13edo scales
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 | 184.615 | 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 | 1015.385 | Minor seventh (m7) |
| 12 | 1107.69 | Major seventh (M7) |
| 13 | 1200 | Octave (P8) |
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, 1200 | J | 1/1, 2/1 | 0 |
| 2 | 184.615 | 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 | 1015.385 | 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, 1200 | J | 1/1, 2/1 | 0 |
| 2 | 184.615 | 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.
Like in archaeotonic, seconds and thirds are similar in consonance to 12edo seconds and thirds, and similarly sixths and sevenths are similar to diatonic sixths and sevenths. Minor fourths (21/16) are dissonant, but they work a lot like diatonic perfect fourths do e.g. in "sus24" chords that resolve down to thirds, and can also be spread out to make convincing 4:9:21 chords which are common in oneirotonic.
Minor tritones (approximating 11/8) work like tritones and they like to resolve inward to a third. Major tritones (16/11) are the opposite: they like to resolve outward to a sixth. Unlike in 12edo, fourths and tritones, and their octave inversions are very different in quality. Minor fourths and minor tritones are more consonant than their inversions major tritones and major fifths; they can also both be spread out to make them more consonant, whereas their inversions cannot.
The diminished fourth can work either like the diatonic diminished fourth, or (uniquely in 13edo among all oneirotonic tunings) serve as an extra 5/4 in the scale and can be part of extra consonant chords (such as the aforementioned BEST that represents both 8:10:11:13 and 13:16:18:21, which occurs as Ob-J-K-M in J Ilarnekian, but it only represents 13:16:18:21 in other oneirotonic-supporting tunings such as 31edo).
As in archeotonic harmony, root-third-ninth chords may be considered basic harmonic triads; oneirotonic scales have 5 such triads, 2 "major" and 3 "minor". J-L-K (4:5:9) and its minor counterpart J-Lb-K work well with an added sixth or seventh, even when the resulting chord does not approximate an obvious JI chord.
Oneirotonic 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).
Oneirotonic provides two Orwell tetrads, made of three stacked minor thirds making one minor sixth; we get them by taking every second degree of the scale, JLNP or KMOQ. They sound like squashed diminished chords, but not quite. One could play an earbending trick where a movement up a major third and up 3 minor thirds will get you back to where you started unlike in 12edo. The two Orwell tetrads contain the two copies of 8:11:13 in oneirotonic, Q-M-Ob or L-P-J in J Ilarnekian
Modal harmony
How I think about the 8 oneirotonic modes:
- Dylathian: 2 2 1 2 2 1 2 1 (major with hints of Mixolydian and "#5")
- Ilarnekian: 2 2 1 2 1 2 2 1 (major with hints of "b6")
- 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.)
- Ultharian: 2 1 2 2 1 2 1 2 (the oneirotonic Dorian)
- Mnarian: 2 1 2 1 2 2 1 2 (half-diminished + Dorian)
- Kadathian: 1 2 2 1 2 2 1 2 (Locrian + Dorian)
- Hlanithian: 1 2 2 1 2 1 2 2 (Locrian + natural minor)
- Sarnathian: 1 2 1 2 2 1 2 2 (diminished + natural minor)
Modes with a flat 5th degree sound weirder to diatonic ears, but ironically this can sound more stable since the flat 5th degree approximates an 11/8. Mnarian has a 4:9:11 on the tonic, Hlanithian has an 8:11:13:17 on the tonic, and Sarnathian has the Beloved Extra Special Tetrad 4:5:11:13 on the tonic. Ironically Sarnathian makes the 5/4 in the chord sound very dark since it's a diminished fourth, rather than a major third.
Hlanithian also has the tetrad 8:13:17:21 on the tonic (in the 13edo tuning).
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:
| 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, 1200 | J | 1/1, 2/1 | 0 |
| 2 | 184.615 | 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 | 1015.385 | 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 |
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 LsssLssLss mode is the only mode that has a 4:5:9:13:21 on the tonic. 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. 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.):
| Degree | Cents | Note name on J | Approximate ratios | # generators up |
|---|---|---|---|---|
| 1, 11 | 0, 1200 | J | 1/1, 2/1 | 0 |
| 2 | 184.615 | 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 | 1015.385 | 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. Alternatively, it can be viewed as an extension of Magic[7] (generated by a major third).
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
Magic[7], 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