13edo: Difference between revisions
+dual fifths in infobox |
Some cleanup |
||
| Line 14: | Line 14: | ||
}} | }} | ||
'''13 equal divisions of the octave''' ('''13edo''') is a tuning system which divides the [[octave]] into 13 equal parts of approximately 92.3 [[cent]]s each | '''13 equal divisions of the octave''' ('''13edo''') is a tuning system which divides the [[octave]] into 13 equal parts of approximately 92.3 [[cent]]s each. | ||
== Theory == | == Theory == | ||
In 13edo, 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 [[harmonic]]s, 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. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|13}} | {{Harmonics in equal|13}} | ||
=== Miscellaneous properties === | |||
13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]]. | |||
{| class="wikitable center-all right-2" | == Intervals == | ||
{| class="wikitable center-all right-2 left-3" | |||
|- | |- | ||
! | ! # | ||
! Cents | ! Cents | ||
! 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> | ! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref> | ||
! [[Erv Wilson's Linear Notations|Erv Wilson]] | ! [[Erv Wilson's Linear Notations|Erv Wilson]] | ||
! Archaeotonic | ! Archaeotonic | ||
! Oneirotonic | ! Oneirotonic | ||
! [[26edo]] names | ! [[26edo]] names | ||
! Fox-Raven Notation (J = 360Hz) | ! Fox-Raven<br>Notation (J = 360Hz) | ||
! Pseudo-Diatonic Category | ! Pseudo-Diatonic<br>Category | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 175: | Line 182: | ||
<references/> | <references/> | ||
== Notation == | |||
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. | 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. | ||
| Line 181: | Line 189: | ||
{| class="wikitable center-all right-2" | {| class="wikitable center-all right-2" | ||
|- | |- | ||
! | ! # | ||
! Cents | ! Cents | ||
! colspan="3" | [[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor | ! colspan="3" | [[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor | ||
| Line 349: | Line 357: | ||
[[File:13_Edo_chromatic_scale_on_J.mp3]] | [[File:13_Edo_chromatic_scale_on_J.mp3]] | ||
== JI approximation == | |||
[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
== Tuning by ear == | == Tuning by ear == | ||
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. | 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 | == Scales == | ||
{{Main | 13edo scales }} | |||
Important | Important [[mos]]ses (values in parentheses are (''period'', ''generator'')): | ||
* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1) | * [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1) | ||
* archeotonic [[6L 1s]] 2222221 (2\13, 1\1) | * archeotonic [[6L 1s]] 2222221 (2\13, 1\1) | ||
| Line 406: | Line 413: | ||
[[File:13_edo_45921_chord.mp3]] | [[File:13_edo_45921_chord.mp3]] | ||
== Notational and | == 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 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. | ||
| Line 756: | Line 763: | ||
* [[File:WT13C Prelude XII chip.mp3]] [[WT13C]]: Prelude XII in F# ([[:File:WT13C Prelude XII.pdf|Score]]) | * [[File:WT13C Prelude XII chip.mp3]] [[WT13C]]: Prelude XII in F# ([[:File:WT13C Prelude XII.pdf|Score]]) | ||
* [[File:WT13C Fugue XII chip.mp3]] [[WT13C]]: Fugue XII in F# ([[:File:13edo_Fugue_in_J_Oneirominor Score.pdf|Score]]) | * [[File:WT13C Fugue XII chip.mp3]] [[WT13C]]: Fugue XII in F# ([[:File:13edo_Fugue_in_J_Oneirominor Score.pdf|Score]]) | ||
==== Oneirotonic Modal Studies ==== | ==== Oneirotonic Modal Studies ==== | ||
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian | * [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian | ||
| Line 819: | Line 827: | ||
[[Category:Pages with internal sound examples]] | [[Category:Pages with internal sound examples]] | ||
[[Category:Oneirotonic]] | [[Category:Oneirotonic]] | ||
{{Todo| Cleanup }} | |||