205edo: Difference between revisions

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205edo can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limits, and for 13-limit amity, as well as other temperaments tempering out the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.

In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]] ~~and~~ [[847/845]] in the 13-limit.

In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]], [[640/637]], [[729/728]], [[847/845]], and [[1188/1183]] in the 13-limit.

Using its alternative mapping {{val| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]], and [[364/363]] to the comma list.

Anyway, assume the patent val, 205et tempers out [[540/539]], so that it allows [[~~swetismic~~ chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; [[1188/1183~~]] and 540/539~~, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. This makes it a tuning of exceptional fludity for its degree of accuracy.

Anyway, assume the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.

~~205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 [[mina]]s~~.

=== Odd harmonics ===

== Temperament generators and Tonal Plexus ==

=== Temperament generators and Tonal Plexus ===

205edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.

The 119\205 meantone fifth is extremely close to the 1/4-comma fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

=== Miscellaneous properties ===

205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.

== Regular temperament properties ==

@@ Line 5: / Line 5: @@
 edo can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limits, and for 13-limit amity, as well as other temperaments tempering out the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.
-In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]] and [[847/845]] in the 13-limit.
+In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]], [[640/637]], [[729/728]], [[847/845]], and [[1188/1183]] in the 13-limit.
 Using its alternative mapping {{val| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]], and [[364/363]] to the comma list.
-Anyway, assume the patent val, 205et tempers out [[540/539]], so that it allows [[swetismic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; [[1188/1183]] and 540/539, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. This makes it a tuning of exceptional fludity for its degree of accuracy.
+Anyway, assume the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.
-factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 [[mina]]s.
 === Odd harmonics ===
 {{Harmonics in equal|205}}
-== Temperament generators and Tonal Plexus ==
+=== Temperament generators and Tonal Plexus ===
 edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.
 The 119\205 meantone fifth is extremely close to the 1/4-comma fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.
+=== Miscellaneous properties ===
+factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.
 == Regular temperament properties ==