612edo: Difference between revisions

Line 3:

== Theory ==

612edo is a very strong [[5-limit]] system, a fact noted by [[Bosanquet]] and [[Barbour]]. It tempers out the {{monzo| 485 -306 }} (sasktel comma) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} (kwazy comma), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

612edo is a very strong [[5-limit]] system, a fact noted by [[R. H. M. Bosanquet]] and [[James Murray Barbour]]. The equal temperament [[tempering out|tempers out]] the {{monzo| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} (kwazy comma), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].

Line 63:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

Line 122:

| [[Hemiennealimmal]] (11-limit)

|}

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

== Music ==

* [https://www.youtube.com/watch?v=_DrkrgkiaAY Theme and Variations in Hemiennealimmal] ~~by [[Eliora]]~~

; [[Eliora]]

* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)

[[Category:Ennealimmal]]

[[Category:Hemiennealimmal]]

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a very strong [[5-limit]] system, a fact noted by [[Bosanquet]] and [[Barbour]]. It tempers out the {{monzo| 485 -306 }} (sasktel comma) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} (kwazy comma), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].
+edo is a very strong [[5-limit]] system, a fact noted by [[R. H. M. Bosanquet]] and [[James Murray Barbour]]. The equal temperament [[tempering out|tempers out]] the {{monzo| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} (kwazy comma), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].
 The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].
@@ 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 122: / Line 122: @@
 | [[Hemiennealimmal]] (11-limit)
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Music ==
-* [https://www.youtube.com/watch?v=_DrkrgkiaAY Theme and Variations in Hemiennealimmal] by [[Eliora]]
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)
 [[Category:Ennealimmal]]
 [[Category:Hemiennealimmal]]
 [[Category:Listen]]