121edo: Difference between revisions

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The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]].

== Theory ==

~~121EDO~~ has a distinctly sharp tendency, in that the odd ~~primes~~ from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[~~Mirkwai_clan #Grendel|~~grendel ~~temperament~~]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[~~marveltwin_triad|~~marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[~~gentle~~ chords]], because it tempers out 676/675 it allows [[~~island tetrad|~~island chords]] and because it tempers out 1575/1573 it allows ~~the~~ [[nicolic ~~tetrad~~]]. That makes for a very flexible system, and since this suite of commas defines 13-limit ~~121ET~~, it is a system only associated with 121.

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== ~~Prime~~ harmonics ===

=== Odd harmonics ===

=== Subsets and supersets ===

Since 121 factors into 112, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==

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

! rowspan="2" | Subgroup

|-

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

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

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

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

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

! colspan="2" | Tuning error

|-

Line 23:

Line 27:

| 2.3

| {{monzo| 192 -121 }}

| [{{~~val~~| 121 192 }}]

| {{mapping| 121 192 }}

| -0.687

| −0.687

| 0.687

| 6.93

Line 30:

Line 34:

| 2.3.5

| 15625/15552, {{monzo| 31 -21 1 }}

| [{{~~val~~| 121 192 281 }}]

| {{mapping| 121 192 281 }}

| -0.524

| −0.524

| 0.606

| 6.11

Line 37:

Line 41:

| 2.3.5.7

| 4000/3969, 6144/6125, 10976/10935

| [{{~~val~~| 121 192 281 340 }}]

| {{mapping| 121 192 281 340 }}

| -0.667

| −0.667

| 0.580

| 5.85

Line 44:

Line 48:

| 2.3.5.7.11

| 540/539, 896/891, 1375/1372, 4375/4356

| [{{~~val~~| 121 192 281 340 419 }}]

| {{mapping| 121 192 281 340 419 }}

| -0.768

| −0.768

| 0.556

| 5.61

Line 51:

Line 55:

| 2.3.5.7.11.13

| 325/324, 352/351, 364/363, 540/539, 625/624

| [{{~~val~~| 121 192 281 340 419 448 }}]

| {{mapping| 121 192 281 340 419 448 }}

| -0.750

| −0.750

| 0.510

| 5.14

Line 58:

Line 62:

| 2.3.5.7.11.13.17

| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441

| [{{~~val~~| 121 192 281 340 419 448 495 }}]

| {{mapping| 121 192 281 340 419 448 495 }}

| -0.787

| −0.787

| 0.480

| 4.85

Line 65:

Line 69:

| 2.3.5.7.11.13.17.19

| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374

| [{{~~val~~| 121 192 281 340 419 448 495 514 }}]

| {{mapping| 121 192 281 340 419 448 495 514 }}

| -0.689

| −0.689

| 0.519

| 5.23

|}

* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]].

=== 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 ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! ~~Temperaments~~

! Associated ratio*

! Temperament

|-

| 1

Line 120:

Line 126:

| 317.36

| 6/5

| [[~~Hanson]] / [[metakleismic~~]]

| [[Metakleismic]]

|-

| 1

Line 183:

Line 189:

|-

| 11

| 50\121 (5\121)

| 50\121 (5\121)

| 495.87 (49.59)

| 495.87 (49.59)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Hendecatonic]]

|}

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

== 13-limit detempering of ~~121ET~~ ==

== 13-limit detempering of 121et ==

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

==Miscellany==

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

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

~~[[Category:Gentle]]~~

[[Category:Grendel]]

~~[[Category:Huntmic]]~~

~~[[Category:Island]]~~

~~[[Category:Marveltwin]]~~

~~[[Category:Minthmic]]~~

[[Category:Quintupole]]

~~[[Category:Swetismic]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]].
+{{ED intro}}
 == Theory ==
-EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].
+edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].
-Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.
+Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
-=== Prime harmonics ===
+=== Odd harmonics ===
-{{Primes in edo|121|columns=10}}
+{{Harmonics in equal|121}}
+=== Subsets and supersets ===
+Since 121 factors into 11<sup>2</sup>, 121edo contains [[11edo]] as its only nontrivial subset.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 23: / Line 27: @@
 | 2.3
 | {{monzo| 192 -121 }}
-| [{{val| 121 192 }}]
+| {{mapping| 121 192 }}
-| -0.687
+| −0.687
 | 0.687
 | 6.93
@@ Line 30: / Line 34: @@
 | 2.3.5
 | 15625/15552, {{monzo| 31 -21 1 }}
-| [{{val| 121 192 281 }}]
+| {{mapping| 121 192 281 }}
-| -0.524
+| −0.524
 | 0.606
 | 6.11
@@ Line 37: / Line 41: @@
 | 2.3.5.7
 | 4000/3969, 6144/6125, 10976/10935
-| [{{val| 121 192 281 340 }}]
+| {{mapping| 121 192 281 340 }}
-| -0.667
+| −0.667
 | 0.580
 | 5.85
@@ Line 44: / Line 48: @@
 | 2.3.5.7.11
 | 540/539, 896/891, 1375/1372, 4375/4356
-| [{{val| 121 192 281 340 419 }}]
+| {{mapping| 121 192 281 340 419 }}
-| -0.768
+| −0.768
 | 0.556
 | 5.61
@@ Line 51: / Line 55: @@
 | 2.3.5.7.11.13
 | 325/324, 352/351, 364/363, 540/539, 625/624
-| [{{val| 121 192 281 340 419 448 }}]
+| {{mapping| 121 192 281 340 419 448 }}
-| -0.750
+| −0.750
 | 0.510
 | 5.14
@@ Line 58: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
-| [{{val| 121 192 281 340 419 448 495 }}]
+| {{mapping| 121 192 281 340 419 448 495 }}
-| -0.787
+| −0.787
 | 0.480
 | 4.85
@@ Line 65: / Line 69: @@
 | 2.3.5.7.11.13.17.19
 | 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
-| [{{val| 121 192 281 340 419 448 495 514 }}]
+| {{mapping| 121 192 281 340 419 448 495 514 }}
-| -0.689
+| −0.689
 | 0.519
 | 5.23
 |}
+* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]].
 === 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
-! Temperaments
+! Associated<br />ratio*
+! Temperament
 |-
 | 1
@@ Line 120: / Line 126: @@
 | 317.36
 | 6/5
-| [[Hanson]] / [[metakleismic]]
+| [[Metakleismic]]
 |-
 | 1
@@ Line 183: / Line 189: @@
 |-
 | 11
-| 50\121<br>(5\121)
+| 50\121<br />(5\121)
-| 495.87<br>(49.59)
+| 495.87<br />(49.59)
-| 4/3<br>(36/35)
+| 4/3<br />(36/35)
 | [[Hendecatonic]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-== 13-limit detempering of 121ET ==
+== 13-limit detempering of 121et ==
 {{See also| Detempering }}
 [100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]
-==Miscellany==
+== Miscellany ==
+Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.
-Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Gentle]]
 [[Category:Grendel]]
-[[Category:Huntmic]]
-[[Category:Island]]
-[[Category:Marveltwin]]
-[[Category:Minthmic]]
 [[Category:Quintupole]]
-[[Category:Swetismic]]