13edo: Difference between revisions

 

 

 

 

In 13edo, the steps less than 600{{c}} 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.  

The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes the proximity of 135/128 to 1\13 through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.)  A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.

! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.9.5.21.11.13.17.19 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

{{Sagittal chart|Evo}}

== 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.

=== 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.

To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.

==== 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]]

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 ====

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]]

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.

=== Selected 13-odd-limit intervals ===

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

=== Local zeta peak and octave stretch ===

{{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.

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]]

 

 

 

 

 

 

[[File:13_edo_459_chord.mp3]]

 

Play the 4:5:9:11 chord:

== Regular temperament properties ==

=== Commas ===

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"

|-

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

! [[Cent]]s

! Name(s)

|-

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

| {{monzo| -21 3 7 }}

| 10.06

| [[Semicomma]], Fokker comma

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

| 49.49

| Keega

| [[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

|}

 

{{Main | 13edo scales }}

 

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

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

* [[No-threes subgroup temperaments#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\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]]

 

 

 

 

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:

 

 

 

 

* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]'')

* 461.5cET: 5 5 5... (''nonoctave'')

* 738.5cET: 8 8 8... (''nonoctave'')

 

 

 

 

{| class="wikitable"

| 1

| 6

| 11

| 3

| 8

| *

|  

| *

| *

|  

| *

| *

|  

| *

| *

|  

| *

|  

| X

|  

| *

|  

| X

| *

|  

| *

| *

|  

| *

|  

| *

|  

|  

|  

|  

| *

| *

| *

|  

| *

|  

|  

| '''D'''

| Eb

| E

| '''C'''

| Db

| '''D'''

| Keeps the pentatonic scale on the white keys

| '''A'''

| Bb

| B

| '''C'''

| Db

|  

| '''D'''

| Eb

| '''E'''

| F

| Gb

| '''G'''

| Ab

| '''A'''

|  

| '''E'''

| F

| Gb

| '''G'''

| Ab

|  

| '''A'''

| Bb

| '''B'''

| C

| Db

| '''D'''

| Eb

| '''E'''

|  

| '''B'''

| C

| Db

| '''D'''

| Eb

|  

| '''E'''

| F

| '''Gb'''

| G

| Ab

| '''A'''

| Bb

| '''B'''

|  

| C

| Db

| D

| Eb

| E

|  

| F

| Gb

| G

| Ab

| A

| Bb

| B

| C

| Puts the missing key between a semitone

| G

| Ab

| A

| Bb

| B

|  

| C

| Db

| D

| Eb

| E

| F

| Gb

| G

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

|}

 

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.

 

== Music ==

{{Main| 13edo/Music}}

{{Catrel|13edo tracks}}

''Oneirotonic Modal Studies''