205edo: Difference between revisions

Line 1:

'''~~205EDO~~''' divides the octave into 205 parts of 5.854 cents each. It can serve as a tuning for various temperaments, such as [[Ragismic microtemperaments|amity]] or [[Hemifamity family|laka]], and supplies the optimal patent val for [[Hemifamity temperaments#Quanic|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 five temperament for which it also supplies the optimal patent val. In the 5-limit it tempers out 1600000/1594323, the amity comma, and |38 -2 -15~~>~~, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val tempers out 4375/4374, 5120/5103, 6144/6125 and 540/539. Using its alternative mapping ~~<~~205 325 476 ~~525|~~ it can also be used for [[Gamelismic clan|hemithirds temperament]]; the 13-limit version of this, ~~<~~205 325 476 575 709 759|, is especially noteworthy. It 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.

'''205edo''' divides the octave into 205 parts of 5.854 cents each.

== Theory ==

205edo can serve as a tuning for various temperaments, such as [[Ragismic microtemperaments #Amity|amity]] or [[Hemifamity family #Laka|laka]], and supplies the [[optimal patent val]] for [[Hemifamity temperaments #Quanic|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 five 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 tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[540/539]]. Using its alternative mapping {{val| 205 325 476 575 }} (205d) it can also be used for [[Gamelismic clan #Hemithirds|hemithirds temperament]]; the 13-limit version of this, {{val| 205 325 476 575 709 759 }} (205d), is especially noteworthy. It 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.

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.

=Primary intervals =

=== Primary intervals ===

~~=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|myna]] generator in 53\205, and a very good [[Porcupine|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~~ [http://en.~~wikipedia~~.~~org~~/~~wiki/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.

== 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.

~~=205edo MOS=~~

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.

==Quanic (24\205) MOS==

== 205edo MOS ==

=== Quanic (24\205) MOS ===

===17 note===

==== 17 note ====

11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13

===26 note===

==== 26 note ====

11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2

==Amity (58\205) MOS==

=== Amity (58\205) MOS ===

===11 note===

==== 11 note ====

27 27 4 27 27 4 27 27 4 27 4

===18 note===

==== 18 note ====

23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4

===25 note===

==== 25 note ====

19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4

==Hemithirds (33\205) MOS==

=== Hemithirds (33\205) MOS ===

==== 13 note ====

===13 note===

26 7 26 7 26 7 26 7 26 7 26 7 7

===19 note===

==== 19 note ====

19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7

===25 note===

==== 25 note ====

12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7

===31 notes===

==== 31 notes ====

5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7

==Meantone (119\205) MOS==

=== Meantone (119\205) MOS ===

==== 12 note ====

===12 note===

13 20 13 20 13 20 20 13 20 13 20 20

===19 note===

==== 19 note ====

13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7

===31 notes===

==== 31 notes ====

6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7

==Myna (53\205) MOS==

=== Myna (53\205) MOS ===

==== 11 note ====

===11 note===

7 7 39 7 7 39 7 7 39 7 39

===15 note===

==== 15 note ====

7 7 7 32 7 7 7 32 7 7 7 32 7 7 32

===19 note===

==== 19 note ====

7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25

===23 note===

==== 23 note ====

7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18

===27 note===

==== 27 note ====

7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11

===31 notes===

==== 31 notes ====

7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4

==Porcupine (28\205) MOS==

=== Porcupine (28\205) MOS ===

==== 15 note ====

===15 note===

19 9 19 9 19 9 19 9 19 9 19 9 19 9 9

===22 note===

==== 22 note ====

10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9

===29 note===

==== 29 note ====

1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 9

@@ Line 1: / Line 1: @@
-'''205EDO''' divides the octave into 205 parts of 5.854 cents each. It can serve as a tuning for various temperaments, such as [[Ragismic microtemperaments|amity]] or [[Hemifamity family|laka]], and supplies the optimal patent val for [[Hemifamity temperaments#Quanic|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 five temperament for which it also supplies the optimal patent val. In the 5-limit it tempers out 1600000/1594323, the amity comma, and |38 -2 -15&gt;, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val tempers out 4375/4374, 5120/5103, 6144/6125 and 540/539. Using its alternative mapping &lt;205 325 476 525| it can also be used for [[Gamelismic clan|hemithirds temperament]]; the 13-limit version of this, &lt;205 325 476 575 709 759|, is especially noteworthy. It 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.
+'''205edo''' divides the octave into 205 parts of 5.854 cents each.
+== Theory ==
+edo can serve as a tuning for various temperaments, such as [[Ragismic microtemperaments #Amity|amity]] or [[Hemifamity family #Laka|laka]], and supplies the [[optimal patent val]] for [[Hemifamity temperaments #Quanic|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 five 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 tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[540/539]]. Using its alternative mapping {{val| 205 325 476 575 }} (205d) it can also be used for [[Gamelismic clan #Hemithirds|hemithirds temperament]]; the 13-limit version of this, {{val| 205 325 476 575 709 759 }} (205d), is especially noteworthy. It 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.
 et 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.
-=Primary intervals =
+=== Primary intervals ===
-{{Primes in edo|205}}
+{{Primes in edo|205|prec=2}}
-=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|myna]] generator in 53\205, and a very good [[Porcupine|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 [http://en.wikipedia.org/wiki/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.
+== 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.
-=205edo MOS=
+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.
-==Quanic (24\205) MOS==
+== 205edo MOS ==
+=== Quanic (24\205) MOS ===
-===17 note===
+==== 17 note ====
 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13
-===26 note===
+==== 26 note ====
 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2
-==Amity (58\205) MOS==
+=== Amity (58\205) MOS ===
-===11 note===
+==== 11 note ====
 27 4 27 27 4 27 27 4 27 4
-===18 note===
+==== 18 note ====
 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4
-===25 note===
+==== 25 note ====
 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4
-==Hemithirds (33\205) MOS==
+=== Hemithirds (33\205) MOS ===
+==== 13 note ====
-===13 note===
 7 26 7 26 7 26 7 26 7 26 7 7
-===19 note===
+==== 19 note ====
 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7
-===25 note===
+==== 25 note ====
 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7
-===31 notes===
+==== 31 notes ====
 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7
-==Meantone (119\205) MOS==
+=== Meantone (119\205) MOS ===
+==== 12 note ====
-===12 note===
 20 13 20 13 20 20 13 20 13 20 20
-===19 note===
+==== 19 note ====
 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7
-===31 notes===
+==== 31 notes ====
 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7
-==Myna (53\205) MOS==
+=== Myna (53\205) MOS ===
+==== 11 note ====
-===11 note===
 7 39 7 7 39 7 7 39 7 39
-===15 note===
+==== 15 note ====
 7 7 32 7 7 7 32 7 7 7 32 7 7 32
-===19 note===
+==== 19 note ====
 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25
-===23 note===
+==== 23 note ====
 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18
-===27 note===
+==== 27 note ====
 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11
-===31 notes===
+==== 31 notes ====
 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4
-==Porcupine (28\205) MOS==
+=== Porcupine (28\205) MOS ===
+==== 15 note ====
-===15 note===
 9 19 9 19 9 19 9 19 9 19 9 19 9 9
-===22 note===
+==== 22 note ====
 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9
-===29 note===
+==== 29 note ====
 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 9

205edo: Difference between revisions

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