13edo: Difference between revisions

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.

One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]). The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes this proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI.

! 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>

! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the wide 5th of 8\13

! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor

rect 20 80 447 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]

== Approximation to JI ==

[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Local zeta peak ===

{{Main | 13edo and optimal octave stretching }}

At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].

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.

== Approximation to irrational intervals ==

=== Golden ratio ===

13edo has a very good approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. The next better approximations are in [[23edo]] and [[36edo]]. As a coincidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. Logarithmic phi has some interesting applications in [[Metallic MOS]].

Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.

See also: [[9edϕ]]

{| class="wikitable center-all"

|-

! Error (abs, [[Cent|¢]])

|-

|-

|-

| 3.179

Important [[mos]]ses (values in parentheses are (''period'', ''generator'')):

* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)

* [[No-threes subgroup temperaments#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)

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.

[[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]]

[[:File:13edo_horograms.pdf|13edo horograms.pdf]]

~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson

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 ===

2 1 1 1 1 2 1 1 1 1 1 [[2L 9s]] MOS

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

2 1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS

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.

 

Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.

 

Play the 4:5:9 chord:

== Notational and compositional approaches ==

=== The Cryptic Ruse Methods ===

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.

 

==== Modes and harmony in the archaeotonic scale ====

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]]

 

 

 

The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.

 

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]]

 

 

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.

 

| 10

| 2

| 12

| 4

| 9

| 1

| Place in Chain of 738.5 cent intervals

| X

| *

|  

| *

|  

| X

| Marked are the octatonic scales (X=Sarnathian)

|  

| *

|  

| *

| *

|  

|  

|  

| *

|  

| X

| *

|  

| *

| *

|  

| *

|  

| *

| *

|  

|  

|  

| *

| *

|  

| *

|  

| *

| *

|  

| *

|  

| X

| *

|  

|  

|  

| *

| *

|  

| *

|  

| X

| *

|  

| *

| *

|  

| *

|  

|  

| '''D'''

| Eb

| E

| '''C'''

| Db

| '''D'''

| Keeps the pentatonic scale on the white keys

| '''A'''

| Bb

| B

| '''G'''

| Ab

| '''A'''

|  

| '''E'''

| F

| Gb

| '''G'''

| Ab

|  

| '''A'''

| Bb

| '''B'''

| C

| Db

| '''D'''

| Eb

| '''E'''

|-

| '''B'''

| '''E'''

| '''Gb'''

| '''B'''

|  

| C

| Db

| D

| Eb

| E

| F

| Gb

| G

| Ab

| A

| Bb

| B

| C

| Puts the missing key between a semitone

| C

| Db

| D

| Eb

| E

| F

| Gb

| G

| (if that were to be valuable in any way)

|}

 

 

{{Uniform map|13|12.5|13.5}}

 

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

 

{| class="commatable wikitable center-1 center-2 right-4 center-5"

! [[Harmonic limit|Prime<br>limit]]

| 5

| [[2109375/2097152|(14 digits)]]

| {{monzo| -21 3 7 }}

| 10.06

| Lasepyo

| [[Semicomma]], Fokker comma

|-

| 7

| [[1029/1000]]

| {{monzo| -3 1 -3 3 }}

| 49.49

| Trizogu

| Keega

| 7

| [[525/512]]

| {{monzo| -9 1 2 1 }}

| 43.41

| Lazoyoyo

| Avicennma, Avicenna's enharmonic diesis

|-

| 7

| [[64/63]]

| {{monzo| 6 -2 0 -1 }}

| 27.26

| Ru

| Septimal comma, Archytas' comma, Leipziger Komma

| 7

| [[64827/64000]]

| {{monzo| -9 3 -3 4 }}

| 22.23

| Laquadzo-atrigu

| Squalentine comma

| 7

| [[3125/3087]]

| {{monzo| 0 -2 5 -3 }}

| 21.18

| Triru-aquinyo

| Gariboh comma

|-

| 7

| [[3136/3125]]

| {{monzo| 6 0 -5 2 }}

| 6.08

| Zozoquingu

| Hemimean comma

| 11

| [[56/55]]

| {{monzo| 3 0 -1 1 -1 }}

| 31.19

| Luzogu

| Undecimal diesis

|-

| 11

| [[121/120]]

| {{monzo| -3 -1 -1 0 2 }}

| 14.37

| Lologu

| Biyatisma

| 11

| [[441/440]]

| {{monzo| -3 2 -1 2 -1 }}

| 3.93

| Luzozogu

| Werckisma

|-

| 13

| [[40/39]]

| {{monzo| 3 -1 1 0 0 -1 }}

| 43.83

| Thuyo

| Tridecimal minor diesis

| 13

| [[105/104]]

| {{monzo| -3 1 1 1 0 -1 }}

| 16.57

| Thuzoyo

| Animist comma

|-

| 13

| [[169/168]]

| {{monzo| -3 -1 0 -1 0 2 }}

| 10.27

| Thothoru

| Buzurgisma

|}

<references/>

 

=== Animism ===

The animist comma, 105/104, appears whenever {{nowrap| ~3 × ~5 × ~7 = ~2³ × ~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:

 

 

 

==== Oneirotonic Modal Studies ====