136/135: Difference between revisions

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[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''diatismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''diatic''' temperament.

Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth.

Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called [[fiventeen]], which is a rank-2 temperament generated by an octave and a perfect fifth.

~~=== Fiventeen ===~~

In fiventeen, [[17/15]] is equated with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

~~[[Subgroup]]: 2.3.17/5~~

~~{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}~~

~~: mapping generators: ~2, ~3~~

~~[[Optimal tuning]]s:~~

* [[Tp tuning|subgroup]] [[CEE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.440{{c}}

* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}

~~{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}~~

=== Diatic ===

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: mapping generators: ~2, ~3, ~5

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.~~1088~~{{c}}, ~5/4 = ~~387~~.~~8544~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}

[[Badness]] (Sintel): 0.139

=== Diatismic ===

The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering 136/135 is [[46edo]], which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then [[80edo]] makes a good and more accurate tuning ~~that extends to the [[23-limit]]~~. Alternatively, if you ~~don't~~ care (as much) about prime 11, [[68edo]] makes a great tuning ~~in the no-11's [[19-limit]] and no-11's no-29's [[31-limit]]~~.

The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering out 136/135 is [[46edo]], which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then [[80edo]] makes for a good and more accurate tuning. Alternatively, if you do not care as much about prime 11, [[68edo]] makes for a great tuning.

[[Subgroup]]: 2.3.5.7.11.13.17

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: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.~~1088~~{{c}}, ~5/4 = ~~387~~.~~8544~~{{c}}, ~7/4, ~11/8, ~13/8

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}, ~7/4 = 970.2512{{c}}, ~11/8 = 553.4578{{c}}, ~13/8 = 842.6669{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}, ~7/4 = 969.9161{{c}}, ~11/8 = 552.6614{{c}}, ~13/8 = 841.9647{{c}}

{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*

{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }} *

<nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]

[[Badness]] (Sintel): 1.15

== Etymology ==

@@ Line 9: / Line 9: @@
 [[Tempering out]] this comma in the full 17-limit results in the rank-6 '''diatismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''diatic''' temperament.
-Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth.
+Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called [[fiventeen]], which is a rank-2 temperament generated by an octave and a perfect fifth.
-=== Fiventeen ===
-In fiventeen, [[17/15]] is equated with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
-[[Subgroup]]: 2.3.17/5
-{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
-: mapping generators: ~2, ~3
-[[Optimal tuning]]s:
-* [[Tp tuning|subgroup]] [[CEE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.440{{c}}
-* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}
-{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
 === Diatic ===
@@ Line 31: / Line 17: @@
 : mapping generators: ~2, ~3, ~5
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}, ~5/4 = 387.8544{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}
 {{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}
+[[Badness]] (Sintel): 0.139
 === Diatismic ===
-The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering 136/135 is [[46edo]], which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then [[80edo]] makes a good and more accurate tuning that extends to the [[23-limit]]. Alternatively, if you don't care (as much) about prime 11, [[68edo]] makes a great tuning in the no-11's [[19-limit]] and no-11's no-29's [[31-limit]].
+The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering out 136/135 is [[46edo]], which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then [[80edo]] makes for a good and more accurate tuning. Alternatively, if you do not care as much about prime 11, [[68edo]] makes for a great tuning.
 [[Subgroup]]: 2.3.5.7.11.13.17
@@ Line 57: / Line 47: @@
 : mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}, ~5/4 = 387.8544{{c}}, ~7/4, ~11/8, ~13/8
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}, ~7/4 = 970.2512{{c}}, ~11/8 = 553.4578{{c}}, ~13/8 = 842.6669{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}, ~7/4 = 969.9161{{c}}, ~11/8 = 552.6614{{c}}, ~13/8 = 841.9647{{c}}
-{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*
+{{Optimal ET sequence|legend=1| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }} *
 <nowiki>*</nowiki> [[optimal patent val]]: [[177edo|177]]
+[[Badness]] (Sintel): 1.15
 == Etymology ==