[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.
[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.
If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[zpi|296zpi]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.
If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.
What follows is a comparison of stretched- and compressed-octave 59edo tunings.
; [[93edt]]
* Octave size: 1206.62{{c}}
Stretching the octave of 59edo by around 6.5{{c}} results in improved primes 3, 7 and 11 but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 8.22{{c}}. The tuning 93edt does this. So does the tuning [[equal tuning|203ed11]] whose octaves are identical within 0.1{{c}}.
{{Harmonics in equal|93|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 93edt}}
{{Harmonics in equal|93|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93edt (continued)}}
; [[ed6|152ed6]]
* Octave size: 1204.05{{c}}
Stretching the octave of 59edo by around 4{{c}} results in improved primes 3 and 7, but worse primes 2, 5, 11 and 13. This approximates all harmonics up to 16 within 9.53{{c}}. The tuning 152ed6 does this.
{{Harmonics in equal|152|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed6}}
{{Harmonics in equal|152|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed6 (continued)}}
Stretching the octave of 59edo by around 3.5{{c}} results in slightly improved primes 3, 7 and 13, but slightly worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 10.08{{c}}. The tuning 294zpi does this.
{{Harmonics in cet|20.399|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 294zpi}}
{{Harmonics in cet|20.399|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 294zpi (continued)}}
; [[ed12|211ed12]]
* Octave size: 1202.92{{c}}
Stretching the octave of 59edo by around 3{{c}} results in improved primes 3 and 7, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.82{{c}}. The tuning 211ed12 does this.
{{Harmonics in equal|211|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 211ed12}}
{{Harmonics in equal|211|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 211ed12 (continued)}}
Stretching the octave of 59edo by around a fifth of a cent results in slightly improved primes 11 and 13, but slightly worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 9.97{{c}}. The tuning 294zpi does this. 294zpi shares error equally between the two mappings of harmonic 3, so it is the best [[dual-fifth]] option for 59edo.
{{Harmonics in cet|20.342|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 295zpi}}
{{Harmonics in cet|20.342|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 295zpi (continued)}}
Pure-octaves 59edo approximates all harmonics up to 16 within 10.04{{c}}. So does the tuning [[ed5|137ed5]] whose octave is identical within 0.05{{c}}.
{{Harmonics in equal|59|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59edo}}
{{Harmonics in equal|59|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59edo (continued)}}
Compressing the octave of 59edo by around 1{{c}} results in slightly improved primes 3, 7 and 13, but slightly worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.95{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|20.320|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59et, 13-limit WE tuning}}
{{Harmonics in cet|20.320|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59et, 13-limit WE tuning (continued)}}
Compressing the octave of 59edo by around 2{{c}} results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.91{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
{{Harmonics in cet|20.301|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, 59et, 7-limit WE tuning}}
{{Harmonics in cet|20.301|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59et, 7-limit WE tuning (continued)}}
; [[ed7|166ed7]]
* Octave size: 1197.35{{c}}
Compressing the octave of 59edo by around 2.5{{c}} results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.71{{c}}. The tuning 166ed7 does this.
{{Harmonics in equal|166|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166ed7}}
{{Harmonics in equal|166|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166ed7 (continued)}}
; [[ed12|212ed12]]
* Octave size: 1197.24{{c}}
Compressing the octave of 59edo by around 3{{c}} results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.26{{c}}. The tuning 212ed12 does this.
{{Harmonics in equal|212|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 212ed12}}
{{Harmonics in equal|212|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 212ed12 (continued)}}
Compressing the octave of 59edo by around 3.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 10.09{{c}}. The tuning 296zpi does this.
{{Harmonics in cet|20.282|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 296zpi}}
{{Harmonics in cet|20.282|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 296zpi (continued)}}
; [[ed6|153ed6]]
* Octave size: 1196.18{{c}}
Compressing the octave of 59edo by around 4{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 8.81{{c}}. The tuning 153ed6 does this.
{{Harmonics in equal|153|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 153ed6}}
{{Harmonics in equal|153|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 153ed6 (continued)}}
59 equal divisions of the octave (abbreviated 59edo or 59ed2), also called 59-tone equal temperament (59tet) or 59 equal temperament (59et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each. Each step represents a frequency ratio of 21/59, or the 59th root of 2.
Using the flat fifth instead of the sharp one allows for the 12 & 35 temperament, which is a kind of bizarre cousin to garibaldi with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for flattertone temperament in the 59bcd val, a variant of meantone with very flat fifths.
As every other step of 118edo, 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50 & 59 temperament with a subminor third generator provides an interesting temperament.
59edo is the 17th prime edo, following 53edo and before 61edo. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.
This notation uses the same sagittal sequence as 66-EDO.
Evo flavor
Revo flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
Second-best fifth notation
This notation uses the same sagittal sequence as EDOs 45 and 52.
Evo flavor
Revo flavor
Evo-SZ flavor
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
Octave stretch or compression
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by 93edt, a stretched-octave version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.
211ed12 is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.
If one prefers compressed octaves, then 153ed6 is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.
