13edo: Difference between revisions

=== Local zeta peak ===

== Tuning by ear ==

== Notational and compositional approaches ==

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.

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 '''archaeotonic''' after the "Old Ones" that rule the Dreamlands. Modes of the archaeotonic are named after the individual Old Ones.

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 13-edo.

[[File:Archaeotonic.png|Archaeotonic.png|link=Special:FilePath/Archaeotonic.png]]

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.

==== Modes and harmony in the oneirotonic scale ====

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.

[[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]]

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.

== Mapping to standard keyboards ==

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.

== Regular temperament properties ==

=== Uniform maps ===

{{Uniform map|edo=13}}

 

 

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

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