Revision as of 00:00, 17 July 2018

21 equal divisions of the octave

Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo augmented triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO <26.

21-EDO as a temperament:

In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit laconic temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

Degree	Cents	Up/down Notation			5L3s Octotonic Notation	D.-R. Interval Types	Approximate *Ratios 1**	*ApproximateRatios 2**	*ApproximateRatios 3**
0	0	1	unison	C	C	Unison	1/1	1/1	1/1
1	57.14	^1 vv2	up unison, double-down 2nd	C^ Dvv	C#	Subminor 2nd	28/27, 30/29	35/34, 36/35	64/63
2	114.29	^^1 v2	double-up unison, down 2nd	C^^ Dv	Db	Minor 2nd	16/15, 15/14, 29/27	18/17	16/15, 25/24
3	171.43	2	2nd	D	D	Submajor 2nd	10/9, 32/29	10/9,11/10	9/8
4	228.57	^2 vv3	up 2nd, double-down 3rd	D^ Evv	D#	Supermajor 2nd	8/7	8/7	8/7, 10/9, 11/10
5	285.71	^^2 v3	double-up 2nd, down 3rd	D^^ Ev	Eb	Subminor 3rd	27/23, 32/27	13/11, 20/17	6/5, 7/6
6	342.86	3	3rd	E	E	Neutral 3rd	28/23	11/9	16/13
7	400	^3 vv4	up 3rd, double-down 4th	E^ Fvv	E#/Fb	Major 3rd	29/23	44/35	5/4, 9/7, 11/9, 14/11
8	457.14	^^3 v4	double-up 3rd, down 4th	E^^ Fv	F	Third-Fourth	30/23	13/10, 17/13, 22/17	13/10
9	514.29	4	4th	F	F#	Acute 4th	161/120, 256/189	35/26	4/3, 18/13
10	571.43	^4 vv5	up 4th, double-down 5th	F^ Gvv	Gb	Narrow Tritone	32/23	18/13	7/5, 11/8
11	628.57	^^4 v5	double-up 4th, down 5th	F^^ Gv	G	Wide Tritone	23/16	13/9	10/7, 16/11
12	685.71	5	5th	G	G#	Grave 5th	189/128, 240/161	52/35	3/2, 13/9
13	742.86	^5 vv6	up 5th, double-down 6th	G^ Avv	Hb	Fifth-Sixth	23/15	17/11, 20/13, 26/17	20/13
14	800	^^5 v6	double-up 5th, down 6th	G^^ Av	H	Minor 6th	46/29	35/22	8/5, 11/7, 14/9, 18/11
15	857.14	6	6th	A	H#/Ab	Neutral 6th	23/14	18/11	13/8
16	914.29	^6 vv7	up 6th, double-down 7th	A^ Bvv	A	Supermajor 6th	27/16, 46/27	17/10, 22/13	5/3, 12/7
17	971.43	^^6 v7	double-up 6th, down 7th	A^^ Bv	A#	Subminor 7th	7/4	7/4	7/4, 9/5, 20/11
18	1028.57	7	7th	B	Bb	Supraminor 7th	29/16, 9/5	9/5, 20/11	16/9
19	1085.71	^7 vv8	up 7th, double-down 8ve	B^ Cvv	B	Major 7th	15/8	17/9	15/8, 48/25
20	1142.86	^^7 v8	double-up 7th, down 8ve	B^^ Cv	B#/Cb	Supermajor 7th	27/14, 29/15	35/18, 68/35	63/32
21	1200	8	8ve	C	C	Octave	2/1	2/1	2/1

1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament

2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament

3: based on treating 21-EDO as 13-limit laconic temperament

Chord Names

Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.

0-6-12 = C E G = C = C or C perfect

0-5-12 = C Ev G = C(v3) = C down-three

0-7-12 = C E^ G = C(^3) = C up-three

0-6-11 = C E Gv = C(v5) = C down-five

0-7-13 = C E^ G^ = C(^3,^5) = C up-three up-five

0-6-12-18 = C E G B = C7 = C seven

0-6-12-17 = C E G Bv = C(v7) = C down-seven

0-5-12-18 = C Ev G B = C7(v3) = C seven down-three

0-5-12-17 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

21-tone scales:

augment6

augment9

augment12

Triadic Harmony in 21-EDO:

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:

Steps	Cents	Ratio	Example in C	Written name	Spoken name
0-5-10	0-286-571	23:27:32	C Ev Gvv	C.v(vv5)	C dot down, double-down five
0-4-11	0-229-629	7:8:10	C Evv Gv	C.vv(v5)	C dot double-down, down five
0-6-11	0-343-629	9:11:13	C E Gv	C(v5)	C down-five
0-5-13	0-286-743	11:13:17	C Ev G^	C.v(^5)	C dot down up-five
0-8-13	0-457-743	13:17:20	C Fv G^	C.v4(^5)	C (sus) down-four up-five
0-5-15	0-286-857	11:13:18	C Ev A	A(v5)	(inversion of 9:11:13)

Moment-of-Symmetry Scales in 21-EDO:

Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.

For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.

For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.

Tetrachordal Scales in 21-EDO

While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:

Step Pattern	Cents	Example	Name*	Ups/downs name
3, 3, 3	(0)-171-343-(514)	C D E F	Equable diatonic	C perfect
4, 3, 2	(0)-229-400-(514)	C D^ E^ F	Soft diatonic	C upperfect up-2
4, 4, 1	(0)-229-457-(514)	C D^ E^^ F	Intense diatonic	C up-2 & 6, double-up-3 & 7
5, 3, 1	(0)-286-457-(514)	C D^^ E^^ F	Archytas chromatic	C double-up-2, 3, 6 and 7
5, 2, 2	(0)-286-400-(514)	C D^^ E^ F	Weak chromatic	C double-up 2 & 6, up-3 & 7
6, 2, 1	(0)-343-457-(514)	C D^3 E^^ F	Strong enharmonic	C triple-up 2 & 6, double-up 3 & 7
7, 1, 1	(0)-400-457-(514)	C D^4 E^^ F	Pythagorean enharmonic	C quadruple-up 2 & 6, double-up 3 & 7

these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!

The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.

Rank two temperaments

List of 21edo rank two temperaments by badness

Periods per octave	Generator	Temperaments
1	1\21	Escapade
1	2\21	Miracle
1	4\21	Slendric/Gorgo/Gidorah
1	5\21	Subklei
1	8\21	Tridec
1	10\21	Triton
3	1\21
3	2\21	Augmented/August
3	3\21	Oodako
7	1\21	Whitewood

13-limit Commas

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)

Comma	Monzo	Value (Cents)	Name 1	Name 2
2187/2048	\| -11 7 >	113.69	Apotome
128/125	\| 7 0 -3 >	41.06	Diesis	Augmented Comma
	\| -25 7 6 >	31.57	Ampersand's Comma
	\| 32 -7 -9 >	9.49	Escapade Comma
1029/1000	\| -3 1 -3 3 >	49.49	Keega
36/35	\| 2 2 -1 -1 >	48.77	Septimal Quarter Tone
	\| -10 7 8 -7 >	22.41	Blackjackisma
1029/1024	\| -10 1 0 3 >	8.43	Gamelisma
225/224	\| -5 2 2 -1 >	7.71	Septimal Kleisma	Marvel Comma
16875/16807	\| 0 3 4 -5 >	6.99	Mirkwai
2401/2400	\| -5 -1 -2 4 >	0.72	Breedsma
	\| 47 -7 -7 -7 >	0.34	Akjaysma	5\7 Octave Comma
99/98	\| -1 2 0 -2 1 >	17.58	Mothwellsma
176/175	\| 4 0 -2 -1 1 >	9.86	Valinorsma
4000/3993	\| 5 -1 3 0 -3 >	3.03	Wizardharry

Books / Literature:

Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.

External image: http://www.ronsword.com/images/ron1.jpg

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External image: http://www.swordguitars.com/21tetsm.JPG

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21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)

Compositions/Listening:

Iridescent Wenge Fugue by Stephen Weigel, accepted to SEAMUS 2018 and Electroacoustic Barn Dance 2018

21-edo Trio for Organ, by Claudi Meneghin

21-penny jingle, by Claudi Meneghin

Short Clip of 21-edo Acoustic by Ron Sword

Open tuning Drone Improvisation in 21-edo by Ron Sword

Anomalous Readings play by Andrew Heathwaite

Comets Over Flatland 15 by Randy Winchester

Comets Over Flatland 18 by Randy Winchester

L'esatonale ubriaco (the drunk hexatonal), ALIENAMENTE by Fabrizio Fulvio Fausto Fiale

21edo: Difference between revisions

Revision as of 00:00, 17 July 2018

Contents

21 equal divisions of the octave

21-EDO as a temperament:

Chord Names

Triadic Harmony in 21-EDO:

Moment-of-Symmetry Scales in 21-EDO:

Tetrachordal Scales in 21-EDO

Rank two temperaments

13-limit Commas

Books / Literature:

Compositions/Listening:

Navigation menu

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-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+<span style="display: block; text-align: right;">[[21平均律|日本語]]</span>
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Stephen_Weigel|Stephen_Weigel]] and made on <tt>2018-01-29 12:52:38 UTC</tt>.<br>
+=21 equal divisions of the octave=
-: The original revision id was <tt>625539519</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">&lt;span style="display: block; text-align: right;"&gt;[[21平均律|日本語]]
-&lt;/span&gt;
-=21 equal divisions of the octave=
 Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &lt;26.
 ==21-EDO as a temperament:==
 In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.
 In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.
-The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
+The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic_Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
+{| class="wikitable"
+|-
+| style="text-align:center;" | '''Degree'''
+| style="text-align:center;" | '''Cents'''
+| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Up/down]]
+[[Ups_and_Downs_Notation|Notation]]
+| style="text-align:center;" | '''5L3s'''
+'''Octotonic'''
+'''Notation'''
+| style="text-align:center;" | '''D.-R. Interval'''
+'''Types'''
+| style="text-align:center;" | '''Approximate'''
+'''Ratios *1'''
+| style="text-align:center;" | <span style="display: block; text-align: center;">'''Approximate'''</span><span style="display: block; text-align: center;">'''Ratios *2'''</span>
+| style="text-align:center;" | <span style="display: block; text-align: center;">'''Approximate'''</span>'''Ratios *3'''
+|-
+| style="text-align:center;" | 0
+| style="text-align:center;" | 0
+| style="text-align:right;" | 1
+| style="text-align:center;" | unison
+| | C
+| style="text-align:center;" | C
+| style="text-align:center;" | Unison
+| style="text-align:center;" | 1/1
+| style="text-align:center;" | 1/1
+| style="text-align:center;" | 1/1
+|-
+| style="text-align:center;" | 1
+| style="text-align:center;" | 57.14
+| style="text-align:right;" | ^1
+vv2
+| style="text-align:center;" | up unison,
+double-down 2nd
+| | C^
+Dvv
+| style="text-align:center;" | C#
+| style="text-align:center;" | Subminor 2nd
+| style="text-align:center;" | 28/27, 30/29
+| style="text-align:center;" | 35/34, 36/35
+| style="text-align:center;" | 64/63
+|-
+| style="text-align:center;" | 2
+| style="text-align:center;" | 114.29
+| style="text-align:right;" | ^^1
+v2
+| style="text-align:center;" | double-up unison,
+down 2nd
+| | C^^
+Dv
+| style="text-align:center;" | Db
+| style="text-align:center;" | Minor 2nd
+| style="text-align:center;" | 16/15, 15/14, 29/27
+| style="text-align:center;" | 18/17
+| style="text-align:center;" | 16/15, 25/24
+|-
+| style="text-align:center;" | 3
+| style="text-align:center;" | 171.43
+| style="text-align:right;" | 2
+| style="text-align:center;" | 2nd
+| | D
+| style="text-align:center;" | D
+| style="text-align:center;" | Submajor 2nd
+| style="text-align:center;" | 10/9, 32/29
+| style="text-align:center;" | 10/9,11/10
+| style="text-align:center;" | 9/8
+|-
+| style="text-align:center;" | 4
+| style="text-align:center;" | 228.57
+| style="text-align:right;" | ^2
+vv3
+| style="text-align:center;" | up 2nd,
+double-down 3rd
+| | D^
+Evv
+| style="text-align:center;" | D#
+| style="text-align:center;" | Supermajor 2nd
+| style="text-align:center;" | 8/7
+| style="text-align:center;" | 8/7
+| style="text-align:center;" | 8/7, 10/9, 11/10
+|-
+| style="text-align:center;" | 5
+| style="text-align:center;" | 285.71
+| style="text-align:right;" | ^^2
+v3
+| style="text-align:center;" | double-up 2nd,
+down 3rd
+| | D^^
-||= **Degree** ||= **Cents** ||||||= [[Ups and Downs Notation|Up/down]]
+Ev
-[[Ups and Downs Notation|Notation]] ||= **5L3s**
+| style="text-align:center;" | Eb
-**Octotonic**
+| style="text-align:center;" | Subminor 3rd
-**Notation** ||= **D.-R. Interval**
+| style="text-align:center;" | 27/23, 32/27
-**Types** ||= **Approximate**
+| style="text-align:center;" | 13/11, 20/17
-**Ratios *1** ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;**Ratios *2**&lt;/span&gt; ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;**Ratios *3** ||
+| style="text-align:center;" | 6/5, 7/6
-||= 0 ||= 0 ||&gt; 1 ||= unison ||&lt; C ||= C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
+|-
-||= 1 ||= 57.14 ||&gt; ^1
+| style="text-align:center;" | 6
-vv2 ||= up unison,
+| style="text-align:center;" | 342.86
-double-down 2nd ||&lt; C^
+| style="text-align:right;" | 3
-Dvv ||= C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
+| style="text-align:center;" | 3rd
-||= 2 ||= 114.29 ||&gt; ^^1
+| | E
-v2 ||= double-up unison,
+| style="text-align:center;" | E
-down 2nd ||&lt; C^^
+| style="text-align:center;" | Neutral 3rd
-Dv ||= Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
+| style="text-align:center;" | 28/23
-||= 3 ||= 171.43 ||&gt; 2 ||= 2nd ||&lt; D ||= D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
+| style="text-align:center;" | 11/9
-||= 4 ||= 228.57 ||&gt; ^2
+| style="text-align:center;" | 16/13
-vv3 ||= up 2nd,
+|-
-double-down 3rd ||&lt; D^
+| style="text-align:center;" | 7
-Evv ||= D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
+| style="text-align:center;" | 400
-||= 5 ||= 285.71 ||&gt; ^^2
+| style="text-align:right;" | ^3
-v3 ||= double-up 2nd,
-down 3rd ||&lt; D^^
+vv4
-Ev ||= Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
+| style="text-align:center;" | up 3rd,
-||= 6 ||= 342.86 ||&gt; 3 ||= 3rd ||&lt; E ||= E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
-||= 7 ||= 400 ||&gt; ^3
+double-down 4th
-vv4 ||= up 3rd,
+| | E^
-double-down 4th ||&lt; E^
-Fvv ||= E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
+Fvv
-||= 8 ||= 457.14 ||&gt; ^^3
+| style="text-align:center;" | E#/Fb
-v4 ||= double-up 3rd,
+| style="text-align:center;" | Major 3rd
-down 4th ||&lt; E^^
+| style="text-align:center;" | 29/23
-Fv ||= F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
+| style="text-align:center;" | 44/35
-||= 9 ||= 514.29 ||&gt; 4 ||= 4th ||&lt; F ||= F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
+| style="text-align:center;" | 5/4, 9/7, 11/9, 14/11
-||= 10 ||= 571.43 ||&gt; ^4
+|-
-vv5 ||= up 4th,
+| style="text-align:center;" | 8
-double-down 5th ||&lt; F^
+| style="text-align:center;" | 457.14
-Gvv ||= Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
+| style="text-align:right;" | ^^3
-||= 11 ||= 628.57 ||&gt; ^^4
-v5 ||= double-up 4th,
+v4
-down 5th ||&lt; F^^
+| style="text-align:center;" | double-up 3rd,
-Gv ||= G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
-||= 12 ||= 685.71 ||&gt; 5 ||= 5th ||&lt; G ||= G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
+down 4th
-||= 13 ||= 742.86 ||&gt; ^5
+| | E^^
-vv6 ||= up 5th,
-double-down 6th ||&lt; G^
+Fv
-Avv ||= Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
+| style="text-align:center;" | F
-||= 14 ||= 800 ||&gt; ^^5
+| style="text-align:center;" | Third-Fourth
-v6 ||= double-up 5th,
+| style="text-align:center;" | 30/23
-down 6th ||&lt; G^^
+| style="text-align:center;" | 13/10, 17/13, 22/17
-Av ||= H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
+| style="text-align:center;" | 13/10
-||= 15 ||= 857.14 ||&gt; 6 ||= 6th ||&lt; A ||= H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
+|-
-||= 16 ||= 914.29 ||&gt; ^6
+| style="text-align:center;" | 9
-vv7 ||= up 6th,
+| style="text-align:center;" | 514.29
-double-down 7th ||&lt; A^
+| style="text-align:right;" | 4
-Bvv ||= A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
+| style="text-align:center;" | 4th
-||= 17 ||= 971.43 ||&gt; ^^6
+| | F
-v7 ||= double-up 6th,
+| style="text-align:center;" | F#
-down 7th ||&lt; A^^
+| style="text-align:center;" | Acute 4th
-Bv ||= A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
+| style="text-align:center;" | 161/120, 256/189
-||= 18 ||= 1028.57 ||&gt; 7 ||= 7th ||&lt; B ||= Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
+| style="text-align:center;" | 35/26
-||= 19 ||= 1085.71 ||&gt; ^7
+| style="text-align:center;" | 4/3, 18/13
-vv8 ||= up 7th,
+|-
-double-down 8ve ||&lt; B^
+| style="text-align:center;" | 10
-Cvv ||= B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
+| style="text-align:center;" | 571.43
-||= 20 ||= 1142.86 ||&gt; ^^7
+| style="text-align:right;" | ^4
-v8 ||= double-up 7th,
-down 8ve ||&lt; B^^
+vv5
-Cv ||= B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
+| style="text-align:center;" | up 4th,
-||= 21 ||= 1200 ||&gt; 8 ||= 8ve ||&lt; C ||= C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||
+double-down 5th
+| | F^
+Gvv
+| style="text-align:center;" | Gb
+| style="text-align:center;" | Narrow Tritone
+| style="text-align:center;" | 32/23
+| style="text-align:center;" | 18/13
+| style="text-align:center;" | 7/5, 11/8
+|-
+| style="text-align:center;" | 11
+| style="text-align:center;" | 628.57
+| style="text-align:right;" | ^^4
+v5
+| style="text-align:center;" | double-up 4th,
+down 5th
+| | F^^
+Gv
+| style="text-align:center;" | G
+| style="text-align:center;" | Wide Tritone
+| style="text-align:center;" | 23/16
+| style="text-align:center;" | 13/9
+| style="text-align:center;" | 10/7, 16/11
+|-
+| style="text-align:center;" | 12
+| style="text-align:center;" | 685.71
+| style="text-align:right;" | 5
+| style="text-align:center;" | 5th
+| | G
+| style="text-align:center;" | G#
+| style="text-align:center;" | Grave 5th
+| style="text-align:center;" | 189/128, 240/161
+| style="text-align:center;" | 52/35
+| style="text-align:center;" | 3/2, 13/9
+|-
+| style="text-align:center;" | 13
+| style="text-align:center;" | 742.86
+| style="text-align:right;" | ^5
+vv6
+| style="text-align:center;" | up 5th,
+double-down 6th
+| | G^
+Avv
+| style="text-align:center;" | Hb
+| style="text-align:center;" | Fifth-Sixth
+| style="text-align:center;" | 23/15
+| style="text-align:center;" | 17/11, 20/13, 26/17
+| style="text-align:center;" | 20/13
+|-
+| style="text-align:center;" | 14
+| style="text-align:center;" | 800
+| style="text-align:right;" | ^^5
+v6
+| style="text-align:center;" | double-up 5th,
+down 6th
+| | G^^
+Av
+| style="text-align:center;" | H
+| style="text-align:center;" | Minor 6th
+| style="text-align:center;" | 46/29
+| style="text-align:center;" | 35/22
+| style="text-align:center;" | 8/5, 11/7, 14/9, 18/11
+|-
+| style="text-align:center;" | 15
+| style="text-align:center;" | 857.14
+| style="text-align:right;" | 6
+| style="text-align:center;" | 6th
+| | A
+| style="text-align:center;" | H#/Ab
+| style="text-align:center;" | Neutral 6th
+| style="text-align:center;" | 23/14
+| style="text-align:center;" | 18/11
+| style="text-align:center;" | 13/8
+|-
+| style="text-align:center;" | 16
+| style="text-align:center;" | 914.29
+| style="text-align:right;" | ^6
+vv7
+| style="text-align:center;" | up 6th,
+double-down 7th
+| | A^
+Bvv
+| style="text-align:center;" | A
+| style="text-align:center;" | Supermajor 6th
+| style="text-align:center;" | 27/16, 46/27
+| style="text-align:center;" | 17/10, 22/13
+| style="text-align:center;" | 5/3, 12/7
+|-
+| style="text-align:center;" | 17
+| style="text-align:center;" | 971.43
+| style="text-align:right;" | ^^6
+v7
+| style="text-align:center;" | double-up 6th,
+down 7th
+| | A^^
+Bv
+| style="text-align:center;" | A#
+| style="text-align:center;" | Subminor 7th
+| style="text-align:center;" | 7/4
+| style="text-align:center;" | 7/4
+| style="text-align:center;" | 7/4, 9/5, 20/11
+|-
+| style="text-align:center;" | 18
+| style="text-align:center;" | 1028.57
+| style="text-align:right;" | 7
+| style="text-align:center;" | 7th
+| | B
+| style="text-align:center;" | Bb
+| style="text-align:center;" | Supraminor 7th
+| style="text-align:center;" | 29/16, 9/5
+| style="text-align:center;" | 9/5, 20/11
+| style="text-align:center;" | 16/9
+|-
+| style="text-align:center;" | 19
+| style="text-align:center;" | 1085.71
+| style="text-align:right;" | ^7
+vv8
+| style="text-align:center;" | up 7th,
+double-down 8ve
+| | B^
+Cvv
+| style="text-align:center;" | B
+| style="text-align:center;" | Major 7th
+| style="text-align:center;" | 15/8
+| style="text-align:center;" | 17/9
+| style="text-align:center;" | 15/8, 48/25
+|-
+| style="text-align:center;" | 20
+| style="text-align:center;" | 1142.86
+| style="text-align:right;" | ^^7
+v8
+| style="text-align:center;" | double-up 7th,
+down 8ve
+| | B^^
+Cv
+| style="text-align:center;" | B#/Cb
+| style="text-align:center;" | Supermajor 7th
+| style="text-align:center;" | 27/14, 29/15
+| style="text-align:center;" | 35/18, 68/35
+| style="text-align:center;" | 63/32
+|-
+| style="text-align:center;" | 21
+| style="text-align:center;" | 1200
+| style="text-align:right;" | 8
+| style="text-align:center;" | 8ve
+| | C
+| style="text-align:center;" | C
+| style="text-align:center;" | Octave
+| style="text-align:center;" | 2/1
+| style="text-align:center;" | 2/1
+| style="text-align:center;" | 2/1
+|}
 *1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
 *2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
 *3: based on treating 21-EDO as 13-limit laconic temperament
 =Chord Names=
 Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.
 -6-12 = C E G = C = C or C perfect
 -5-12 = C Ev G = C(v3) = C down-three
 -7-12 = C E^ G = C(^3) = C up-three
 -6-11 = C E Gv = C(v5) = C down-five
 -7-13 = C E^ G^ = C(^3,^5) = C up-three up-five
 -6-12-18 = C E G B = C7 = C seven
 -6-12-17 = C E G Bv = C(v7) = C down-seven
 -5-12-18 = C Ev G B = C7(v3) = C seven down-three
 -5-12-17 = C Ev G Bv = C.v7 = C dot down seven
-For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].
+For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
+'''21-tone scales:'''
+[[augment6|augment6]]
+[[augment9|augment9]]
-**21-tone scales:**
+[[augment12|augment12]]
-[[augment6]]
-[[augment9]]
-[[augment12]]
 ==Triadic Harmony in 21-EDO:==
 One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:
-||= **Steps** ||= **Cents** ||= **Ratio** || **Example in C** || **Written name** || **Spoken name** ||
+{| class="wikitable"
-||= 0-5-10 ||= 0-286-571 ||= 23:27:32 || C Ev Gvv || C.v(vv5) || C dot down, double-down five ||
+|-
-||= 0-4-11 ||= 0-229-629 ||= 7:8:10 || C Evv Gv || C.vv(v5) || C dot double-down, down five ||
+| style="text-align:center;" | '''Steps'''
-||= 0-6-11 ||= 0-343-629 ||= 9:11:13 || C E Gv || C(v5) || C down-five ||
+| style="text-align:center;" | '''Cents'''
-||= 0-5-13 ||= 0-286-743 ||= 11:13:17 || C Ev G^ || C.v(^5) || C dot down up-five ||
+| style="text-align:center;" | '''Ratio'''
-||= 0-8-13 ||= 0-457-743 ||= 13:17:20 || C Fv G^ || C.v4(^5) || C (sus) down-four up-five ||
+| | '''Example in C'''
-||= 0-5-15 ||= 0-286-857 ||= 11:13:18 || C Ev A || A(v5) || (inversion of 9:11:13) ||
+| | '''Written name'''
+| | '''Spoken name'''
+|-
+| style="text-align:center;" | 0-5-10
+| style="text-align:center;" | 0-286-571
+| style="text-align:center;" | 23:27:32
+| | C Ev Gvv
+| | C.v(vv5)
+| | C dot down, double-down five
+|-
+| style="text-align:center;" | 0-4-11
+| style="text-align:center;" | 0-229-629
+| style="text-align:center;" | 7:8:10
+| | C Evv Gv
+| | C.vv(v5)
+| | C dot double-down, down five
+|-
+| style="text-align:center;" | 0-6-11
+| style="text-align:center;" | 0-343-629
+| style="text-align:center;" | 9:11:13
+| | C E Gv
+| | C(v5)
+| | C down-five
+|-
+| style="text-align:center;" | 0-5-13
+| style="text-align:center;" | 0-286-743
+| style="text-align:center;" | 11:13:17
+| | C Ev G^
+| | C.v(^5)
+| | C dot down up-five
+|-
+| style="text-align:center;" | 0-8-13
+| style="text-align:center;" | 0-457-743
+| style="text-align:center;" | 13:17:20
+| | C Fv G^
+| | C.v4(^5)
+| | C (sus) down-four up-five
+|-
+| style="text-align:center;" | 0-5-15
+| style="text-align:center;" | 0-286-857
+| style="text-align:center;" | 11:13:18
+| | C Ev A
+| | A(v5)
+| | (inversion of 9:11:13)
+|}
 ==Moment-of-Symmetry Scales in 21-EDO:==
 Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.
 For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.
 For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.
 ==Tetrachordal Scales in 21-EDO==
 While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:
-||= Step Pattern ||= Cents || Example ||= Name* ||&lt; Ups/downs name ||
+{| class="wikitable"
-||= 3, 3, 3 ||= (0)-171-343-(514) || C D E F ||= Equable diatonic ||&lt; C perfect ||
+|-
-||= 4, 3, 2 ||= (0)-229-400-(514) || C D^ E^ F ||= Soft diatonic ||&lt; C upperfect up-2 ||
+| style="text-align:center;" | Step Pattern
-||= 4, 4, 1 ||= (0)-229-457-(514) || C D^ E^^ F ||= Intense diatonic ||&lt; C up-2 &amp; 6, double-up-3 &amp; 7 ||
+| style="text-align:center;" | Cents
-||= 5, 3, 1 ||= (0)-286-457-(514) || C D^^ E^^ F ||= Archytas chromatic ||&lt; C double-up-2, 3, 6 and 7 ||
+| | Example
-||= 5, 2, 2 ||= (0)-286-400-(514) || C D^^ E^ F ||= Weak chromatic ||&lt; C double-up 2 &amp; 6, up-3 &amp; 7 ||
+| style="text-align:center;" | Name*
-||= 6, 2, 1 ||= (0)-343-457-(514) || C D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; E^^ F ||= Strong enharmonic ||&lt; C triple-up 2 &amp; 6, double-up 3 &amp; 7 ||
+| | Ups/downs name
-||= 7, 1, 1 ||= (0)-400-457-(514) || C D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;4&lt;/span&gt; E^^ F ||= Pythagorean enharmonic ||&lt; C quadruple-up 2 &amp; 6, double-up 3 &amp; 7 ||
+|-
+| style="text-align:center;" | 3, 3, 3
+| style="text-align:center;" | (0)-171-343-(514)
+| | C D E F
+| style="text-align:center;" | Equable diatonic
+| | C perfect
+|-
+| style="text-align:center;" | 4, 3, 2
+| style="text-align:center;" | (0)-229-400-(514)
+| | C D^ E^ F
+| style="text-align:center;" | Soft diatonic
+| | C upperfect up-2
+|-
+| style="text-align:center;" | 4, 4, 1
+| style="text-align:center;" | (0)-229-457-(514)
+| | C D^ E^^ F
+| style="text-align:center;" | Intense diatonic
+| | C up-2 &amp; 6, double-up-3 &amp; 7
+|-
+| style="text-align:center;" | 5, 3, 1
+| style="text-align:center;" | (0)-286-457-(514)
+| | C D^^ E^^ F
+| style="text-align:center;" | Archytas chromatic
+| | C double-up-2, 3, 6 and 7
+|-
+| style="text-align:center;" | 5, 2, 2
+| style="text-align:center;" | (0)-286-400-(514)
+| | C D^^ E^ F
+| style="text-align:center;" | Weak chromatic
+| | C double-up 2 &amp; 6, up-3 &amp; 7
+|-
+| style="text-align:center;" | 6, 2, 1
+| style="text-align:center;" | (0)-343-457-(514)
+| | C D^<span style="font-size: 90%; vertical-align: super;">3</span> E^^ F
+| style="text-align:center;" | Strong enharmonic
+| | C triple-up 2 &amp; 6, double-up 3 &amp; 7
+|-
+| style="text-align:center;" | 7, 1, 1
+| style="text-align:center;" | (0)-400-457-(514)
+| | C D^<span style="font-size: 90%; vertical-align: super;">4</span> E^^ F
+| style="text-align:center;" | Pythagorean enharmonic
+| | C quadruple-up 2 &amp; 6, double-up 3 &amp; 7
+|}
 *these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
 The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.
 ==Rank two temperaments==
-[[List of 21edo rank two temperaments by badness]]
+[[List_of_21edo_rank_two_temperaments_by_badness|List of 21edo rank two temperaments by badness]]
-||~ Periods
-per octave ||~ Generator ||~ Temperaments ||
+{| class="wikitable"
-|| 1 || 1\21 || [[Escapade family#Escapade|Escapade]] ||
+|-
-|| 1 || 2\21 || [[Gamelismic clan#Miracle|Miracle]] ||
+! | Periods
-|| 1 || 4\21 || [[Slendric]]/[[Gamelismic clan#Gorgo|Gorgo]]/[[Gamelismic clan#Gidorah|Gidorah]] ||
-|| 1 || 5\21 || [[Mint temperaments#Subklei|Subklei]] ||
-|| 1 || 8\21 || [[Chromatic pairs#Tridec|Tridec]] ||
-|| 1 || 10\21 || [[Marvel temperaments#Triton|Triton]] ||
-|| 3 || 1\21 ||   ||
-|| 3 || 2\21 || [[Augmented family|Augmented]]/[[August]] ||
-|| 3 || 3\21 || [[Oodako]] ||
-|| 7 || 1\21 || [[Apotome family|Whitewood]] ||
+per octave
+! | Generator
+! | Temperaments
+|-
+| | 1
+| | 1\21
+| | [[Escapade_family#Escapade|Escapade]]
+|-
+| | 1
+| | 2\21
+| | [[Gamelismic_clan#Miracle|Miracle]]
+|-
+| | 1
+| | 4\21
+| | [[Slendric|Slendric]]/[[Gamelismic_clan#Gorgo|Gorgo]]/[[Gamelismic_clan#Gidorah|Gidorah]]
+|-
+| | 1
+| | 5\21
+| | [[Mint_temperaments#Subklei|Subklei]]
+|-
+| | 1
+| | 8\21
+| | [[Chromatic_pairs#Tridec|Tridec]]
+|-
+| | 1
+| | 10\21
+| | [[Marvel_temperaments#Triton|Triton]]
+|-
+| | 3
+| | 1\21
+| |
+|-
+| | 3
+| | 2\21
+| | [[Augmented_family|Augmented]]/[[august|August]]
+|-
+| | 3
+| | 3\21
+| | [[Oodako|Oodako]]
+|-
+| | 7
+| | 1\21
+| | [[Apotome_family|Whitewood]]
+|}
 ==13-limit Commas==
 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)
-||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
-||= 2187/2048 ||&lt; | -11 7 &gt; ||&gt; 113.69 ||= Apotome ||=   ||
-||= 128/125 ||&lt; | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||
-||=   ||&lt; | -25 7 6 &gt; ||&gt; 31.57 ||= Ampersand's Comma ||   ||
-||=   ||&lt; | 32 -7 -9 &gt; ||&gt; 9.49 ||= Escapade Comma ||   ||
-||= 1029/1000 ||&lt; | -3 1 -3 3 &gt; ||&gt; 49.49 ||= Keega ||=   ||
-||= 36/35 ||&lt; | 2 2 -1 -1 &gt; ||&gt; 48.77 ||= Septimal Quarter Tone ||   ||
-||=   ||&lt; | -10 7 8 -7 &gt; ||&gt; 22.41 ||= Blackjackisma ||=   ||
-||= 1029/1024 ||&lt; | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=   ||
-||= 225/224 ||&lt; | -5 2 2 -1 &gt; ||&gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||
-||= 16875/16807 ||&lt; | 0 3 4 -5 &gt; ||&gt; 6.99 ||= Mirkwai ||=   ||
-||= 2401/2400 ||&lt; | -5 -1 -2 4 &gt; ||&gt; 0.72 ||= Breedsma ||=   ||
-||=   ||&lt; | 47 -7 -7 -7 &gt; ||&gt; 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
-||= 99/98 ||&lt; | -1 2 0 -2 1 &gt; ||&gt; 17.58 ||= Mothwellsma ||=   ||
-||= 176/175 ||&lt; | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=   ||
-||= 4000/3993 ||&lt; | 5 -1 3 0 -3 &gt; ||&gt; 3.03 ||= Wizardharry ||=   ||
+{| class="wikitable"
+|-
+! | Comma
+! | Monzo
+! | Value (Cents)
+! | Name 1
+! | Name 2
+|-
+| style="text-align:center;" | 2187/2048
+| | | -11 7 &gt;
+| style="text-align:right;" | 113.69
+| style="text-align:center;" | Apotome
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 128/125
+| | | 7 0 -3 &gt;
+| style="text-align:right;" | 41.06
+| style="text-align:center;" | Diesis
+| style="text-align:center;" | Augmented Comma
+|-
+| style="text-align:center;" |
+| | | -25 7 6 &gt;
+| style="text-align:right;" | 31.57
+| style="text-align:center;" | Ampersand's Comma
+| |
+|-
+| style="text-align:center;" |
+| | | 32 -7 -9 &gt;
+| style="text-align:right;" | 9.49
+| style="text-align:center;" | Escapade Comma
+| |
+|-
+| style="text-align:center;" | 1029/1000
+| | | -3 1 -3 3 &gt;
+| style="text-align:right;" | 49.49
+| style="text-align:center;" | Keega
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 36/35
+| | | 2 2 -1 -1 &gt;
+| style="text-align:right;" | 48.77
+| style="text-align:center;" | Septimal Quarter Tone
+| |
+|-
+| style="text-align:center;" |
+| | | -10 7 8 -7 &gt;
+| style="text-align:right;" | 22.41
+| style="text-align:center;" | Blackjackisma
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 1029/1024
+| | | -10 1 0 3 &gt;
+| style="text-align:right;" | 8.43
+| style="text-align:center;" | Gamelisma
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 225/224
+| | | -5 2 2 -1 &gt;
+| style="text-align:right;" | 7.71
+| style="text-align:center;" | Septimal Kleisma
+| style="text-align:center;" | Marvel Comma
+|-
+| style="text-align:center;" | 16875/16807
+| | | 0 3 4 -5 &gt;
+| style="text-align:right;" | 6.99
+| style="text-align:center;" | Mirkwai
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 2401/2400
+| | | -5 -1 -2 4 &gt;
+| style="text-align:right;" | 0.72
+| style="text-align:center;" | Breedsma
+| style="text-align:center;" |
+|-
+| style="text-align:center;" |
+| | | 47 -7 -7 -7 &gt;
+| style="text-align:right;" | 0.34
+| style="text-align:center;" | Akjaysma
+| style="text-align:center;" | 5\7 Octave Comma
+|-
+| style="text-align:center;" | 99/98
+| | | -1 2 0 -2 1 &gt;
+| style="text-align:right;" | 17.58
+| style="text-align:center;" | Mothwellsma
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 176/175
+| | | 4 0 -2 -1 1 &gt;
+| style="text-align:right;" | 9.86
+| style="text-align:center;" | Valinorsma
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 4000/3993
+| | | 5 -1 3 0 -3 &gt;
+| style="text-align:right;" | 3.03
+| style="text-align:center;" | Wizardharry
+| style="text-align:center;" |
+|}
-=**Books / Literature:**=
+='''Books / Literature:'''=
 Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
-[[image:http://www.ronsword.com/images/ron1.jpg width="254" height="188"]][[image:http://www.swordguitars.com/21tetsm.JPG width="363" height="191"]]
-**//21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)//**
-=**Compositions/Listening:**=
-**[[https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal|Iridescent Wenge Fugue]] by Stephen Weigel, accepted to SEAMUS 2018 and Electroacoustic Barn Dance 2018**
-[[@http://soonlabel.com/xenharmonic/archives/2494|21-edo Trio for Organ, by Claudi Meneghin]]
-[[@http://soonlabel.com/xenharmonic/archives/2336|21-penny jingle, by Claudi Meneghin]]
-[[@http://www.ronsword.com/sounds/21_improv.mp3|Short Clip of 21-edo Acoustic]] by [[Ron Sword]]
-[[@http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3|Open tuning Drone Improvisation in 21-edo]] by Ron Sword
-[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933715|Anomalous Readings]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3|play]] by [[Andrew Heathwaite]]
-[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3|Comets Over Flatland 15]] by [[Randy Winchester]]
-[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3|Comets Over Flatland 18]] by [[Randy Winchester]]
-[[@http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco|L'esatonale ubriaco (the drunk hexatonal)]], ALIENAMENTE by [[xenharmonic/Fabrizio Fiale|Fabrizio Fulvio Fausto Fiale]]</pre></div>
-<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;21edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="/21%E5%B9%B3%E5%9D%87%E5%BE%8B"&gt;日本語&lt;/a&gt;&lt;br /&gt;
-&lt;/span&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x21 equal divisions of the octave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;21 equal divisions of the octave&lt;/h1&gt;
- &lt;br /&gt;
-Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &amp;quot;equi-heptatonic&amp;quot; scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &amp;lt;26.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x21 equal divisions of the octave-21-EDO as a temperament:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;21-EDO as a temperament:&lt;/h2&gt;
- In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a &amp;quot;third-fourth&amp;quot; (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.&lt;br /&gt;
-&lt;br /&gt;
-In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.&lt;br /&gt;
-&lt;br /&gt;
-The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit &lt;a class="wiki_link" href="/Laconic%20Family"&gt;laconic&lt;/a&gt; temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some &amp;quot;interesting&amp;quot; mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.&lt;br /&gt;
-&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Cents&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td colspan="3" style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;Up/down&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;Notation&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;5L3s&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;Octotonic&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;Notation&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;D.-R. Interval&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;Types&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Approximate&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;Ratios *1&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;Approximate&lt;/strong&gt;&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;Ratios *2&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;Approximate&lt;/strong&gt;&lt;/span&gt;&lt;strong&gt;Ratios *3&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;unison&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Unison&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;57.14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^1&lt;br /&gt;
-vv2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up unison,&lt;br /&gt;
-double-down 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C^&lt;br /&gt;
-Dvv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Subminor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;28/27, 30/29&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;35/34, 36/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;114.29&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^1&lt;br /&gt;
-v2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up unison,&lt;br /&gt;
-down 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C^^&lt;br /&gt;
-Dv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Minor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;16/15, 15/14, 29/27&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;18/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;16/15, 25/24&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;171.43&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;D&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Submajor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;10/9, 32/29&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;10/9,11/10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;228.57&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^2&lt;br /&gt;
-vv3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 2nd,&lt;br /&gt;
-double-down 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;D^&lt;br /&gt;
-Evv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Supermajor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8/7, 10/9, 11/10&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;285.71&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^2&lt;br /&gt;
-v3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 2nd,&lt;br /&gt;
-down 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;D^^&lt;br /&gt;
-Ev&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Subminor 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;27/23, 32/27&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13/11, 20/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;6/5, 7/6&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;342.86&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;E&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Neutral 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;28/23&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;11/9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;16/13&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;400&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^3&lt;br /&gt;
-vv4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 3rd,&lt;br /&gt;
-double-down 4th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;E^&lt;br /&gt;
-Fvv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;E#/Fb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Major 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;29/23&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;44/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5/4, 9/7, 11/9, 14/11&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;457.14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^3&lt;br /&gt;
-v4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 3rd,&lt;br /&gt;
-down 4th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;E^^&lt;br /&gt;
-Fv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Third-Fourth&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;30/23&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13/10, 17/13, 22/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13/10&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;514.29&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;4th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Acute 4th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;161/120, 256/189&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;35/26&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;4/3, 18/13&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;571.43&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^4&lt;br /&gt;
-vv5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 4th,&lt;br /&gt;
-double-down 5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;F^&lt;br /&gt;
-Gvv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Gb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Narrow Tritone&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;32/23&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;18/13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7/5, 11/8&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;628.57&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^4&lt;br /&gt;
-v5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 4th,&lt;br /&gt;
-down 5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;F^^&lt;br /&gt;
-Gv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Wide Tritone&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;23/16&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13/9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;10/7, 16/11&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;685.71&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;G#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Grave 5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;189/128, 240/161&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;52/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;3/2, 13/9&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;742.86&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^5&lt;br /&gt;
-vv6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 5th,&lt;br /&gt;
-double-down 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;G^&lt;br /&gt;
-Avv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Hb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Fifth-Sixth&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;23/15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;17/11, 20/13, 26/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;20/13&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;800&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^5&lt;br /&gt;
-v6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 5th,&lt;br /&gt;
-down 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;G^^&lt;br /&gt;
-Av&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;H&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Minor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;46/29&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;35/22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8/5, 11/7, 14/9, 18/11&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;857.14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;A&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;H#/Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Neutral 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;23/14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;18/11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13/8&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;914.29&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^6&lt;br /&gt;
-vv7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 6th,&lt;br /&gt;
-double-down 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;A^&lt;br /&gt;
-Bvv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Supermajor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;27/16, 46/27&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;17/10, 22/13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5/3, 12/7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;971.43&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^6&lt;br /&gt;
-v7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 6th,&lt;br /&gt;
-down 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;A^^&lt;br /&gt;
-Bv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;A#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Subminor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7/4, 9/5, 20/11&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1028.57&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;B&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Supraminor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;29/16, 9/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9/5, 20/11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;16/9&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1085.71&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^7&lt;br /&gt;
-vv8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 7th,&lt;br /&gt;
-double-down 8ve&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;B^&lt;br /&gt;
-Cvv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Major 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;15/8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;17/9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;15/8, 48/25&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;20&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1142.86&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;^^7&lt;br /&gt;
-v8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 7th,&lt;br /&gt;
-down 8ve&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;B^^&lt;br /&gt;
-Cv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;B#/Cb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Supermajor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;27/14, 29/15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;35/18, 68/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;63/32&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1200&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8ve&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Octave&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
+<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
-*1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament&lt;br /&gt;
+External image: http://www.ronsword.com/images/ron1.jpg<br>
-*2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament&lt;br /&gt;
+: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
-*3: based on treating 21-EDO as 13-limit laconic temperament&lt;br /&gt;
+: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
-&lt;br /&gt;
+</div>
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Chord Names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Chord Names&lt;/h1&gt;
+[[Category:IMPORTDEBUG - Change External Images]]
- &lt;br /&gt;
-Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.&lt;br /&gt;
-&lt;br /&gt;
--6-12 = C E G = C = C or C perfect&lt;br /&gt;
--5-12 = C Ev G = C(v3) = C down-three&lt;br /&gt;
--7-12 = C E^ G = C(^3) = C up-three&lt;br /&gt;
--6-11 = C E Gv = C(v5) = C down-five&lt;br /&gt;
--7-13 = C E^ G^ = C(^3,^5) = C up-three up-five&lt;br /&gt;
-&lt;br /&gt;
--6-12-18 = C E G B = C7 = C seven&lt;br /&gt;
--6-12-17 = C E G Bv = C(v7) = C down-seven&lt;br /&gt;
--5-12-18 = C Ev G B = C7(v3) = C seven down-three&lt;br /&gt;
--5-12-17 = C Ev G Bv = C.v7 = C dot down seven&lt;br /&gt;
-&lt;br /&gt;
-For a more complete list, see &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs"&gt;Ups and Downs Notation - Chord names in other EDOs&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-&lt;strong&gt;21-tone scales:&lt;/strong&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/augment6"&gt;augment6&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/augment9"&gt;augment9&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/augment12"&gt;augment12&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Chord Names-Triadic Harmony in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Triadic Harmony in 21-EDO:&lt;/h2&gt;
- &lt;br /&gt;
-One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &amp;quot;3rds&amp;quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &amp;quot;altered&amp;quot; triads that stand out as representations to parts of the overtone series:&lt;br /&gt;
-&lt;br /&gt;
+<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
+External image: http://www.swordguitars.com/21tetsm.JPG<br>
+: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
+: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
+</div>
+[[Category:IMPORTDEBUG - Change External Images]]
-&lt;table class="wiki_table"&gt;
+'''''21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)'''''
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Steps&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Cents&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Ratio&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;Example in C&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;Written name&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;Spoken name&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0-5-10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0-286-571&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;23:27:32&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Ev Gvv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C.v(vv5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C dot down, double-down five&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0-4-11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0-229-629&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7:8:10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Evv Gv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C.vv(v5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C dot double-down, down five&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0-6-11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0-343-629&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9:11:13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C E Gv&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C(v5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C down-five&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0-5-13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0-286-743&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;11:13:17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Ev G^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C.v(^5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C dot down up-five&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0-8-13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0-457-743&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;13:17:20&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Fv G^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C.v4(^5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C (sus) down-four up-five&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0-5-15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0-286-857&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;11:13:18&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Ev A&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;A(v5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;(inversion of 9:11:13)&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
+='''Compositions/Listening:'''=
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Chord Names-Moment-of-Symmetry Scales in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Moment-of-Symmetry Scales in 21-EDO:&lt;/h2&gt;
- &lt;br /&gt;
-Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.&lt;br /&gt;
-For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.&lt;br /&gt;
-&lt;br /&gt;
-For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Chord Names-Tetrachordal Scales in 21-EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Tetrachordal Scales in 21-EDO&lt;/h2&gt;
- While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:&lt;br /&gt;
-&lt;br /&gt;
+'''[https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal Iridescent Wenge Fugue] by Stephen Weigel, accepted to SEAMUS 2018 and Electroacoustic Barn Dance 2018'''
-&lt;table class="wiki_table"&gt;
+[http://soonlabel.com/xenharmonic/archives/2494 21-edo Trio for Organ, by Claudi Meneghin]
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;Step Pattern&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Cents&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Example&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Name*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;Ups/downs name&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3, 3, 3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-171-343-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D E F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Equable diatonic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C perfect&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4, 3, 2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-229-400-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D^ E^ F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Soft diatonic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C upperfect up-2&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4, 4, 1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-229-457-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D^ E^^ F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Intense diatonic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C up-2 &amp;amp; 6, double-up-3 &amp;amp; 7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5, 3, 1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-286-457-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D^^ E^^ F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Archytas chromatic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C double-up-2, 3, 6 and 7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5, 2, 2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-286-400-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D^^ E^ F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Weak chromatic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C double-up 2 &amp;amp; 6, up-3 &amp;amp; 7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;6, 2, 1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-343-457-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; E^^ F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Strong enharmonic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C triple-up 2 &amp;amp; 6, double-up 3 &amp;amp; 7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;7, 1, 1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(0)-400-457-(514)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;4&lt;/span&gt; E^^ F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Pythagorean enharmonic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;C quadruple-up 2 &amp;amp; 6, double-up 3 &amp;amp; 7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!&lt;br /&gt;
+[http://soonlabel.com/xenharmonic/archives/2336 21-penny jingle, by Claudi Meneghin]
-&lt;br /&gt;
-The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Chord Names-Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Rank two temperaments&lt;/h2&gt;
- &lt;a class="wiki_link" href="/List%20of%2021edo%20rank%20two%20temperaments%20by%20badness"&gt;List of 21edo rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
+[http://www.ronsword.com/sounds/21_improv.mp3 Short Clip of 21-edo Acoustic] by [[Ron_Sword|Ron Sword]]
-&lt;table class="wiki_table"&gt;
+[http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3 Open tuning Drone Improvisation in 21-edo] by Ron Sword
-    &lt;tr&gt;
-        &lt;th&gt;Periods&lt;br /&gt;
-per octave&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Generator&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Temperaments&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Escapade%20family#Escapade"&gt;Escapade&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&gt;Miracle&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;4\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Slendric"&gt;Slendric&lt;/a&gt;/&lt;a class="wiki_link" href="/Gamelismic%20clan#Gorgo"&gt;Gorgo&lt;/a&gt;/&lt;a class="wiki_link" href="/Gamelismic%20clan#Gidorah"&gt;Gidorah&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;5\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Mint%20temperaments#Subklei"&gt;Subklei&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;8\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Chromatic%20pairs#Tridec"&gt;Tridec&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;10\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Marvel%20temperaments#Triton"&gt;Triton&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Augmented%20family"&gt;Augmented&lt;/a&gt;/&lt;a class="wiki_link" href="/August"&gt;August&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;3\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Oodako"&gt;Oodako&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1\21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;a class="wiki_link" href="/Apotome%20family"&gt;Whitewood&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
+[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715 Anomalous Readings] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]]
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Chord Names-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit Commas&lt;/h2&gt;
-EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; 21 33 49 59 73 78 |.)&lt;br /&gt;
+[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3 Comets Over Flatland 15] by [[Randy_Winchester|Randy Winchester]]
-&lt;table class="wiki_table"&gt;
+[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3 Comets Over Flatland 18] by [[Randy_Winchester|Randy Winchester]]
-    &lt;tr&gt;
-        &lt;th&gt;Comma&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Monzo&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Value (Cents)&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Name 1&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Name 2&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2187/2048&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -11 7 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;113.69&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Apotome&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;128/125&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 7 0 -3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;41.06&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Diesis&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Augmented Comma&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -25 7 6 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;31.57&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Ampersand's Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 32 -7 -9 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;9.49&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Escapade Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1029/1000&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -3 1 -3 3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;49.49&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Keega&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;36/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;48.77&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Septimal Quarter Tone&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -10 7 8 -7 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;22.41&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Blackjackisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;8.43&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Gamelisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;225/224&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -5 2 2 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;7.71&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Septimal Kleisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Marvel Comma&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;16875/16807&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 0 3 4 -5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;6.99&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Mirkwai&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2401/2400&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -5 -1 -2 4 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;0.72&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Breedsma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 47 -7 -7 -7 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;0.34&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Akjaysma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5\7 Octave Comma&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;99/98&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -1 2 0 -2 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;17.58&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Mothwellsma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 5 -1 3 0 -3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;3.03&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Wizardharry&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
+[http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco L'esatonale ubriaco (the drunk hexatonal)], ALIENAMENTE by [[Fabrizio_Fiale|Fabrizio Fulvio Fausto Fiale]]      [[Category:21edo]]
-&lt;br /&gt;
+[[Category:edo]]
-&lt;br /&gt;
+[[Category:listen]]
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;strong&gt;Books / Literature:&lt;/strong&gt;&lt;/h1&gt;
+[[Category:todo:unify_precision]]
- Sword, Ron. &amp;quot;Icosihenaphonic Scales for Guitar&amp;quot;. IAAA Press. 1st ed: July 2009.&lt;br /&gt;
+[[Category:twentuning]]
-&lt;!-- ws:start:WikiTextRemoteImageRule:1006:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1006 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:1007:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1007 --&gt;&lt;br /&gt;
-&lt;strong&gt;&lt;em&gt;21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)&lt;/em&gt;&lt;/strong&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
- &lt;br /&gt;
-&lt;strong&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/overtoneshock/little-fugue-21-edo?in=overtoneshock/sets/xenharmonic-microtonal" rel="nofollow"&gt;Iridescent Wenge Fugue&lt;/a&gt; by Stephen Weigel, accepted to SEAMUS 2018 and Electroacoustic Barn Dance 2018&lt;/strong&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2494" rel="nofollow" target="_blank"&gt;21-edo Trio for Organ, by Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2336" rel="nofollow" target="_blank"&gt;21-penny jingle, by Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank"&gt;Short Clip of 21-edo Acoustic&lt;/a&gt; by &lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank"&gt;Open tuning Drone Improvisation in 21-edo&lt;/a&gt; by Ron Sword&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933715" rel="nofollow"&gt;Anomalous Readings&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3" rel="nofollow"&gt;Comets Over Flatland 15&lt;/a&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3" rel="nofollow"&gt;Comets Over Flatland 18&lt;/a&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco" rel="nofollow" target="_blank"&gt;L'esatonale ubriaco (the drunk hexatonal)&lt;/a&gt;, ALIENAMENTE by &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Fabrizio%20Fiale"&gt;Fabrizio Fulvio Fausto Fiale&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

21edo: Difference between revisions

Revision as of 00:00, 17 July 2018

21 equal divisions of the octave

21-EDO as a temperament:

Chord Names

Triadic Harmony in 21-EDO:

Moment-of-Symmetry Scales in 21-EDO:

Tetrachordal Scales in 21-EDO

Rank two temperaments

13-limit Commas

Books / Literature:

Compositions/Listening:

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