612edo: Difference between revisions

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The '''612 equal divisions of the octave''' ('''612edo'''), or the '''612(-tone) equal temperament''' ('''612tet''', '''612et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 612 parts of about 1.96 [[cent]]s each, a size close to [[32805/32768]], the schisma.

== Theory ==

612edo is a very strong [[5-limit]] system, a fact noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}{{citation needed}} and {{w|James Murray Barbour}}{{citation needed}}. ~~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]].

612edo is a very strong [[5-limit]] system, a fact noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}{{citation needed}} and {{w|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[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]].

The 612edo step 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]].

=== Prime harmonics ===

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3.5

| {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }}

| {{~~mapping~~| 612 970 1421 }}

| {{Mapping| 612 970 1421 }}

| +0.0044

| 0.0089

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| 2.3.5.7

| 2401/2400, 4375/4374, {{monzo| -53 10 16 }}

| {{~~mapping~~| 612 970 1421 1718 }}

| {{Mapping| 612 970 1421 1718 }}

| +0.0210

| 0.0297

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }}

| {{~~mapping~~| 612 970 1421 1718 2117 }}

| {{Mapping| 612 970 1421 1718 2117 }}

| +0.0363

| 0.0406

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| 2.3.5.7.11.13

| 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374

| {{~~mapping~~| 612 970 1421 1718 2117 2265 }}

| {{Mapping| 612 970 1421 1718 2117 2265 }}

| +0.0010

| 0.0871

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| 2.3.5.7.11.13.19

| 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095

| {{~~mapping~~| 612 970 1421 1718 2117 2265 2600 }}

| {{Mapping| 612 970 1421 1718 2117 2265 2600 }}

| -0.0168

| −0.0168

| 0.0917

| 4.68

|}

* 612et has a lower relative error than any previous equal temperaments in the 5-limit. Not until [[1171edo|1171]] do we find a better equal temperament in terms of either absolute error or relative error.

* ~~Besides, it~~ has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by [[935edo|935]] and [[836edo|836]], respectively.

* It also has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by [[935edo|935]] and [[836edo|836]], respectively.

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 162.75

| 1125/1024

| [[~~Kwazy~~]]

| [[Crazy]]

|-

| 4

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| [[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

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

== Music ==

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== Notes ==

[[Category:Ennealimmal]]

[[Category:Hemiennealimmal]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''612 equal divisions of the octave''' ('''612edo'''), or the '''612(-tone) equal temperament''' ('''612tet''', '''612et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 612 parts of about 1.96 [[cent]]s each, a size close to [[32805/32768]], the schisma.
+{{ED intro}}
 == Theory ==
-edo is a very strong [[5-limit]] system, a fact noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}{{citation needed}} and {{w|James Murray Barbour}}{{citation needed}}. 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]].
+edo is a very strong [[5-limit]] system, a fact noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}{{citation needed}} and {{w|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[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]].
+The 612edo step 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]].
 === Prime harmonics ===
@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 23: / Line 24: @@
 | 2.3.5
 | {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }}
-| {{mapping| 612 970 1421 }}
+| {{Mapping| 612 970 1421 }}
 | +0.0044
 | 0.0089
@@ Line 30: / Line 31: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, {{monzo| -53 10 16 }}
-| {{mapping| 612 970 1421 1718 }}
+| {{Mapping| 612 970 1421 1718 }}
 | +0.0210
 | 0.0297
@@ Line 37: / Line 38: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }}
-| {{mapping| 612 970 1421 1718 2117 }}
+| {{Mapping| 612 970 1421 1718 2117 }}
 | +0.0363
 | 0.0406
@@ Line 44: / Line 45: @@
 | 2.3.5.7.11.13
 | 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374
-| {{mapping| 612 970 1421 1718 2117 2265 }}
+| {{Mapping| 612 970 1421 1718 2117 2265 }}
 | +0.0010
 | 0.0871
@@ Line 51: / Line 52: @@
 | 2.3.5.7.11.13.19
 | 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095
-| {{mapping| 612 970 1421 1718 2117 2265 2600 }}
+| {{Mapping| 612 970 1421 1718 2117 2265 2600 }}
-| -0.0168
+| −0.0168
 | 0.0917
 | 4.68
 |}
 * 612et has a lower relative error than any previous equal temperaments in the 5-limit. Not until [[1171edo|1171]] do we find a better equal temperament in terms of either absolute error or relative error.
-* Besides, it has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by [[935edo|935]] and [[836edo|836]], respectively.
+* It also has a lower absolute error in the 7- and 11-limit than any previous equal temperaments, and is only bettered by [[935edo|935]] and [[836edo|836]], respectively.
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 84: / Line 86: @@
 | 162.75
 | 1125/1024
-| [[Kwazy]]
+| [[Crazy]]
 |-
 | 4
@@ Line 122: / Line 124: @@
 | [[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
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 == Music ==
@@ Line 129: / Line 131: @@
 == Notes ==
+<references />
 [[Category:Ennealimmal]]
 [[Category:Hemiennealimmal]]
 [[Category:Listen]]