Xenharmonic Wiki

23edo and octave stretching: Difference between revisions

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,295 edits
Hstraub (talk | contribs)
autopatrolled, Sysops
260 edits
Audio example for superdiatonic scale in stretched 23edo
 
(3 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 || 33:32
| Quarter-tone || 1 || 52.45 || [[33/32]]
|-
|-
| Semitone || 2 || 105.89 || 16:15, 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
| 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 || 14:11, 9:7
| Septimal supermajor third || 8 || 418.58 || [[14/11]], [[9/7]]
|-
|-
| Minor fourth || 9 || 472.92 || 13:10
| Minor fourth || 9 || 472.92 || [[13/10]]
|-
|-
| Major fourth || 10 || 524.47 || 15:11
| 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 || 10:7, 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 || 20:13
| 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 || 12:7
| Septimal subminor seventh;<br />septimal supermajor sixth || 18 || 944.04 || [[12/7]]
|-
|-
| Minor seventh || 19 || 996.49 || 7:4, 9:5
| Minor seventh || 19 || 996.49 || [[7/4]], [[9/5]]
|-
|-
| Neutral seventh || 20 || 1048.94 || 11:6
| Neutral seventh || 20 || 1048.94 || [[11/6]]
|-
|-
| Major seventh || 21 || 1101.38 || 15:8
| 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 92: Line 97:
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 ===
=== (Compressed) ===


; [[zpi|86zpi]]  
; [[zpi|86zpi]]  
Line 115: Line 120:
{{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 ===
=== (Pure octaves) ===


; 23edo
; 23edo
Line 124: Line 129:
{{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 ===
=== (Stretched) ===


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