125edo: Difference between revisions

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-The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] in the 11-limit; and [[275/273]] in the 13-limit. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the cubic (fine) relative cent of [[1edo]].
+The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.
+edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)
+<nowiki>*</nowiki> including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.
 === Prime harmonics ===
-{{Primes in edo|125|prec=2}}
+{{Harmonics in equal|125|columns=12}}
+{{Harmonics in equal|125|start=13|columns=7}}
+{{Harmonics in equal|125|start=20|columns=10|collapsed=true}}
+=== Octave stretch ===
+edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this.
+=== Subsets and supersets ===
+Since 125 factors into primes as 5<sup>3</sup>, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].
+Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
+{{harmonics in cet|86.4|prec=1}}
+{{harmonics in cet|86.4|prec=1|start=11}}
 == Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{Monzo| -198 125 }}
+| {{Mapping| 125 198 }}
+| +0.364
+| 0.364
+| 3.80
+|-
+| 2.3.5
+| 15625/15552, 17433922005/17179869184
+| {{Mapping| 125 198 290 }}
+| +0.575
+| 0.421
+| 4.39
+|-
+| 2.3.5.7
+| 225/224, 4375/4374, 589824/588245
+| {{Mapping| 125 198 290 351 }}
+| +0.362
+| 0.519
+| 5.40
+|-
+| 2.3.5.7.11
+| 225/224, 385/384, 1331/1323, 4375/4374
+| {{Mapping| 125 198 290 351 432 }}
+| +0.528
+| 0.570
+| 5.94
+|-
+| 2.3.5.7.11.13
+| 169/168, 225/224, 325/324, 385/384, 1331/1323
+| {{Mapping| 125 198 290 351 432 462 }} (125f)
+| +0.680
+| 0.622
+| 6.47
+|}
+=== Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br>per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 21: / Line 87: @@
 | 49/48
 | [[Slender]]
+|-
+| 1
+| 12\125
+| 115.2
+| 77/72
+| [[Semigamera]]
 |-
 | 1
@@ Line 38: / Line 110: @@
 | 316.8
 | 6/5
-| [[Hanson]] / [[catakleismic]]
+| [[Catakleismic]]
 |-
 | 1
@@ Line 51: / Line 123: @@
 | 7/5
 | [[Merman]]
+|-
+| 5
+| 26\125<br>(1\125)
+| 249.6<br>(9.6)
+| 81/70<br>(176/175)
+| [[Hemiquintile]]
+|-
+| 5
+| 52\125<br>(2\125)
+| 499.2<br>(19.2)
+| 4/3<br>(81/80)
+| [[Quintile]]
+|-
+| 5
+| 33\125<br>(8\125)
+| 316.8<br/>(76.8)
+| 6/5<br/>(24/23~23/22~22/21)
+| [[Thunderclysmic]]
 |}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)
-[[Category:Equal divisions of the octave]]
-[[Category:Theory]]
 [[Category:Catakleismic]]