27edo: Difference between revisions

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Assuming pure octaves, 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. Its fifth and harmonic seventh are both sharp by 9{{c}}, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7{{c}}. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from [[superpyth|1/3-septimal-comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3-comma meantone|1/3-syntonic-comma meantone]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 and 5:7:9, ~~making 27~~ the ~~smallest edo that can simulate tritave harmony~~, although it rapidly ~~becomes~~ rough if extended to ~~the~~ 11 and above~~, unlike a true tritave based system~~.

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.

27edo, with its 400{{c}} major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

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; 27edo

* Step size: 44.444{{c}}, octave size: 1200.000{{c}}

Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}.

Pure-octaves 27edo approximates all harmonics up to 16 within 22.8{{c}}, with the greatest error occurring at 15/8 and 16/15, mapped inconsistently to 25\27 and 2\27, respectively.

{{Harmonics in equal|27|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 27edo}}

{{Harmonics in equal|27|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edo (continued)}}

~~; [[WE|27et, 11-limit WE tuning]]~~

* Step size: 44.370{{c}}, octave size: 1197.985{{c}}

Compressing the octave of 27edo by around 2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.

~~{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 11-limit WE tuning}}~~

~~{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 11-limit WE tuning (continued)}}~~

; [[97ed12]]

* Step size: 44.350{{c}}, octave size: 1197.451{{c}}

Compressing the octave of 27edo by around 2.5{{c}} ~~has the same benefits~~ and ~~drawbacks as the 11-limit tuning, but both are slightly amplified~~. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 97ed12 does this.

Compressing the octave of 27edo by around 2.5{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 but 11 within 12.8{{c}}. The tuning 97ed12 does this.

{{Harmonics in equal|97|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 97ed12}}

{{Harmonics in equal|97|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 97ed12 (continued)}}

; [[WE|27et, 2.3.5.7.13-subgroup WE tuning]]

* Step size: 44.326{{c}}, octave size: 1196.796{{c}}

Compressing the octave of 27edo by around 3.2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 but 11 within 12.8{{c}}. Both the 2.3.5.7.13-subgroup [[TE]] and WE tunings do this.

{{Harmonics in cet|44.325787|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 2.3.5.7.13-subgroup WE tuning}}

{{Harmonics in cet|44.325787|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 2.3.5.7.13-subgroup WE tuning (continued)}}

; [[ZPI|106zpi]] / [[WE|27et, 7-limit WE tuning]] / [[70ed6]]

* Step size (~~106zpi~~): 44.~~302326~~{{c}}, octave size (~~106zpi~~): 1196.~~163~~{{c}}

* Step size (70ed6): 44.314, octave size (70ed6): 1196.468{{c}}

* ~~Octave~~ size (~~70ed6~~): ~~1196~~.~~468~~{{c}}

* Step size (7-limit WE): 44.306, octave size (7-limit WE): 1196.273{{c}}

* Octave size (~~7-limit WE~~): 1196.~~273~~{{c}}

* Step size (106zpi): 44.302{{c}}, octave size (106zpi): 1196.163{{c}}

Compressing the octave of 27edo by around 3.5{{c}} results in even more improvement to primes 3, 5 and 7 than the 13~~-limit~~ tuning, but now at the cost of moderate damage to 2~~, 11~~ and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.

Compressing the octave of 27edo by around 3.5 to 3.8{{c}} results in even more improvement to primes 3, 5 and 7 than the 2.3.5.7.13 tuning, but now at the cost of moderate damage to 2 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.

{{Harmonics in cet|44.302326|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106zpi}}

{{Harmonics in cet|44.302326|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106zpi (continued)}}

Line 1,165:

; [[90ed10]]

* Step size: 44.292{{c}}, octave size: 1195.894{{c}}

Compressing the octave of 27edo by around 4.1{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.

Compressing the octave of 27edo by around 4.1{{c}} results in improved primes 3, 5, 7 and 11, using the 27e val, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.

{{Harmonics in equal|90|10|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed10}}

{{Harmonics in equal|90|10|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed10 (continued)}}

@@ Line 5: / Line 5: @@
 Assuming pure octaves, 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. Its fifth and harmonic seventh are both sharp by 9{{c}}, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7{{c}}. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from [[superpyth|1/3-septimal-comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3-comma meantone|1/3-syntonic-comma meantone]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].
-Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.
+Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.
 edo, with its 400{{c}} major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.
@@ Line 1,139: / Line 1,139: @@
 ; 27edo
 * Step size: 44.444{{c}}, octave size: 1200.000{{c}}
-Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}.
+Pure-octaves 27edo approximates all harmonics up to 16 within 22.8{{c}}, with the greatest error occurring at 15/8 and 16/15, mapped inconsistently to 25\27 and 2\27, respectively.
 {{Harmonics in equal|27|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 27edo}}
 {{Harmonics in equal|27|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edo (continued)}}
-; [[WE|27et, 11-limit WE tuning]]
-* Step size: 44.370{{c}}, octave size: 1197.985{{c}}
-Compressing the octave of 27edo by around 2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
-{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 11-limit WE tuning}}
-{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 11-limit WE tuning (continued)}}
 ; [[97ed12]]
 * Step size: 44.350{{c}}, octave size: 1197.451{{c}}
-Compressing the octave of 27edo by around 2.5{{c}} has the same benefits and drawbacks as the 11-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 97ed12 does this.
+Compressing the octave of 27edo by around 2.5{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 but 11 within 12.8{{c}}. The tuning 97ed12 does this.
 {{Harmonics in equal|97|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 97ed12}}
 {{Harmonics in equal|97|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 97ed12 (continued)}}
+; [[WE|27et, 2.3.5.7.13-subgroup WE tuning]]
+* Step size: 44.326{{c}}, octave size: 1196.796{{c}}
+Compressing the octave of 27edo by around 3.2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 but 11 within 12.8{{c}}. Both the 2.3.5.7.13-subgroup [[TE]] and WE tunings do this.
+{{Harmonics in cet|44.325787|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 2.3.5.7.13-subgroup WE tuning}}
+{{Harmonics in cet|44.325787|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 2.3.5.7.13-subgroup WE tuning (continued)}}
 ; [[ZPI|106zpi]] / [[WE|27et, 7-limit WE tuning]] / [[70ed6]]
-* Step size (106zpi): 44.302326{{c}}, octave size (106zpi): 1196.163{{c}}
+* Step size (70ed6): 44.314, octave size (70ed6): 1196.468{{c}}
-* Octave size (70ed6): 1196.468{{c}}
+* Step size (7-limit WE): 44.306, octave size (7-limit WE): 1196.273{{c}}
-* Octave size (7-limit WE): 1196.273{{c}}
+* Step size (106zpi): 44.302{{c}}, octave size (106zpi): 1196.163{{c}}
-Compressing the octave of 27edo by around 3.5{{c}} results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.
+Compressing the octave of 27edo by around 3.5 to 3.8{{c}} results in even more improvement to primes 3, 5 and 7 than the 2.3.5.7.13 tuning, but now at the cost of moderate damage to 2 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.
 {{Harmonics in cet|44.302326|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106zpi}}
 {{Harmonics in cet|44.302326|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106zpi (continued)}}
@@ Line 1,165: / Line 1,165: @@
 ; [[90ed10]]
 * Step size: 44.292{{c}}, octave size: 1195.894{{c}}
-Compressing the octave of 27edo by around 4.1{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.
+Compressing the octave of 27edo by around 4.1{{c}} results in improved primes 3, 5, 7 and 11, using the 27e val, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.
 {{Harmonics in equal|90|10|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed10}}
 {{Harmonics in equal|90|10|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed10 (continued)}}