Xenharmonic Wiki

23edo and octave stretching: Difference between revisions

← Older editNewer edit →
VisualWikitext
Contribution (talk | contribs)
3,348 edits
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,280 edits
Changed from 1216 to 1206 as the statement in the synopsis is incorrect about 1216, but true about 1206. Read through the full article and changed anything that no longer held true in 1206 (it was not too much, it's mostly the same). Changed the lists of tunings to more reasonable ones.
Line 1: Line 1:
[[23edo|23edo]] is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.
[[23edo|23edo]] is not often taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.


However, when using a slightly [[stretched tuning|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.
However, when using a slightly [[stretched tuning|stretched octave]] of around 1206 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.


The perfect fifth is sharper than it is in [[7edo]], and thus the width of the perfect fifth falls within the [[syntonic temperament]]'s tuning range. However, stretched-23 is ''not'' a syntonic temperament; using the perfect fifth as [[generator]] results in an [[antidiatonic]] scale, like those of the [[mavila]] and [[pelogic]] temperaments. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.
The perfect fifth is sharper than it is in [[7edo]], and thus the width of the perfect fifth falls within the [[syntonic temperament]]'s tuning range. However, stretched-23 is ''not'' a syntonic temperament; using the perfect fifth as [[generator]] results in an [[antidiatonic]] scale, like those of the [[mavila]] and [[pelogic]] temperaments. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.
Line 8: Line 8:


== Table of intervals ==
== Table of intervals ==
The table below gives the intervals and error values assuming a stretched octave of ''exactly'' 1216 cents.
The table below gives the intervals and error values using a stretched octave of 1206.278 cents (23et's 2.3.5.13 [[WE]] tuning).


{| class="wikitable"
{| class="wikitable"
Line 14: Line 14:
! Interval !! Width in steps !! Width in cents !! Approximations
! Interval !! Width in steps !! Width in cents !! Approximations
|-
|-
| Quarter-tone || 1 || 52.87 ||  
| Quarter-tone || 1 || 52.45 ||  
|-
|-
| Semitone || 2 || 105.74 || 16:1, 15:14
| Semitone || 2 || 105.89 || 16:1, 15:14
|-
|-
| 3/4-tone || 3 || 158.61 || 12:11, 11:10, 10:9
| 3/4-tone || 3 || 157.34 || 12:11, 11:10, 10:9
|-
|-
| Whole tone || 4 || 211.47 || 9:8, 8:7
| Whole tone || 4 || 209.79 || 9:8, 8:7
|-
|-
| Septimal subminor third || 5 || 264.35 || 7:6
| Septimal subminor third || 5 || 262.23 || 7:6
|-
|-
| Minor third || 6 || 317.22 || 6:5
| Minor third || 6 || 314.68 || 6:5
|-
|-
| Major third || 7 || 370.09 || 5:4
| Major third || 7 || 367.13 || 5:4
|-
|-
| Septimal supermajor third || 8 || 422.96 || 9:7
| Septimal supermajor third || 8 || 418.58 || 9:7
|-
|-
| Minor fourth || 9 || 475.83 || 4:3*
| Minor fourth || 9 || 472.92 || 4:3*
|-
|-
| Major fourth || 10 || 528.70 || 4:3*
| Major fourth || 10 || 524.47 || 4:3*
|-
|-
| Septimal tritone || 11 || 581.57 || 7:5
| Septimal tritone || 11 || 576.92 || 7:5
|-
|-
| Tridecimal tritone || 12 || 634.43 || 13:9
| Tridecimal tritone || 12 || 629.36 || 13:9
|-
|-
| Natural fifth || 13 || 687.30 || 3:2
| Natural fifth || 13 || 681.81 || 3:2
|-
|-
| Augmented fifth || 14 || 740.17 ||  
| Augmented fifth || 14 || 734.26 ||  
|-
|-
| Undecimal minor sixth || 15 || 793.04 || 11:7
| Undecimal minor sixth || 15 || 786.70 || 11:7
|-
|-
| Tridecimal neutral sixth || 16 || 845.91 || 13:8
| Tridecimal neutral sixth || 16 || 839.15 || 13:8
|-
|-
| Major sixth || 17 || 898.782 || 5:3
| Major sixth || 17 || 891.60 || 5:3
|-
|-
| Septimal subminor seventh;<br />septimal supermajor sixth || 18 || 951.65 || 7:4
| Septimal subminor seventh;<br />septimal supermajor sixth || 18 || 944.04 || 7:4
|-
|-
| Minor seventh || 19 || 1004.52 || 9:5
| Minor seventh || 19 || 996.49 || 9:5
|-
|-
| Neutral seventh || 20 || 1057.39 || 11:6, 13:7
| Neutral seventh || 20 || 1048.94 || 11:6, 13:7
|-
|-
| Major seventh || 21 || 1110.26 ||  
| Major seventh || 21 || 1101.38 ||  
|-
|-
| Diminished octave || 22 || 1163.13 ||  
| Diminished octave || 22 || 1153.83 ||  
|-
|-
| Natural (stretched) octave || 23 || 1216 || 2:1
| Natural (stretched) octave || 23 || 1206.28 || 2:1
|}
|}


== Xenharmonic and xenmelodic properties ==
== Xenharmonic and xenmelodic properties ==
Stretching the octave by this much weakens (but does not eliminate) the sense of [[octave equivalence]]. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the [[9/8]] whole tone is approximated by 4 rather than 3 steps. This is because the triple octave [[8/1]] is stretched by nearly a [[quarter tone]], and thus this version of stretched 23edo is not [[consistent]] for intervals involving 8.
Stretching the octave by this much weakens (but does not eliminate) the sense of [[octave equivalence]]. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the [[9/8]] whole tone is approximated by 4 rather than 3 steps. This is because the triple octave [[8/1]] is stretched by half a [[quarter tone]], and thus this version of stretched 23edo is not [[consistent]] for intervals involving 8.


Another odd feature of this scale is that the perfect fourth ([[4/3]]) is sandwiched almost exactly between two scale degrees, thus resulting in two fourths (a major and a minor one). This might not actually be a bad thing. In common-practice music, the perfect fourth, despite having low [[harmonic entropy]], was often classified as a [[dissonant]] interval for reasons relating to [https://en.wikipedia.org/wiki/Lipps%E2%80%93Meyer_law Lipps-Meyer's law]. It was considered a dissonance even after thirds and sixths began to be reclassified as consonances. Thus, by splitting the fourth in two we might actually be reducing this dissonance. Because of the split fourth, 23edo is also not consistent for intervals involving the factor 4.
Another odd feature of this scale is that the perfect fourth ([[4/3]]) is sandwiched almost exactly between two scale degrees, thus resulting in two fourths (a major and a minor one). This might not actually be a bad thing. In common-practice music, the perfect fourth, despite having low [[harmonic entropy]], was often classified as a [[dissonant]] interval for reasons relating to [https://en.wikipedia.org/wiki/Lipps%E2%80%93Meyer_law Lipps-Meyer's law]. It was considered a dissonance even after thirds and sixths began to be reclassified as consonances. Thus, by splitting the fourth in two we might actually be reducing this dissonance. Because of the split fourth, 23edo is also not consistent for intervals involving the factor 4.
Line 84: Line 84:
Stretched 23edo provides a good option for those seeking to combine Western-style instruments like piano and guitar (which have nearly harmonic spectra) with more obviously inharmonic ones (idiophones) from other cultures. In particular, the natural fifth in stretched-23 is almost halfway between the very flat fifth of [[9edo]] and many Indonesian [[pelog]] scales, and the harmonic perfect fifth (3/2). Indonesian [[slendro]] scales may be approximated in stretched-23 as 5-5-4-5-4, or (if we use the ''diminished'' octave as our repeat unit instead) 5-4-5-4-4.
Stretched 23edo provides a good option for those seeking to combine Western-style instruments like piano and guitar (which have nearly harmonic spectra) with more obviously inharmonic ones (idiophones) from other cultures. In particular, the natural fifth in stretched-23 is almost halfway between the very flat fifth of [[9edo]] and many Indonesian [[pelog]] scales, and the harmonic perfect fifth (3/2). Indonesian [[slendro]] scales may be approximated in stretched-23 as 5-5-4-5-4, or (if we use the ''diminished'' octave as our repeat unit instead) 5-4-5-4-4.


Some non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not always presemt a problem. Even many of those which do - like the pelog and slendro scales above - still slightly stretch or compress the octave anyway for various purposes. So this is not too unusual of a feature to introduce.
Some non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not always present a problem. Even many of those which do - like the pelog and slendro scales above - still slightly stretch or compress the octave anyway for various purposes. So this is not too unusual of a feature to introduce.
 
== List of stretched 23 tunings ==
These are tunings which may be used as a stretched 23edo.
* [[11ed7/5]]
* [[36edt]]
* [[59ed6]]
* [[68ed8]]
* [[159ed128]]
* [[227ed1024]]
* [[1ed33/32]]
* [[zeta peak index|84zpi]]
* [[Quartkeenlig]] temperament


== Compressed 23edo ==
== Compressed 23edo ==
[[Octave shrinking]] may also be used instead of octave stretching, which results in a scale with very different properties. 23edo’s approximations of harmonics are still improved upon, but by shifting them in the opposite direction. This is a less-studied option with fertile ground to explore.
[[Octave shrinking]] may also be used instead of octave stretching, which results in a scale with very different properties. 23edo’s approximations of harmonics are still improved upon, but by shifting them in the opposite direction. This is a less-studied option with fertile ground to explore.


== List of compressed 23 tunings ==
== Comparison of stretched tunings ==
These are tunings which may be used as a compressed 23edo.
What follows is a comparison of compressed- and stretched-octave 23edo tunings.
* [[60ed6]]
 
* [[zeta peak index|85zpi]]
; [[zpi|86zpi]]
* [[zeta peak index|86zpi]]
* Step size: 51.653{{c}}, octave size: 1188.0{{c}}
* Approximates all harmonics <9 within 24.0{{c}}
Compressing the octave of 23edo by around 12{{c}} results in improved primes 5, 11 and 13, but worse primes 2, 3 and 7. The tuning 86zpi does this.
{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 86zpi}}
{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 86zpi (continued)}}
 
; [[60ed6]]  
* Step size: 51.700{{c}}, octave size: 1189.1{{c}}
* Approximates all harmonics <9 within 21.8{{c}}
Compressing the octave of 23edo by around 11{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2. The tuning 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.
{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60ed6}}
{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}
 
; [[zpi|85zpi]]  
* Step size: 52.114{{c}}, octave size: 1198.6{{c}}
* Approximates all harmonics <9 within 24.9{{c}}
Compressing the octave of 23edo by around 1.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. The tuning 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.
{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 85zpi}}
{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 85zpi (continued)}}
 
; 23edo
* Step size: 52.174{{c}}, octave size: 1200.0{{c}}
* Approximates all harmonics <9 within 23.7{{c}}
Pure-octaves 23edo.
{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23edo}}
{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23edo (continued)}}
 
; [[WE|23et, 13-limit WE tuning]]
* Step size: 52.237{{c}}, octave size: 1201.5{{c}}
* Approximates all harmonics <9 within 25.7{{c}}
Stretching the octave of 23edo by around 1.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|85ed13]] whose octave is identical within 0.1{{c}}.
{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning}}
{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning (continued)}}
 
; [[WE|23et, 2.3.5.13 WE tuning]]
* Step size: 52.447{{c}}, octave size: 1206.3{{c}}
* Approximates all harmonics <9 within 18.8{{c}}
Stretching the octave of 23edo by around 6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. Its 2.3.5.13 WE tuning and 2.3.5.13 [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.
{{Harmonics in cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning}}
{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning (continued)}}
 
; [[59ed6]]
* Step size: 52.575{{c}}, octave size: 1209.2{{c}}
* Approximates all harmonics <9 within 24.9{{c}}
Stretching the octave of 23edo by around 9{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 59ed6 does this. So does the tuning [[ed5|53ed5]] whose octave is identical within 0.01{{c}}.
{{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59ed6}}
{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59ed6 (continued)}}
 
; [[zpi|84zpi]]
* Step size: 52.615{{c}}, octave size: 1210.1{{c}}
* Approximates all harmonics <9 within 22.2{{c}}
Stretching the octave of 23edo by around 10{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 84zpi does this.
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84zpi}}
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84zpi (continued)}}
 
; [[36edt]]
* Step size: 52.832{{c}}, octave size: 1215.1{{c}}
* Approximates all harmonics <9 within 22.6{{c}}
Stretching the octave of 23edo by around 15{{c}} results in improved primes 3, 5, 7 and 13, but a worse prime 2. The tuning 36edt does this.
{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36edt}}
{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edt (continued)}}
 
; [[equal tuning|84ed13]]  
* Step size: 52.863{{c}}, octave size: 1215.9{{c}}
* Approximates all harmonics <9 within 21.1{{c}}
Stretching the octave of 23edo by around 16{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. The tuning 84ed13 does this.
{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84ed13}}
{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84ed13 (continued)}}


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/23edo_and_octave_stretching"