13edo: Difference between revisions

'''13EDO''' is a tuning system which divides the [[octave]] into 13 equal parts of approximately 92.3 [[cent]]s each. It is the sixth [[prime EDO]], following [[11edo|11EDO]] and coming before [[17edo|17EDO]]. The steps less than 600¢ are narrower than their nearest 12EDO approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12EDO can quickly arrive at an unfamiliar place.

As a temperament of 21-odd-limit Just Intonation, 13EDO has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13EDO unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12EDO. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the '''2.5.9.11.13.17.19.21''' subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.

! Approximated 21-limit Ratios<ref>Ratios are based on treating 13EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>

! [[26edo|26EDO]] names

13EDO can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 13EDO "on the fly".

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, and heptatonic 2nd-generated.

=== Differences between distributionally-even scales and smaller EDOs ===

! N

! s-Nedo

| 2

| 46.154¢

| 3

| 61.538¢

| 4

| 5

| 36.923¢

| -65.615¢

| 6

| -23.077¢

| 7

| 13.187¢

| 8

| 34.615¢

| 9

| 10

| 64.615¢

| 11

| 75.524¢

| 12

| 84.615¢

13EDO can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-EDO's step size.

== Scales in 13EDO ==

:''Main article: [[13edo scales|13EDO scales]]''

Due to the prime character of the number 13, 13EDO can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two [[degree]]s of 13EDO), 3\13, 4\13, 5\13, &amp; 6\13, respectively.

Another neat facet of 13EDO is the fact that any 12EDO scale can be "turned into" a 13EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.

=== Pathological Modes ===

3 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Harmony in 13EDO ==

Contrary to popular belief, consonant harmony is possible in 13EDO, but it requires a radically different approach than that used in 12EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12EDO within 13EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12EDO, since the strongest dissonances in 13EDO are near the middle of the octave (<u>[[13edo#top|degree]]s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N_subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26EDO.

By this, we can assume that the major ninth of 13EDO can be thought of as analogous to the perfect fifth in 12EDO and other meantone EDOs. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13EDO followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13EDO.

The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13EDO and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.

== Notational and Compositional Approaches to 13EDO ==

13EDO has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.

13EDO offers two main candidates for diatonic-like scales: the 6L 1s heptatonic MOS generated by 2\13, and the 5L 3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12EDO diatonic scale. Specifically, the 6L 1s scale resembles the 12EDO diatonic with one of its semitones replaced with a whole-tone, while the 5L 3s scale resembles the 12EDO diatonic with an extra semitone inserted between two adjacent whole-tones.

Treating 13EDO as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.

Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13EDO.

13EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)

The animist comma, 105/104, appears whenever 3*5*7 = 2^3*13... 13EDO does not approximate 3 and 7 individually (26EDO does), but 13EDO has 21/16 (21 = 3*7) and is also an animist temperament. In 13EDO, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:

* [[File:13edo_1MC.mp3|270px]] 13EDO example composition by [[User:IlL|Inthar]] ([[File:13edo_1MC_score.pdf|score]])

* [https://cityoftheasleep.bandcamp.com/track/wintermint-13edo Wintermint (13EDO)] by [[City of the Asleep]]

* [https://cityoftheasleep.bandcamp.com/track/future-dust-13edo Future Dust (13EDO) &#124; City of the Asleep]

* [https://cityoftheasleep.bandcamp.com/track/fugitive-from-sleep-13edo Fugitive from Sleep (13EDO) &#124; City of the Asleep]

* [[Well-Tempered 13-Tone Clavier|The Well-Tempered 13-Tone Clavier]]: A collab project to create 26 preludes and 26 fugues, one in each 13EDO key.