13edo scales

Inthar's thoughts on 13edo

Suggested tuning standard

J (F#) = 180 Hz

Archaeotonic

The 6L 1s archaeotonic scale 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, and extremely dissonant intervals such as 16/11 and 32/21 are less common.

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.

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 basic harmonic triads approximately like 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 4 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 Chord. The reason it is 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.

Scales and 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 MOS scales is otherwise simpler than in diatonic due to the dearth of small steps. To get more interesting modal harmony you can 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 can make 12edo-like chromatic modifications to spice things up.

Oneirotonic

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.

Chords

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. Perfect 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.

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 provides 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. One could use it to 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.

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. Perfect fourths and minor tritones are more consonant than their inversions major tritones and perfect 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) serve as an extra 5/4 in the scale and can be part of extra consonant chords (such as the aforementioned Beloved Extra Special Chord representing both 8:10:11:13 and 13:16:18:21, which occurs as O-J-K-M in J Ilarnekian, but it only represents 13:16:18:21 in other oneirotonic-supporting tunings such as 31edo).

Basic chord progressions can move by perfect fourths or major seconds: J major-M minor-P minor-O major-J major (in Ilarnekian) or J major-K major-O# major-M major-J major (in Dylathian)

Scales and modal harmony

You can also view oneirotonic as scales made of two tetrachords each spanning a 21/16 and one trichord spanning a 7/6. 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)

Chords

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