consonant triad Most common consonant tetrad(s)
archeotonic (LLLLLLs) major second 4:5:9 4:5:9:11, 4:5:9:13
oneirotonic (LLsLsLLs) minor fourth 4:9:21. Also important: 4:5:9 and its minor counterpart 4:19:9. Basic triads with added 6ths and 7ths
nonatonic (LsLsLsLss) minor third 4:11:13 4:11:13:19, 4:9:11:13
decatonic (LssLsssLss) major third 4:5:13 4:5:13:21

Archaeotonic (6L 1s)

The archaeotonic scale is one of the two pseudo-diatonic scale types in 13edo. It is overall brighter, more "majory" and more concordant than the oneirotonic scale: there are more 4:5:9 chords and chords involving the 11th and 13th harmonics.

Being a 7-note scale, the unison to octave interval categories remain the same as in the diatonic scale, except that we now have major fourths (6\13, approx. 11/8) and minor fourths (5\13, approx. 21/16), and their inversions minor and major fifths. An interesting feature is that you can switch whether you perceive an interval as minor or major by approaching it from opposite directions: for example, a minor sixth can be made to sound like a diatonic major sixth by walking up whole-whole-half-whole-whole steps from the tonic or like a diatonic minor sixth by walking down two whole steps.

Scale

Sortable table of intervals in the Lobonian mode. (Harmonics are in bold; this is useful for seeing a chord's complexity when you sort the intervals according to the generator chain.)

Degree Cents Note name on J Approximate ratios #Gens up
1, 8 0.00, 1200.00 J 1/1, 2/1 0
2 184.62 K 9/8, 10/9, 11/10, 19/17, 21/19 +1
3 369.23 L 5/4, 11/9, 16/13, 26/21 +2
4 553.85 M# 11/8, 18/13, 26/19 +3
5 738.46 Ob 17/11, 20/13, 26/17, 32/21 +4
6 923.08 Pb 8/5, 13/8, 18/11, 21/13 -2
7 1015.38 Qb 9/5, 16/9, 20/11, 34/19, 38/21 -1

Chords

The root-major third-major ninth (approximating 4:5:9; J-L-K in Kentaku notation) and its minor equivalent root-minor third-major ninth (J-Lb-K in Kentaku notation) may be considered equivalents of root-third-fifth chords in diatonic music. Archeotonic scales have 6 such triads, 5 "major" and 1 "minor". The 11th and 13th harmonics are also plentiful, as already noted by Cryptic Ruse; 4 roots have the 11th harmonic over them and 5 roots have the 13th harmonic over them.

The chord spelled root-major third-major fourth-minor sixth in archeotonic nomenclature occurs twice in archeotonic and I call it The Beloved Extra Special Tetrad (BEST). The reason it's beloved and extra special is that it can be interpreted both as an 8:10:11:13 and as a 13:16:18:21 (which can be revoiced as 8:9:13:21), thanks to the way 13edo conflates higher-limit JI intervals together.

Archeotonic offers fairly familiar-sounding chord progressions by major seconds, thirds, and (both major and minor) fourths. One example is root-major third-two major thirds-root (spelled J major - L major - N# major - J major in J Ryonian), where the (two major thirds) is a 21/16 minor fourth away from the root.

Modal harmony

The 7 archeotonic modes each sound like one part of the scale (the part with the unique small step) is diatonic and thus can evoke various modes of the diatonic scale. The modal harmony of the unmodified archeotonic scales is otherwise simpler than diatonic modal harmony due to the dearth of small steps. To get more complex modal harmony, you could contrast major and minor intervals of the same interval class by playing the same melody in a different mode (like you can do in porcupine), and you could make 12edo-like chromatic modifications to spice things up.

Oneirotonic (5L 3s)

The oneirotonic scale is the darker, damper, more "minory" cousin of archaeotonic. Only 2 out of 8 oneirotonic modes (Dylathian and Ilarnekian) are "major" in the sense of having a major third, and both sound pretty bittersweet.

The names I use for the oneirotonic interval classes are borrowed from diatonic interval categories: "second", "third", "fourth", "tritone" (4-step intervals), "fifth" (5-step intervals), "sixth" (6-step intervals), "seventh" (7-step intervals) and octave. You just have to remember that there's an extra category between fourths and fifths and that fourths and fifths are dissonant. Like in archeotonic you can change the perception of an interval by approaching it from different directions, but in oneirotonic it will change what diatonic interval class you hear it as: say, as both a third and a fourth, rather than both a major and a minor third.

Scale

The Dylathian mode is the most otonal mode, and is the basis for Kentaku note names JKLMNOPQJ (J is approx. 180 Hz, or an 11/8 above middle C). Sortable table of Dylathian (Harmonics are in bold; this is useful for seeing a chord's complexity when you sort the intervals according to the generator chain):

Degree Cents Note name on J Approximate ratios #Gens up
1, 9 0.00, 1200.00 J 1/1, 2/1 0
2 184.62 K 9/8, 10/9, 11/10, 19/17, 21/19 +3
3 369.23 L 5/4, 11/9, 16/13, 26/21 +6
4 461.54 M 13/10, 17/13, 21/16, 22/17 +1
5 646.15 N 16/11, 13/9, 19/13 +4
6 830.77 O 8/5, 13/8, 18/11, 21/13 +7
7 923.08 P 17/10, 12/7, 22/13, 19/11 +2
8 1107.69 Q 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 +5

Chords

Despite being melodically familiar, oneirotonic may be the most harmonically complex of the 13edo scales; the most common consonant triad is a fairly complex 4:9:21. Hence oneirotonic especially benefits from either using inharmonic timbres in addition to harmonic ones, or using a well-tempered or primodal JI version of 13edo. The availability of primes also varies greatly by mode: for example, only Dylathian, Ilarnekian and Sarnathian have a 5/4 on the tonic, and only Mnarian, Kadathian, Hlanithian and Sarnathian have an 11/8 on the tonic.

Modal harmony

How I think about the 8 oneirotonic modes:

  1. Dylathian: 2 2 1 2 2 1 2 1 (major with hints of Mixolydian and "#5")
  2. Ilarnekian: 2 2 1 2 1 2 2 1 (major with hints of "b6")
  3. Celephaïsian: 2 1 2 2 1 2 2 1 (the oneirotonic melodic minor. Very classical-sounding; Easley Blackwood's 13-note etude uses this as its home mode.)
  4. Ultharian: 2 1 2 2 1 2 1 2 (the oneirotonic Dorian)
  5. Mnarian: 2 1 2 1 2 2 1 2 (half-diminished + Dorian)
  6. Kadathian: 1 2 2 1 2 2 1 2 (Locrian + Dorian)
  7. Hlanithian: 1 2 2 1 2 1 2 2 (Locrian + natural minor)
  8. Sarnathian: 1 2 1 2 2 1 2 2 (diminished + natural minor)

Modes with sharp tritone

The brighter modes can be viewed as providing a distorted version of diatonic functional harmony. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context. Basic chord progressions can move by minor fourths, thirds, or major seconds: for example, J major-M minor-P minor-Ob major-J major (in Ilarnekian) or J major-K major-O major-M major-J major (in Dylathian).

Modes with flat tritone

The darker modes are radically different in character than the brighter modes...

Tetrachordal 8-note scales

You can also view oneirotonic as scales made of two tetrachords each spanning a minor fourth and one trichord spanning a minor third. This will let you build 13edo "tetrachordal" scales with a similar structure that is not one of the 8 modes, with tetrachord structures similar to 12edo ones. For example:

  • [2 1 1] [2 1] [1 3 1] is a kind of harmonic minor (also obtained by lowering the 7th degree of the Celephaïsian mode)
  • [1 3 1] [2 1] [1 2 2] is a kind of Phrygian dominant scale (which also contains 1 3 1 2 2 2 2, a chromatic modification of the Zo-Kalarian mode of the archeotonic scale).
    • Harmonically this will give you an 8:10:13 over the first degree, an 8:10:11 over the second degree, a "minor" key and an 8:9:10:11:13 over the fourth degree, an 8:9:10:11 over the fifth degree and an 8:9:10:13 over the seventh degree.
    • Melodically you can play tricks by going up 5 scale steps which will be a fifth instead of a sixth, the same note as down 3 steps.

Samples

‎(A rather classical-sounding 3-part harmonization of the ascending J Ilarnekian scale; tuning is 13edo)

Switching between archeo- and oneirotonic

Pseudo-diatonic music in 13edo can easily use both archeotonic and oneirotonic, switching back and forth between a 7-note mode and a corresponding 8-note one as the situation requires.

Twin modes

The most obvious way to do this is to exploit the fact that an oneirotonic mode and an archeotonic mode based on the same tonic may share up to 6 notes; replace the "1 2 1" in the oneirotonic mode with a "2 2". Six of the 8 oneirotonic modes have a "twin" archeotonic mode that keeps the same tonic, listed below from brightest to darkest:

Oneiro-Archeo Twin Modes
Oneirotonic Archeotonic
Dylathian 2 2 1 2 2 1 2 1 Oukranian 2 2 1 2 2 2 2
Ilarnekian 2 2 1 2 1 2 2 1 Ryonian 2 2 2 2 2 2 1
Ultharian 2 1 2 2 1 2 1 2 Tamashian 2 1 2 2 2 2 2
Mnarian 2 1 2 1 2 2 1 2 Karakalian 2 2 2 2 2 1 2
Hlanithian 1 2 2 1 2 1 2 2 Zo-Kalarian 1 2 2 2 2 2 2
Sarnathian 1 2 1 2 2 1 2 2 Lobonian 2 2 2 2 1 2 2

This operation might change the mood of the scale drastically. For example, Mnarian, a "minor" mode, becomes Karakalian, a "major" mode.

Nonatonic (4L 5s)

Generated by 3\13, the 276.9-cent minor third approximating 13/11, this scale sounds a little like the octatonic scale in 12edo with an extra small step inserted. Two of these make an 11/8 and three make a 13/8, making this scale very good for 4:11:13 triads. (In terms of regular temperament theory, this makes 13edo a tuning for the bithotrilu temperament that tempers out the bithotrilu comma 1352/1331 = [3 0 0 0 -3 2, aka "lovecraft temperament".) 17edo also supports bithotrilu temperament and thus has a similar 4L 5s scale, generated by the 4\17 minor third. Similar scales also exist in 22edo and 31edo with flatter generators, but they use a different temperament and won't approximate the 13th harmonic.

Scale

The brightest mode is LsLsLsLss or 0-2-3-5-6-8-9-11-12-13. The triad 4:11:13 occurs on degrees 1, 2, 3, 5, 7 and 9; these can be extended to either 4:10:11:13:17, 4:9:11:13:21, or 4:5:9:11:13 depending on what degree you're on. Since you get 21/16 as the minor version of 11/8, you also get two 8:13:17:21's with the same interval classes, on degrees 6 and 8. Degree 4 has a 4:5:11.

Sortable table of LsLsLsLss (Harmonics are in bold; this is useful for seeing a chord's complexity when you sort the intervals according to the generator chain.):

Degree Cents Note name on J Approximate ratios # generators up
1, 10 0.00, 1200.00 J 1/1, 2/1 0
2 184.62 K 9/8, 10/9, 11/10, 19/17, 21/19 +5
3 276.92 Lb 7/6, 13/11, 20/17, 19/16, 22/19 +1
4 461.54 M 13/10, 17/13, 21/16, 22/17 +6
5 553.85 M#/Nb 11/8, 18/13, 26/19 +2
6 738.46 N#/Ob 17/11, 20/13, 26/17, 32/21 +7
7 830.77 O 8/5, 13/8, 18/11, 21/13 +3
8 1015.38 P# 9/5, 16/9, 20/11, 34/19, 38/21 +8
9 1107.69 Q 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 +4

Modal harmony

The nonatonic scale can be thought of as a (chromatically altered) octatonic scale with one note added.

The 6-generators-up mode 0-2-3-5-6-7-9-10-12-13 contains an octatonic MODMOS 0-2-3-5-7-9-10-12-13 (Celephaïsian with a sharpened 6th degree).

Musical examples

Brusselator Sprouts (by Xotla) (The main riffs are in this scale, although key changes and notes outside the 9 note subset are used too.)

Decatonic (3L 7s)

The decatonic scale is excellent for 4:5:13 triads. It's generated by a major third, and two of them span a 4:5:13 triad, spanning degrees 1-4-8, and three of them span a 4:5:13:21 tetrad. This means that 8 of the 10 degrees have a 4:5:13 triad, and 7 of them in turn have a 4:5:13:21.

The following is a sortable table of LsssLssLss. (Harmonics are in bold; this is useful for seeing a chord's complexity when you sort the intervals according to the generator chain.) If you want an 11/8 instead of a 21/16 you can sharpen the 5th degree to get LssLsssLsss which is the only mode to have a 4:5:9:11:13 on the tonic.

Degree Cents Note name on J Approximate ratios # generators up
1, 11 0.00, 1200.00 J 1/1, 2/1 0
2 184.62 K 9/8, 10/9, 11/10, 19/17, 21/19 +7
3 276.92 Lb 7/6, 13/11, 20/17, 19/16, 22/19 +4
4 369.23 L 5/4, 11/9, 16/13, 26/21 +1
5 461.54 M 13/10, 17/13, 21/16, 22/17 -2
6 646.15 N 16/11, 13/9, 19/13 +5
7 738.46 Ob 17/11, 20/13, 26/17, 32/21 +2
8 830.77 O 8/5, 13/8, 18/11, 21/13 -1
9 1015.38 P# 9/5, 16/9, 20/11, 34/19, 38/21 +6
10 1107.69 Q 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 +3

Melodic properties

The decatonic scale can be viewed as containing archeotonic or oneirotonic scales with possible chromatic alterations, containing different degrees of some intervals. For example, the LsssLssLss mode can be viewed as:

  • From a 7-tone POV: P1 - M2 - m3~M3 - m4 - m5~M5 - m6 - m7~M7 - P8
  • From an 8-tone POV: P1 - M2 - m3~M3 - m4 - Mᴛ - M5 - m6 - m7~M7 - P8 or P1 - M2 - m3~M3 - m4 - Mᴛ - M5~A5 - A6 - M7 - P8, thus containing chromatically altered versions of Dylathian, Ilarnekian, Ultharian, and Celephaïsian

The 3113131 subset (P1-A2-M3-m4-M5-m6-M7-P8), is more important from the regular temperament POV, in that you can modulate up by major thirds by using 4:5:13 triads on it. The subset is also melodically interesting and pleasing.

Other stuff

todo: try added fifths or tritones, describe chords with two additions or more