59edo: Difference between revisions

(One intermediate revision by the same user not shown)

Line 86:

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 [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[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.

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)}}

; [[zpi|294zpi]]

* Step size: 20.399{{c}}, octave size: 1203.54{{c}}

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)}}

; [[zpi|295zpi]]

* Step size: 20.342{{c}}, octave size: 1200.18{{c}}

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)}}

; 59edo

* Step size: 20.339{{c}}, octave size: 1200.00{{c}}

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)}}

; [[WE|59et, 13-limit WE tuning]]

* Step size: 20.320{{c}}, octave size: 1198.88{{c}}

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)}}

; [[WE|59et, 7-limit WE tuning]]

* Step size: 20.301{{c}}, octave size: 1197.76{{c}}

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)}}

; [[zpi|296zpi]]

* Step size: 20.282{{c}}, octave size: 1196.64{{c}}

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)}}

== Instruments ==

; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==

@@ Line 86: / Line 86: @@
 Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
+== Octave stretch or compression ==
+edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.
+[[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.
+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)}}
+; [[zpi|294zpi]]
+* Step size: 20.399{{c}}, octave size: 1203.54{{c}}
+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)}}
+; [[zpi|295zpi]]
+* Step size: 20.342{{c}}, octave size: 1200.18{{c}}
+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)}}
+; 59edo
+* Step size: 20.339{{c}}, octave size: 1200.00{{c}}
+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)}}
+; [[WE|59et, 13-limit WE tuning]]
+* Step size: 20.320{{c}}, octave size: 1198.88{{c}}
+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)}}
+; [[WE|59et, 7-limit WE tuning]]
+* Step size: 20.301{{c}}, octave size: 1197.76{{c}}
+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)}}
+; [[zpi|296zpi]]
+* Step size: 20.282{{c}}, octave size: 1196.64{{c}}
+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)}}
 == Instruments ==
 ; Lumatone
 See [[Lumatone mapping for 59edo]].
 == Music ==