121edo: Difference between revisions

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← 120edo

121edo

122edo →

Prime factorization

11²

Step size

9.91736 ¢

Fifth

71\121 (704.132 ¢)

Semitones (A1:m2)

13:8 (128.9 ¢ : 79.34 ¢)

Consistency limit

19

Distinct consistency limit

15

121 equal divisions of the octave (abbreviated 121edo or 121ed2), also called 121-tone equal temperament (121tet) or 121 equal temperament (121et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 121 equal parts of about 9.92 ¢ each. Each step represents a frequency ratio of 2^1/121, or the 121st root of 2.

Theory

121edo has a distinctly sharp tendency, in that the odd harmonics from 3 to 19 all have sharp tunings. It tempers out 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 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.

Odd harmonics

Approximation of odd harmonics in 121edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+2.18	+0.46	+3.07	+4.35	+4.05	+2.45	+2.64	+4.14	+0.01	-4.67	-3.48
Error	Relative (%)	+22.0	+4.7	+31.0	+43.9	+40.9	+24.7	+26.6	+41.7	+0.1	-47.0	-35.1
Steps (reduced)		192 (71)	281 (39)	340 (98)	384 (21)	419 (56)	448 (85)	473 (110)	495 (11)	514 (30)	531 (47)	547 (63)

Subsets and supersets

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

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[192 -121⟩	[⟨121 192]]	−0.687	0.687	6.93
2.3.5	15625/15552, [31 -21 1⟩	[⟨121 192 281]]	−0.524	0.606	6.11
2.3.5.7	4000/3969, 6144/6125, 10976/10935	[⟨121 192 281 340]]	−0.667	0.580	5.85
2.3.5.7.11	540/539, 896/891, 1375/1372, 4375/4356	[⟨121 192 281 340 419]]	−0.768	0.556	5.61
2.3.5.7.11.13	325/324, 352/351, 364/363, 540/539, 625/624	[⟨121 192 281 340 419 448]]	−0.750	0.510	5.14
2.3.5.7.11.13.17	256/255, 325/324, 352/351, 364/363, 375/374, 442/441	[⟨121 192 281 340 419 448 495]]	−0.787	0.480	4.85
2.3.5.7.11.13.17.19	190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374	[⟨121 192 281 340 419 448 495 514]]	−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 111 before being superseded by 130 in all those limits except for the 17-limit, where it is superseded by 140.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	9\121	89.26	21/20	Slithy
1	10\121	99.17	18/17	Quintupole
1	12\121	119.01	15/14	Subsedia
1	13\121	128.93	14/13	Tertiathirds
1	16\121	158.68	35/32	Hemikleismic
1	27\121	267.77	7/6	Hemimaquila
1	32\121	317.36	6/5	Metakleismic
1	39\121	386.78	5/4	Grendel
1	40\121	396.69	44/35	Squarschmidt
1	42\121	416.53	14/11	Sqrtphi
1	46\121	456.20	125/96	Qak
1	47\121	466.12	55/42	Hemiseptisix
1	48\121	476.03	21/16	Subfourth
1	50\121	495.87	4/3	Leapday / polypyth
1	51\121	505.79	75/56	Marfifths / marf / diatessic
1	54\121	535.54	512/375	Maquila
1	59\121	585.12	7/5	Pluto
11	50\121 (5\121)	495.87 (49.59)	4/3 (36/35)	Hendecatonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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

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.

121edo: Difference between revisions

Latest revision as of 15:38, 24 June 2025

Contents

Theory

Odd harmonics

Subsets and supersets

Regular temperament properties

Rank-2 temperaments

13-limit detempering of 121et

Miscellany

Navigation menu

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-01-26 13:37:06 UTC</tt>.<br>
-: The original revision id was <tt>485361746</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It 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 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.
+== Theory ==
+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]].
-=13-limit detempering of 121et=
+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.
-[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]</pre></div>
-<h4>Original HTML content:</h4>
+=== Odd harmonics ===
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;121edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It 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 &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 13-limit &lt;a class="wiki_link" href="/Mirkwai%20clan"&gt;grendel temperament&lt;/a&gt;. It is &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through to the 19 odd limit and uniquely consistent to the 15 odd limit.&lt;br /&gt;
+{{Harmonics in equal|121}}
-&lt;br /&gt;
-Because it tempers out 540/539 it allows &lt;a class="wiki_link" href="/swetismic%20chords"&gt;swetismic chords&lt;/a&gt;, because it tempers out 325/324 it allows &lt;a class="wiki_link" href="/marveltwin%20triad"&gt;marveltwin chords&lt;/a&gt;, because it tempers out 640/637 it allows &lt;a class="wiki_link" href="/huntmic%20chords"&gt;huntmic chords&lt;/a&gt;, because it tempers out 352/351 it allows &lt;a class="wiki_link" href="/minthmic%20chords"&gt;minthmic chords&lt;/a&gt;, because it tempers out 364/363 it allows &lt;a class="wiki_link" href="/gentle%20chords"&gt;gentle chords&lt;/a&gt;, because it tempers out 676/675 it allows &lt;a class="wiki_link" href="/island%20tetrad"&gt;island chords&lt;/a&gt; and because it tempers out 1575/1573 it allows the &lt;a class="wiki_link" href="/nicolic%20tetrad"&gt;nicolic tetrad&lt;/a&gt;. 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.&lt;br /&gt;
+=== Subsets and supersets ===
-&lt;br /&gt;
+Since 121 factors into 11<sup>2</sup>, 121edo contains [[11edo]] as its only nontrivial subset.
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x13-limit detempering of 121et"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;13-limit detempering of 121et&lt;/h1&gt;
-[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]&lt;/body&gt;&lt;/html&gt;</pre></div>
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| 192 -121 }}
+| {{mapping| 121 192 }}
+| −0.687
+| 0.687
+| 6.93
+|-
+| 2.3.5
+| 15625/15552, {{monzo| 31 -21 1 }}
+| {{mapping| 121 192 281 }}
+| −0.524
+| 0.606
+| 6.11
+|-
+| 2.3.5.7
+| 4000/3969, 6144/6125, 10976/10935
+| {{mapping| 121 192 281 340 }}
+| −0.667
+| 0.580
+| 5.85
+|-
+| 2.3.5.7.11
+| 540/539, 896/891, 1375/1372, 4375/4356
+| {{mapping| 121 192 281 340 419 }}
+| −0.768
+| 0.556
+| 5.61
+|-
+| 2.3.5.7.11.13
+| 325/324, 352/351, 364/363, 540/539, 625/624
+| {{mapping| 121 192 281 340 419 448 }}
+| −0.750
+| 0.510
+| 5.14
+|-
+| 2.3.5.7.11.13.17
+| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
+| {{mapping| 121 192 281 340 419 448 495 }}
+| −0.787
+| 0.480
+| 4.85
+|-
+| 2.3.5.7.11.13.17.19
+| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
+| {{mapping| 121 192 281 340 419 448 495 514 }}
+| −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"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperament
+|-
+| 1
+| 9\121
+| 89.26
+| 21/20
+| [[Slithy]]
+|-
+| 1
+| 10\121
+| 99.17
+| 18/17
+| [[Quintupole]]
+|-
+| 1
+| 12\121
+| 119.01
+| 15/14
+| [[Subsedia]]
+|-
+| 1
+| 13\121
+| 128.93
+| 14/13
+| [[Tertiathirds]]
+|-
+| 1
+| 16\121
+| 158.68
+| 35/32
+| [[Hemikleismic]]
+|-
+| 1
+| 27\121
+| 267.77
+| 7/6
+| [[Hemimaquila]]
+|-
+| 1
+| 32\121
+| 317.36
+| 6/5
+| [[Metakleismic]]
+|-
+| 1
+| 39\121
+| 386.78
+| 5/4
+| [[Grendel]]
+|-
+| 1
+| 40\121
+| 396.69
+| 44/35
+| [[Squarschmidt]]
+|-
+| 1
+| 42\121
+| 416.53
+| 14/11
+| [[Sqrtphi]]
+|-
+| 1
+| 46\121
+| 456.20
+| 125/96
+| [[Qak]]
+|-
+| 1
+| 47\121
+| 466.12
+| 55/42
+| [[Hemiseptisix]]
+|-
+| 1
+| 48\121
+| 476.03
+| 21/16
+| [[Subfourth]]
+|-
+| 1
+| 50\121
+| 495.87
+| 4/3
+| [[Leapday]] / [[polypyth]]
+|-
+| 1
+| 51\121
+| 505.79
+| 75/56
+| [[Marfifths]] / marf / diatessic
+|-
+| 1
+| 54\121
+| 535.54
+| 512/375
+| [[Maquila]]
+|-
+| 1
+| 59\121
+| 585.12
+| 7/5
+| [[Pluto]]
+|-
+| 11
+| 50\121<br />(5\121)
+| 495.87<br />(49.59)
+| 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 ==
+{{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 ==
+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:Quintupole]]

121edo: Difference between revisions

Latest revision as of 15:38, 24 June 2025

Theory

Odd harmonics

Subsets and supersets

Regular temperament properties

Rank-2 temperaments

13-limit detempering of 121et

Miscellany

Navigation menu

Search