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121edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|121}}
{{ED intro}}


== Theory ==
== Theory ==
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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.
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 ===
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.
 
=== Odd harmonics ===
{{Harmonics in equal|121}}
{{Harmonics in equal|121}}


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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 192 -121 }}
| {{Monzo| 192 -121 }}
| {{mapping| 121 192 }}
| {{Mapping| 121 192 }}
| −0.687
| −0.687
| 0.687
| 0.687
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| 2.3.5
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| {{Mapping| 121 192 281 }}
| −0.524
| −0.524
| 0.606
| 0.606
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| 2.3.5.7
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| {{Mapping| 121 192 281 340 }}
| −0.667
| −0.667
| 0.580
| 0.580
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| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| {{Mapping| 121 192 281 340 419 }}
| −0.768
| −0.768
| 0.556
| 0.556
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| {{Mapping| 121 192 281 340 419 448 }}
| −0.750
| −0.750
| 0.510
| 0.510
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| {{Mapping| 121 192 281 340 419 448 495 }}
| −0.787
| −0.787
| 0.480
| 0.480
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| {{Mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| −0.689
| 0.519
| 0.519
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
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| 495.87
| 495.87
| 4/3
| 4/3
| [[Leapday]] / [[polypyth]]
| [[Polypyth]]
|-
|-
| 1
| 1
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|-
|-
| 11
| 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]]
| [[Hendecatonic (temperament)|Hendecatonic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== 13-limit detempering of 121et ==
== 13-limit detempering of 121et ==
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[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]
[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 ==
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:Grendel]]
[[Category:Grendel]]
[[Category:Quintupole]]
[[Category:Quintupole]]
Retrieved from "https://en.xen.wiki/w/121edo"