Xenharmonic Wiki

23edo and octave stretching: Difference between revisions

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,320 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.
Hstraub (talk | contribs)
autopatrolled, Sysops
260 edits
Audio example for superdiatonic scale in stretched 23edo
 
(5 intermediate revisions by 3 users not shown)
Line 5: Line 5:
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.


Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[superantidiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.
Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[superdiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.
 
 
[[File:Stretched23edo MavilaSuperdiatonic.mp3|Mavila superdiatonioc scale in stretched 23edo]]
 
Mavila superdiatonic scale in stretched 23edo


== Table of intervals ==
== Table of intervals ==
Line 14: Line 19:
! Interval !! Width in steps !! Width in cents !! Approximations
! Interval !! Width in steps !! Width in cents !! Approximations
|-
|-
| Quarter-tone || 1 || 52.45 ||  
| Quarter-tone || 1 || 52.45 || [[33/32]]
|-
|-
| Semitone || 2 || 105.89 || 16:1, 15:14
| Semitone || 2 || 105.89 || [[16/15]], [[15/14]]
|-
|-
| 3/4-tone || 3 || 157.34 || 12:11, 11:10, 10:9
| 3/4-tone || 3 || 157.34 || [[12/11]], [[11/10]], [[10/9]]
|-
|-
| Whole tone || 4 || 209.79 || 9:8, 8:7
| Whole tone || 4 || 209.79 || [[9/8]]
|-
|-
| Septimal subminor third || 5 || 262.23 || 7:6
| Septimal subminor third || 5 || 262.23 || [[7/6]]
|-
|-
| Minor third || 6 || 314.68 || 6:5
| Minor third || 6 || 314.68 || [[6/5]]
|-
|-
| Major third || 7 || 367.13 || 5:4
| Major third || 7 || 367.13 || [[5/4]]
|-
|-
| Septimal supermajor third || 8 || 418.58 || 9:7
| Septimal supermajor third || 8 || 418.58 || [[14/11]], [[9/7]]
|-
|-
| Minor fourth || 9 || 472.92 || 4:3*
| Minor fourth || 9 || 472.92 || [[13/10]]
|-
|-
| Major fourth || 10 || 524.47 || 4:3*
| Major fourth || 10 || 524.47 || [[15/11]]
|-
|-
| Septimal tritone || 11 || 576.92 || 7:5
| Septimal tritone || 11 || 576.92 || [[7/5]]
|-
|-
| Tridecimal tritone || 12 || 629.36 || 13:9
| Tridecimal tritone || 12 || 629.36 || [[10/7]], [[13/9]]
|-
|-
| Natural fifth || 13 || 681.81 || 3:2
| Natural fifth || 13 || 681.81 || [[3/2]]
|-
|-
| Augmented fifth || 14 || 734.26 ||  
| Augmented fifth || 14 || 734.26 || [[20/13]]
|-
|-
| Undecimal minor sixth || 15 || 786.70 || 11:7
| Undecimal minor sixth || 15 || 786.70 || [[11/7]]
|-
|-
| Tridecimal neutral sixth || 16 || 839.15 || 13:8
| Tridecimal neutral sixth || 16 || 839.15 || [[13/8]]
|-
|-
| Major sixth || 17 || 891.60 || 5:3
| Major sixth || 17 || 891.60 || [[5/3]]
|-
|-
| Septimal subminor seventh;<br />septimal supermajor sixth || 18 || 944.04 || 7:4
| Septimal subminor seventh;<br />septimal supermajor sixth || 18 || 944.04 || [[12/7]]
|-
|-
| Minor seventh || 19 || 996.49 || 9:5
| Minor seventh || 19 || 996.49 || [[7/4]], [[9/5]]
|-
|-
| Neutral seventh || 20 || 1048.94 || 11:6, 13:7
| Neutral seventh || 20 || 1048.94 || [[11/6]]
|-
|-
| Major seventh || 21 || 1101.38 ||  
| Major seventh || 21 || 1101.38 || [[15/8]]
|-
|-
| Diminished octave || 22 || 1153.83 ||  
| Diminished octave || 22 || 1153.83 ||  
|-
|-
| Natural (stretched) octave || 23 || 1206.28 || 2:1
| Natural (stretched) octave || 23 || 1206.28 || [[2/1]]
|}
|}


Line 89: Line 94:
[[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.


== Comparison of stretched tunings ==
== Comparison of tunings ==
What follows is a comparison of compressed- and stretched-octave 23edo tunings.
What follows is a comparison of compressed- and stretched-octave 23edo tunings.
=== (Compressed) ===


; [[zpi|86zpi]]  
; [[zpi|86zpi]]  
Line 112: Line 119:
{{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=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)}}
{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 85zpi (continued)}}
=== (Pure octaves) ===


; 23edo
; 23edo
Line 119: Line 128:
{{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=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)}}
{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23edo (continued)}}
=== (Stretched) ===


; [[WE|23et, 13-limit WE tuning]]  
; [[WE|23et, 13-limit WE tuning]]  
Retrieved from "https://en.xen.wiki/w/23edo_and_octave_stretching"