17edo: Difference between revisions

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{{Harmonics in equal|17|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 17edo (continued)}}

=== ~~Octave stretch~~ ===

=== Subsets and supersets ===

17edo~~'s approximations of harmonics 3, 7, 11~~, and ~~13 are all tempered sharp, so 17edo adapts well to slightly~~ [[~~stretched and compressed tuning|compressing the octave~~]], ~~if that is acceptable.~~ [[~~27edt~~]] and [[~~44ed6~~]] ~~are great demonstrations~~ of ~~this~~, where the ~~octaves~~ are ~~flattened by about 2~~.5 and ~~1.5 cents, respectively~~.

17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. It does not contain any nontrivial subset edos, though it contains [[17ed4]] and [[17ed8]]. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.

~~=== Subsets and supersets ===~~

[[34edo]], which doubles 17edo, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

~~17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]], so it does not contain any nontrivial subset edos, though it contains [[17ed4]].~~ [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

== Intervals ==

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=== Selected 13-limit intervals ===

[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 56~~

~~| steps = 17.0445886606675~~

~~| step size = 70.4035764012981~~

~~| tempered height = 5.056957~~

~~| pure height = 4.528893~~

~~| integral = 1.032175~~

~~| gap = 14.269437~~

~~| octave = 1196.86079882207~~

~~| consistent = 4~~

~~| distinct = 4~~

}}

== Tuning by ear ==

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| [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)

|}

== Octave stretch or compression ==

17edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly [[stretched and compressed tuning|compressing the octave]], if that is acceptable. [[27edt]] and [[44ed6]] are great demonstrations of this, where the octaves are flattened by about 2.5 and 1.5 cents, respectively.

; 17edo

* Step size: 70.588{{c}}, octave size: 1200.000{{c}}

Pure-octaves 17edo approximates the 2.3.11.13 subgroup best. Its approximation to 7 is less good, and it does not really approximate 5. It might make tuning for exploring new harmonies with the 7th, 11th and 13th harmonics not found in 12edo, but its very sharp 7th harmonic might arguably hamper that use case.

{{Harmonics in equal|17|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 17edo}}

{{Harmonics in equal|17|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 17edo (continued)}}

; [[44ed6]]

* Step size: 70.499{{c}}, octave size: 1198.483{{c}}

Compressing the octave of 17edo by around 1.5{{c}} results in much improved primes 3, 7, 11 and 13, but a slightly worse prime 2. The tuning 44ed6 does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.

{{Harmonics in equal|44|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 44ed6}}

{{Harmonics in equal|44|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 44ed6 (continued)}}

; [[27edt]]

* Step size: 70.443{{c}}, octave size: 1197.527{{c}}

Compressing the octave of 17edo by around 2.5{{c}} results in even more improved primes 3, 7, 11 and 13 than 44ed6, but a with more error on prime 2 than 44ed6 also. The tuning 27edt does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.

{{Harmonics in equal|27|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 27edt}}

{{Harmonics in equal|27|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edt (continued)}}

; [[ZPI|56zpi]] / [[WE|17et, 2.3.7.11.13-subgroup WE tuning]]

* Step size: 70.404{{c}}, octave size: 1296.861{{c}}

Compressing the octave of 17edo by around 3{{c}} results in even more improved primes 3, 7, 11 and 13 than 27edt, but a with more error on prime 2 than 27edt also. Tunings that do this include:

* 56zpi

* 17et, 2.3.7.11-subgroup [[TE]] and [[WE]]

* 17et, 2.3.7.11.13-subgroup TE and WE

Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.

{{Harmonics in cet|70.403576|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 56zpi}}

{{Harmonics in cet|70.403576|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 56zpi (continued)}}

== Scales ==

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* [[User:CritDeathX/Sam's 17-note Well Temperament|Nicolai's 17-note well temperament]]

* [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]

~~=== Non-octave scales ===~~

Taking '''every 3 degrees of 17edo''' (or [[17ed8]]) produces a scale in which all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th harmonic. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.

== Introductory materials ==

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~~[[Category:3-limit record edos|##]] ~~

[[Category:Teentuning]]

@@ Line 22: / Line 22: @@
 {{Harmonics in equal|17|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 17edo (continued)}}
-=== Octave stretch ===
+=== Subsets and supersets ===
-edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly [[stretched and compressed tuning|compressing the octave]], if that is acceptable. [[27edt]] and [[44ed6]] are great demonstrations of this, where the octaves are flattened by about 2.5 and 1.5 cents, respectively.
+edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. It does not contain any nontrivial subset edos, though it contains [[17ed4]] and [[17ed8]]. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.
-=== Subsets and supersets ===
+[[34edo]], which doubles 17edo, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
-edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]], so it does not contain any nontrivial subset edos, though it contains [[17ed4]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
 == Intervals ==
@@ Line 523: / Line 522: @@
 === Selected 13-limit intervals ===
 [[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
-=== Zeta peak index ===
-{{ZPI
-| zpi = 56
-| steps = 17.0445886606675
-| step size = 70.4035764012981
-| tempered height = 5.056957
-| pure height = 4.528893
-| integral = 1.032175
-| gap = 14.269437
-| octave = 1196.86079882207
-| consistent = 4
-| distinct = 4
-}}
 == Tuning by ear ==
@@ Line 761: / Line 746: @@
 | [[Lee]] / [[liese]] (17c) / [[pycnic]] (17)<br>[[Progress]] (17c)
 |}
+== Octave stretch or compression ==
+edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly [[stretched and compressed tuning|compressing the octave]], if that is acceptable. [[27edt]] and [[44ed6]] are great demonstrations of this, where the octaves are flattened by about 2.5 and 1.5 cents, respectively.
+; 17edo
+* Step size: 70.588{{c}}, octave size: 1200.000{{c}}
+Pure-octaves 17edo approximates the 2.3.11.13 subgroup best. Its approximation to 7 is less good, and it does not really approximate 5. It might make tuning for exploring new harmonies with the 7th, 11th and 13th harmonics not found in 12edo, but its very sharp 7th harmonic might arguably hamper that use case.
+{{Harmonics in equal|17|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 17edo}}
+{{Harmonics in equal|17|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 17edo (continued)}}
+; [[44ed6]]
+* Step size: 70.499{{c}}, octave size: 1198.483{{c}}
+Compressing the octave of 17edo by around 1.5{{c}} results in much improved primes 3, 7, 11 and 13, but a slightly worse prime 2. The tuning 44ed6 does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.
+{{Harmonics in equal|44|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 44ed6}}
+{{Harmonics in equal|44|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 44ed6 (continued)}}
+; [[27edt]]
+* Step size: 70.443{{c}}, octave size: 1197.527{{c}}
+Compressing the octave of 17edo by around 2.5{{c}} results in even more improved primes 3, 7, 11 and 13 than 44ed6, but a with more error on prime 2 than 44ed6 also. The tuning 27edt does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.
+{{Harmonics in equal|27|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 27edt}}
+{{Harmonics in equal|27|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edt (continued)}}
+; [[ZPI|56zpi]] / [[WE|17et, 2.3.7.11.13-subgroup WE tuning]]
+* Step size: 70.404{{c}}, octave size: 1296.861{{c}}
+Compressing the octave of 17edo by around 3{{c}} results in even more improved primes 3, 7, 11 and 13 than 27edt, but a with more error on prime 2 than 27edt also. Tunings that do this include:
+* 56zpi
+* 17et, 2.3.7.11-subgroup [[TE]] and [[WE]]
+* 17et, 2.3.7.11.13-subgroup TE and WE
+Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5's [[13-limit]] tuning for its size.
+{{Harmonics in cet|70.403576|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 56zpi}}
+{{Harmonics in cet|70.403576|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 56zpi (continued)}}
 == Scales ==
@@ Line 786: / Line 802: @@
 * [[User:CritDeathX/Sam's 17-note Well Temperament|Nicolai's 17-note well temperament]]
 * [[User:FloraC/Flora's 17-note well temperament|Flora's 17-note well temperament]]
-=== Non-octave scales ===
-Taking '''every 3 degrees of 17edo''' (or [[17ed8]]) produces a scale in which all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th harmonic. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.
 == Introductory materials ==
@@ Line 826: / Line 839: @@
 <references group="note" />
-[[Category:3-limit record edos|##]] <!-- 2-digit number -->
 [[Category:Teentuning]]