125edo: Difference between revisions

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== Theory ==

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.

125edo 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 ===

{{Harmonics in equal|125~~|start=13~~|columns=~~7}}~~

{{Harmonics in equal|125|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

~~{{Harmonics in equal|125~~|start~~=20|columns~~=10|collapsed=true}}

=== Octave stretch ===

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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|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==

=== Higher-limit JI ===

125edo 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.

== Regular temperament properties ==

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| 5

| 33\125 (8\125)

| 316.8 (76.8)

| 316.8 (76.8)

| 6/5 (~~24/23~~~23/22~~~22/21~~)

| 6/5 (23/22)

| [[Thunderclysmic]]

|}

@@ Line 4: / Line 4: @@
 == Theory ==
 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 ===
-{{Harmonics in equal|125|columns=12}}
+{{Harmonics in equal|125|columns=9}}
-{{Harmonics in equal|125|start=13|columns=7}}
+{{Harmonics in equal|125|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}
-{{Harmonics in equal|125|start=20|columns=10|collapsed=true}}
 === Octave stretch ===
@@ Line 21: / Line 16: @@
 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|columns=11|collapsed=true}}
-{{harmonics in cet|86.4|prec=1|start=11}}
+{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}
+== Approximation to JI ==
+=== Higher-limit JI ===
+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.
 == Regular temperament properties ==
@@ Line 138: / Line 139: @@
 | 5
 | 33\125<br>(8\125)
-| 316.8<br/>(76.8)
+| 316.8<br>(76.8)
-| 6/5<br/>(24/23~23/22~22/21)
+| 6/5<br>(23/22)
 | [[Thunderclysmic]]
 |}