Xenharmonic Wiki

User:BudjarnLambeth/Draft related tunings section: Difference between revisions

← Older editNewer edit →
VisualWikitext
One or two subgroup tunings are enough
→Octave stretch or compression: we need to look at the individual harmonics, not just primes
Line 43: Line 43:
{| class="wikitable sortable center-all"
{| class="wikitable sortable center-all"
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents)  
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents)  
! rowspan="2" |Mapping of primes 2-13 (steps)
! rowspan="2" | Mapping of primes 2-13 (steps)
! rowspan="2" |Stretch
! rowspan="2" | Stretch
|-
|-
! 2 !! 3 !! 5 !! 7 !! 11  
! 2 !! 3 !! 5 !! 7 !! 11  
!13
! 13
|-
|-
! 154zpi
! 154zpi
| 33.547 || +7.7|| +10.2|| -1.9|| -14.1||+8.5
| 33.547 || +7.7 || +10.2 || -1.9 || -14.1 || +8.5
| -12.3
| -12.3
|36, 57, 83, 100, 124, 132
| 36, 57, 83, 100, 124, 132
| +23.1%
| +23.1%
|-
|-
!21edf
! 21edf
|33.426
| 33.426
| +3.3
| +3.3
| +3.3
| +3.3
Line 63: Line 63:
| -6.5
| -6.5
| +5.1
| +5.1
|36, 57, 83, 101, 124, 133
| 36, 57, 83, 101, 124, 133
| +10.2%
| +10.2%
|-
|-
! 57edt
! 57edt
| 33.368 || +1.2|| 0|| +16.6|| +1.3||-13.7
| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || -13.7
| -2.6
| -2.6
|36, 57, 84, 101, 124, 133
| 36, 57, 84, 101, 124, 133
| +3.6%
| +3.6%
|-
|-
! 36edo
! 36edo
| '''33.333'''|| '''0'''|| '''-2.0'''|| '''+13.7'''|| '''-2.2'''||'''+15.3'''
| '''33.333''' || '''0.0''' || '''-2.0''' || '''+13.7''' || '''-2.2''' || '''+15.3'''
| '''-7.2'''
| '''-7.2'''
|'''36, 57, 84, 101, 125, 133'''
| '''36, 57, 84, 101, 125, 133'''
|'''0%'''
| '''0%'''
|-
|-
!13lim WE
! 13-limit WE
|33.302
| 33.302
| -1.1
| -1.1
| -3.7
| -3.7
Line 86: Line 86:
| +11.4
| +11.4
| -11.4
| -11.4
|36, 57, 84, 101, 125, 133
| 36, 57, 84, 101, 125, 133
| -3.3%
| -3.3%
|-
|-
!11lim WE
! 11-limit WE
|33.286
| 33.286
| -1.7
| -1.7
| -4.7
| -4.7
Line 97: Line 97:
| +9.4
| +9.4
| -13.5
| -13.5
|36, 57, 84, 101, 125, 133
| 36, 57, 84, 101, 125, 133
| -5.1%
| -5.1%
|-
|-
! 156zpi
! 156zpi
| 33.152 || -6.5|| -12.3|| -1.5|| +12.7||-7.3
| 33.152 || -6.5 || -12.3 || -1.5 || +12.7 || -7.3 || +1.8
| +1.8
| 36, 57, 84, 102, 125, 134
|36, 57, 84, 102, 125, 134
| -19.5%
| -19.5%
|}
|}


; [[21edf]]
; [[21edf]]
{{Harmonics in equal|21|3|2|columns=12|collapsed=true}}
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}
Stretching the octave of 36edo by about 3.5 cents results in improved primes 5, 11 and 13, but worse primes 2, 3 and 7. This approximates all primes up to 11 within ''12 cents''. The tuning 21edf does this.
Stretching the octave of 36edo by about 3.5 cents results in improved primes 5, 11 and 13, but worse primes 2, 3 and 7. This approximates all primes up to 11 within ''12 cents''. The tuning 21edf does this.


; [[57edt]] / [[ed7|101ed7]] / [[zpi|155zpi]] / [[WE|2.3.7.13 WE-tuned 36edo]]
; [[57edt]] / [[ed7|101ed7]] / [[zpi|155zpi]] / [[WE|2.3.7.13 WE-tuned 36edo]]
{{Harmonics in equal|57|3|1|columns=12|collapsed=true}}
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1 cent optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11 and 13. This approximates all primes up to 11 within ''16.6 cents''. Four almost identical tunings do this: 57edt, 101ed7, 155zpi, and the 2.3.7.13 subgroup WE tuning of 36edo.
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1 cent optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11 and 13. This approximates all primes up to 11 within ''16.6 cents''. Four almost identical tunings do this: 57edt, 101ed7, 155zpi, and the 2.3.7.13 subgroup WE tuning of 36edo.


Line 117: Line 122:


; [[WE|11-limit WE 36edo / 13-limit WE 36edo]]
; [[WE|11-limit WE 36edo / 13-limit WE 36edo]]
{{Harmonics in cet|33.302|columns=12|collapsed=true}}
{{Harmonics in cet|33.302|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed33.302c (continued)}}
Compressing the octave of 36edo by about 2 cents results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all primes up to 11 within ''9.7 cents''. The 11- and 13-limit WE tunings of 36edo both do this, as do their respective [[TE]] tunings.
Compressing the octave of 36edo by about 2 cents results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all primes up to 11 within ''9.7 cents''. The 11- and 13-limit WE tunings of 36edo both do this, as do their respective [[TE]] tunings.
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Draft_related_tunings_section"