205edo: Difference between revisions

Line 3:

== Theory ==

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.

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-limit, 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]], [[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.

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]]; 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.

Line 46:

| 2.3.5

| 1600000/1594323, {{monzo| 38 -2 -15 }}

| [{{~~val~~| 205 325 476 }}]

| {{mapping| 205 325 476 }}

| -0.106

| 0.141

Line 53:

| 2.3.5.11

| 5632/5625, 14641/14580, 1600000/1594323

| [{{~~val~~| 181 287 420 508 }}]

| {{mapping| 181 287 420 508 }}

| -0.002

| 0.218

Line 63:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

Line 104:

| [[Countercomp]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==

Line 160:

Line 161:

: 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

[[Category:~~Equal divisions of the octave|###~~]] ~~~~

[[Category:Huntmic]]

[[Category:Amity]]

[[Category:Hemithirds]]

[[Category:Laka]]

[[Category:Quanic]]

@@ Line 3: / Line 3: @@
 == Theory ==
-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.
+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-limit, 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]], [[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.
+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]]; 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.
@@ Line 46: / Line 46: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| 38 -2 -15 }}
-| [{{val| 205 325 476 }}]
+| {{mapping| 205 325 476 }}
 | -0.106
 | 0.141
@@ Line 53: / Line 53: @@
 | 2.3.5.11
 | 5632/5625, 14641/14580, 1600000/1594323
-| [{{val| 181 287 420 508 }}]
+| {{mapping| 181 287 420 508 }}
 | -0.002
 | 0.218
@@ Line 63: / Line 63: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 104: / Line 104: @@
 | [[Countercomp]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Scales ==
@@ Line 160: / Line 161: @@
 : 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
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+[[Category:Huntmic]]
 [[Category:Amity]]
 [[Category:Hemithirds]]
 [[Category:Laka]]
 [[Category:Quanic]]