27edo: Difference between revisions

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.

== Octave stretch or compression ==

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], [[90ed10]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, 4.11, and 2.55{{c}}, respectively.

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

; [[97ed12]]  

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

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

* Step size (7-limit WE): 44.306, 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 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)}}

* 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, 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)}}

* Step size: 44.232{{c}}, octave size: 1194.251{{c}}

Compressing the octave of 27edo by around 5.8{{c}} results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2{{c}}. The tuning 43edt does this.

{{Harmonics in equal|43|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 43edt}}

{{Harmonics in equal|43|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 43edt (continued)}}