13edo: Difference between revisions

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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.

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*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.

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".

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[[:File:13ed2-001.svg|13ed2-001.svg]]

:''Main article: [[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 <u>[[Degree|degree]]s of</u> 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.

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

Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO 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 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO 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 13-EDO 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 26et.

[[File:13_edo_45921_chord.mp3]]

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 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are [[Rothenberg_propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.

To facilitate discussion of these scales, Cryptic Ruse has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. The 2\13-based heptatonic has been named "archeotonic" after the "Old Ones" that rule the Dreamlands, and the 5\13-based octatonic has been named "oneirotonic" after the Dreamlands themselves. Modes of the archeotonic are named after the individual Old Ones themselves; modes of the oneirotonic are named after cities in the Dreamlands.

A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.

There may be other concordant harmonies possible in this scale that do not represent segments of the overtone series; further exploration is pending.

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.

There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.

Normally, 13edo can be notated by adding an accidental between E and F. For some reading the same staff with the same letters but in different places can be mind boggling and lead to confusion. That's why some have recommended different options.

[[Kentaku's_Approach_to_13EDO|More on William Lynch's 13 EDO octaton approach]]

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

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The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.

13 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes the val &lt; 13 21 30 36 45 48 |.)

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| |<nowiki> | -21 3 7 </nowiki>&gt;

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| |<nowiki> | -3 1 -3 3 </nowiki>&gt;

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| |<nowiki> | -9 1 2 1 </nowiki>&gt;

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| |<nowiki> | 6 -2 0 -1 </nowiki>&gt;

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| |<nowiki> | -9 3 -3 4 </nowiki>&gt;

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| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;

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| |<nowiki> | 6 0 -5 2 </nowiki>&gt;

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| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;

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| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;

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The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=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:

0 1 3 4 5 8 9 10 12 13 nonatonic

<ul><li>[[13EDO_Scales_and_Chords_for_Guitar|13EDO Scales and Chords for Guitar]]</li></ul>

* [http://www.microtonalmusic.net/audio/slowdance13edo.mp3 Slow Dance] by [http://danielthompson.blogspot.com/ Daniel Thompson]