User:Triethylamine/draft: 22平均律

This page is a draft of JP translation of 22edo.

Prime factorization

2 × 11

Step size

54.5455¢

Fifth

13\22 (709.091¢)

Semitones (A1:m2)

3:1 (163.6¢ : 54.55¢)

Consistency limit

11

Distinct consistency limit

5

Special properties

zeta peak
zeta peak integer

22 平均律、または 22 音平均律（英: 22 equal divisions of the octave, 22 equal temperament, 22EDO, 22ET）は、レギュラー音律の観点から見ると、オクターブを均等な 22 個のステップに分割した調律システムである。

1 ステップあたりの周波数比は 2 の 22 乗根 [math](\sqrt[22]{2})[/math] であり、約 54.545 セントである。9/8 と 10/9 を区別するので、これはミーントーン (en) システムではない。

理論

歴史

オクターブを同じサイズの 22 のステップに分割するという考えは、19 世紀の音楽理論家R.H.M. Bosanquetに由来しているようである。インドの音楽理論 (en) におけるオクターブの 22 の不均等な部分への分割に触発され、Bosanquetは、そのような均等な分割により 5-リミットの音楽を許容できる精度で表現できることに注目した。この点については、20 世紀に理論家のJosé Würschmidtが続き、彼はこれを 19 平均律の次の可能性として指摘した。また、J. Murray Barbourは、調律の歴史に関する古典的な調査書『Tuning and Temperament』の中で、これに続いた。

純正音程近似のクオリティの概観

22 平均律のシステムは実際には、12 と 19 に次ぐ、5-リミット音程をTE誤差 (en) 4 セント/オクターヴ以内に近似することができる 3 番目の平均律である。ゼータ積分やゼータギャップ平均律ではないが、少なくともゼータピーク平均律 (en) ではある。さらに、5-リミットだけではない。12 や 19 とは異なり、 7, 11-リミット音程を 3 セント/オクターヴ以内の誤差で近似できる。31 平均律の方がはるかに優れているが、22 平均律でもこれらのリミットの和声を利用できる。実際、22 は 11-奇数リミットを一貫して表す最小の等分割である。

さらに、22 平均律は 12 や 19 とは異なり、ミーントーン (en) システムではない。最終的な効果は、22 という数字があまり馴染みのない音楽領域の探求を可能にし、ある程度強制することでもあるが、やはり十分に小さいので、22 音ギターなどの適切に設計された楽器を使用したライブパフォーマンスで使用できることであろう。

22 平均律は、11 平均律の 2.7.9.11.15.17 サブグループに倍音 3 と 5 を追加したものとして扱うこともでき、（かなり正確な）2.3.5.7.11.17 サブグループ音律になる。31 倍音の近似値は 0.5 セント以内であり、かなり正確であることも注目に値する。また、特に 29/24 などの 29 倍音を含むいくつかの間隔も近似しており、これも 0.5 セント以内で一致する。これにより、2.3.5.7.11.17.29.31 がもたらされる。

22 平均律は、拡張された「クォーターコンマarchy」に非常に近い。これはシントニックコンマ 81/80 の代わりにアルキュタスコンマ 64/63 をテンパーアウトすることを除いて、クォーターコンマミーントーン (en) に似たチューニングである。このため、ほぼ純粋な 7 倍音系長 3 度（9/7）を持つ。

素数倍音

Approximation of prime harmonics in 22edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.0	+7.1	-4.5	+13.0	-5.9	-22.3	+4.1	-24.8	+26.3	+6.8	+0.4
Error	relative (%)	+0	+13	-8	+24	-11	-41	+8	-45	+48	+12	+1
Steps (reduced)		22 (0)	35 (13)	51 (7)	62 (18)	76 (10)	81 (15)	90 (2)	93 (5)	100 (12)	107 (19)	109 (21)

部分集合と上位集合

22 は 11 で割り切れるため、12 平均律が 6 平均律（全音音階）を演奏できるのと同じように、22 平均律楽器は 11 平均律のあらゆる音楽を演奏できる。11 平均律は、旋律的には 12 平均律（よく知られた 1:2:3 の比率で全音、半音、短 3 度）に聞こえる点で興味深いが、特に完全 5 度/4 度や 5-リミット長 3 度/短 6 度がないため、和声的には大きく異なる。同様に、22 平均律と 24 平均律は、どちらも 4 分音や短/中/長 2 度を含むため、メロディー的に似ている。しかし、22 平均律は 24 よりもはるかに優れた全体的なハーモニーを提供する。サジタルノーテーション (en) では、11 は 22 の 1 つおきの音として記譜できる。

音程

(以下未翻訳)

See also: 22edo solfege

Degree	Cents	Approximate Ratios*	Ups and Downs Notation
0	0.000	1/1	perfect unison	P1	D
1	54.545	36/35, 34/33, 33/32, 32/31	minor 2nd	m2	Eb
2	109.091	18/17, 17/16, 16/15, 15/14	upminor 2nd	^m2	^Eb
3	163.636	12/11, 11/10, 10/9	downmajor 2nd	vM2	vE
4	218.182	9/8, 17/15, 8/7	major 2nd	M2	E
5	272.727	20/17, 7/6	minor 3rd	m3	F
6	327.273	6/5, 17/14, 11/9	upminor 3rd	^m3	^F
7	381.818	5/4, 96/77	downmajor 3rd	vM3	vF#
8	436.364	14/11, 9/7, 22/17	major 3rd	M3	F#
9	490.909	4/3	perfect fourth	P4	G
10	545.455	15/11, 11/8	up-4th, dim 5th	^4, d5	^G, Ab
11	600.000	7/5, 24/17, 17/12, 10/7	downaug 4th, updim 5th	vA4, ^d5	vG#, ^Ab
12	654.545	16/11, 22/15	aug 4th, down-5th	A4, v5	G#, vA
13	709.091	3/2	perfect 5th	P5	A
14	763.636	17/11, 14/9, 11/7	minor 6th	m6	Bb
15	818.182	8/5, 77/48	upminor 6th	^m6	^Bb
16	872.727	18/11, 28/17, 5/3	downmajor 6th	vM6	vB
17	927.273	17/10, 12/7	major 6th	M6	B
18	981.818	7/4, 30/17, 16/9	minor 7th	m7	C
19	1036.364	9/5, 11/6, 20/11	upminor 7th	^m7	^C
20	1090.909	28/15, 15/8, 32/17, 17/9	downmajor 7th	vM7	vC#
21	1145.455	31/16, 64/33, 33/17, 35/18	major 7th	M7	C#
22	1200.000	2/1	perfect octave	P8	D

* some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.

純正音程近似

15-奇数リミット音程のマッピング

The following tables show how 15-odd-limit intervals are represented in 22edo. Prime harmonics are in bold; inconsistent intervals are in italic.

15-odd-limit intervals by direct approximation (even if inconsistent)
Interval, complement	Error (abs, ¢)	Error (rel, %)
9/7, 14/9	1.280	2.3
11/10, 20/11	1.368	2.5
15/8, 16/15	2.640	4.8
5/4, 8/5	4.496	8.2
7/6, 12/7	5.856	10.7
11/8, 16/11	5.863	10.7
3/2, 4/3	7.136	13.1
15/11, 22/15	8.504	15.6
15/14, 28/15	10.352	19.0
5/3, 6/5	11.631	21.3
7/4, 8/7	12.992	23.8
11/6, 12/11	12.999	23.8
9/8, 16/9	14.272	26.2
13/11, 22/13	16.482	30.2
7/5, 10/7	17.488	32.1
13/10, 20/13	17.850	32.7
13/9, 18/13	17.928	32.9
9/5, 10/9	18.767	34.4
11/7, 14/11	18.856	34.6
13/7, 14/13	19.207	35.2
11/9, 18/11	20.135	36.9
13/8, 16/13	22.346	41.0
15/13, 26/15	24.986	45.8
13/12, 24/13	25.064	46.0

15-odd-limit intervals by patent val mapping
Interval, complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
9/7, 14/9	1.280	2.3
11/10, 20/11	1.368	2.5
15/8, 16/15	2.640	4.8
5/4, 8/5	4.496	8.2
7/6, 12/7	5.856	10.7
11/8, 16/11	5.863	10.7
3/2, 4/3	7.136	13.1
15/11, 22/15	8.504	15.6
15/14, 28/15	10.352	19.0
5/3, 6/5	11.631	21.3
7/4, 8/7	12.992	23.8
11/6, 12/11	12.999	23.8
9/8, 16/9	14.272	26.2
13/11, 22/13	16.482	30.2
7/5, 10/7	17.488	32.1
13/10, 20/13	17.850	32.7
9/5, 10/9	18.767	34.4
11/7, 14/11	18.856	34.6
11/9, 18/11	20.135	36.9
13/8, 16/13	22.346	41.0
15/13, 26/15	24.986	45.8
13/12, 24/13	29.482	54.0
13/7, 14/13	35.338	64.8
13/9, 18/13	36.618	67.1

いくつかの 17-リミット音程

決定づける特徴

セプティマル vs シントニックコンマ

Possibly the most striking characteristic of 22edo to those not used to it is that it does not temper out the syntonic comma of 81/80, and therefore is not a system of meantone temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as 34edo, 41edo and 53edo.

The diatonic scale it produces is instead derived from superpyth temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, 5L 2s), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22et. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

ポーキュパインコンマ

It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22edo supports porcupine temperament. The generator for porcupine is a flat minor whole tone of 10/9, two of which is a slightly sharp 6/5, and three of which is a slightly flat 4/3, implying the existence of an equal-step tetrachord, which is characteristic of porcupine. Porcupine is notable for being the 5-limit temperament lowest in badness which is not approximated by the familiar 12-tone equal temperament, and as such represents one excellent point of departure for examining the harmonic properties of 22edo. It forms mos scales of 7 and 8, which in 22edo are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).

その他の 5-リミットコンマ

Other 5-limit commas 22edo tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12et or 22et, the diatonic tritone 45/32, which is a major third above a major whole tone representing 9/8, is equated to its inverted form, 64/45. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

その他の 7-リミットコンマ

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both 50/49, (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the septimal kleisma, so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the orwell comma; and the orwell tetrad is also a chord of 22et.

11-リミットコンマ

In the 11-limit, 22edo tempers out the quartisma, leading to a stack of five 33/32 quartertones being equated with one 7/6 subminor third. This is a trait which, while shared with 24edo, is surprisingly not shared with a number of other relatively small edos such as 17edo, 26edo and 34edo. In fact, not even the famous 53edo has this property – although it should be noted that the related 159edo does.

その他の特徴

The 164¢ "flat minor whole tone" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the 11-limit: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.

22edo also supports the orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms mos scales with step patterns 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. Harmonically, orwell can be tuned more accurately in other temperaments, such as 31edo, 53edo and 84edo. But 22edo orwell has a leg-up on the others melodically, as the large and small steps of orwell[9] are easier to distinguish in 22.

22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In Sagittal notation, 11 can be notated as every other note of 22.

レギュラー音律の性質

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3	[35 -22⟩	[⟨22 35]]	-2.25	2.25	4.12
2.3.5	250/243, 2048/2025	[⟨22 35 51]]	-0.86	2.70	4.94
2.3.5.7	50/49, 64/63, 245/243	[⟨22 35 51 62]]	-1.80	2.85	5.23
2.3.5.7.11	50/49, 55/54, 64/63, 99/98	[⟨22 35 51 62 76]]	-1.11	2.90	5.33
2.3.5.7.11.17	50/49, 55/54, 64/63, 85/84, 99/98	[⟨22 35 51 62 76 90]]	-1.09	2.65	4.87

22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is 31. 22et is even more prominent in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is 46.

一様写像

13-limit uniform maps between 21.5 and 22.5
Min. size	Max. size	Wart notation	Map
21.5000	21.5353	22bccdddeeeeff	⟨22 34 50 60 74 80]
21.5353	21.5505	22bccdddeeff	⟨22 34 50 60 75 80]
21.5505	21.7492	22bccdeeff	⟨22 34 50 61 75 80]
21.7492	21.7542	22bdeeff	⟨22 34 51 61 75 80]
21.7542	21.7671	22bdee	⟨22 34 51 61 75 81]
21.7671	21.8244	22dee	⟨22 35 51 61 75 81]
21.8244	21.9067	22d	⟨22 35 51 61 76 81]
21.9067	22.0244	22	⟨22 35 51 62 76 81]
22.0244	22.1135	22f	⟨22 35 51 62 76 82]
22.1135	22.1798	22ef	⟨22 35 51 62 77 82]
22.1798	22.2629	22cef	⟨22 35 52 62 77 82]
22.2629	22.2946	22cddef	⟨22 35 52 63 77 82]
22.2946	22.3980	22cddefff	⟨22 35 52 63 77 83]
22.3980	22.4025	22bbcddefff	⟨22 36 52 63 77 83]
22.4025	22.5000	22bbcddeeefff	⟨22 36 52 63 78 83]

コンマ

22et tempers out the following commas. (Note: This assumes the val ⟨22 35 51 62 76 81].)

Prime limit	Ratio^[1]	Monzo	Cents	Color name	Name
3	(22 digits)	[35 -22⟩	156.98
5	250/243	[1 -5 3⟩	49.17	Triyo	Porcupine comma
5	3125/3072	[-10 -1 5⟩	29.61	Laquinyo	Magic comma
5	2048/2025	[11 -4 -2⟩	19.55	Sagugu	Diaschisma
5	(14 digits)	[-21 3 7⟩	10.06	Lasepyo	Semicomma
5	(20 digits)	[32 -7 -9⟩	9.49	Sasa-tritrigu	Escapade comma
5	(32 digits)	[-53 10 16⟩	0.57	Quadla-quadquadyo	Kwazy
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Jubilisma
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal comma
7	875/864	[-5 -3 3 1⟩	21.90	Zotriyo	Keema
7	2430/2401	[1 5 1 -4⟩	20.79	Quadru-ayo	Nuwell
7	245/243	[0 -5 1 2⟩	14.19	Zozoyo	Sensamagic
7	1728/1715	[6 3 -1 -3⟩	13.07	Triru-agu	Orwellisma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyo	Marvel comma
7	10976/10935	[5 -7 -1 3⟩	6.48	Trizo-agu	Hemimage
7	6144/6125	[11 1 -3 -2⟩	5.36	Saruru-atrigu	Porwell
7	65625/65536	[-16 1 5 1⟩	2.35	Lazoquinyo	Horwell
7	(12 digits)	[-6 -8 2 5⟩	1.12	Quinzo-ayoyo	Wizma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	896/891	[7 -4 0 1 -1⟩	9.69	Saluzo	Pentacircle
11	65536/65219	[16 0 0 -2 -3⟩	8.39	Satrilu-aruru	Orgonisma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	540/539	[2 3 1 -2 -1⟩	3.21	Lururuyo	Swetisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizardharry
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Bilorugu	Kalisma
13	65/64	[-6 0 1 0 0 1⟩	26.84	Thoyo	Wilsorma
13	78/77	[1 1 0 -1 -1 1⟩	22.34	Tholuru	Negustma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap
31	125/124	[-2 0 3 0 0 0 0 0 0 0 -1⟩	13.91	Thiwutriyo	Twizzler

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

ランク-2 音律

Periods per octave	Generator	Temperaments
1	1\22	Sensa Chromo Ceratitid
1	3\22	Porcupine
1	5\22	Orwell (22) / blair (22) / winston (22f)
1	7\22	Magic / telepathy
1	9\22	Superpyth / suprapyth
2	1\22	Shrutar / hemipaj Comic
2	2\22	Srutal / pajara / pajarous
2	3\22	Hedgehog / echidna
2	4\22	Astrology Antikythera Wizard
2	5\22	Doublewide / fleetwood
11	1\22	Undeka Hendecatonic

スケール

See 22edo modes.

テトラコルド

See 22edo tetrachords.

記譜法

スーパーパイス/ポーキュパイン表記

Superpyth/Porcupine Notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.

ポーキュパイン表記

Porcupine Notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.

The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.

ペンタトニック表記

In Pentatonic Notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals.

The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D.

デカトニック表記

The Decatonic Notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.

Chain 1: C G D A E

Chain 2: γ δ α ε β

The alphabet is, in ascending order: C δ D ε E γ G α A β C

In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.

サジタルノーテーション

When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:

This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.

The division of the apotome into three syntonic commas also indicates 22's tempering out of the porcupine comma (which is equivalent to three syntonic commas minus a Pythagorean apotome).

We also have, from the appendix to The Sagittal Songbook by Jacob A. Barton, this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:

アップ&ダウン表記

Treating ups and downs as "fused" with sharps and flats, and never appearing separately:

Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:

A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.

Shown below is Paul Erlich's "Tibia" in G, with independent ups and downs.

Tibia in G (page 1)
Tibia in G (page 2)

22平均律の各記譜法の比較

Degree	Cents	Superpyth/Porcupine Notation		Porcupine			Pentatonic			Decatonic			Ups and Downs
0	0	Natural Unison	1	perfect unison	P1	D	perfect unison	P1	D	natural 1st	N1	C	perfect unison	P1	D
1	55	s-minor second	sm2	aug unison	A1	D#	aug unison	A1	D#	flat 2nd	f2	C#, δb	minor 2nd	m2	Eb
2	109	p-diminished second	pd2	dim 2nd	d2	Eb	double-aug unison, double-dim sub3rd	AA1, dds3	Dx, Fb3	natural 2nd	N2	δ	upminor 2nd	^m2	^Eb
3	164	p-minor second	pm2	perfect 2nd	P2	E	dim sub3rd	ds3	Fbb	sharp 2nd, flat 3rd	s2, f3	δ#, Db	downmajor 2nd	vM2	vE
4	218	(s/p) Major second	M2	aug 2nd	A2	E#	minor sub3rd	ms3	Fb	natural 3rd	N3	D	major 2nd	M2	E
5	273	s-minor third	sm3	dim 3rd	d3	Fb	major sub3rd	Ms3	F	sharp 3rd	s3	D#	minor 3rd	m3	F
6	327	p-minor third	pm3	minor 3rd	m3	F	aug sub3rd	As3	F#	flat 4th	f4	εb	upminor 3rd	^m3	^F
7	382	p-Major third	pM3	major 3rd	M3	F#	double-aug sub3rd, double-dim 4thoid	AAs3, dd4d	Fx, Gbb	natural 4th	N4	ε	downmajor 3rd	vM3	vF#
8	436	s-Major third	sM3	aug 3rd, dim 4th	A3, d4	Fx, Gb	dim 4thoid	d4d	Gb	sharp 4th, flat 5th	s4, f5	ε#, Eb	major 3rd	M3	F#
9	491	Natural Fourth	4, N4	minor 4th	m4	G	perfect 4thoid	P4d	G	natural 5th	N5	E	perfect fourth	P4	G
10	545	p-Major fourth, s-dim fifth	pM4, sd5	major 4th	M4	G#	aug 4thoid	A4d	G#	sharp 5th, flat 6th	s5, f6	E#, γb	up-4th, dim 5th	^4, d5	^G, Ab
11	600	p-Augmented Fourth, p-diminished Fifth Half-Octave	A4, HO	aug 4th, dim 5th	A4, d5	Gx, Abb	double-aug 4thoid, double-dim 5thoid	AA4d, dd5d	Gx, Abb	natural 6th	N6	γ	downaug 4th, updim 5th	vA4, ^d5	vG#, ^Ab
12	655	p-minor Fifth, s-aug Fourth	pm5, sA4	minor 5th	m5	Ab	dim 5thoid	d5d	Ab	sharp 6th, flat 7th	s6, f7	γ#, Gb	aug 4th, down-5th	A4, v5	G#, vA
13	709	Natural Fifth	5, N5	major 5th	M5	A	perfect 5thoid	P5d	A	natural 7th	N7	G	perfect 5th	P5	A
14	764	s-minor sixth	sm6	aug 5th, dim 6th	A5, d6	A#, Bbb	aug 5thoid	A5d	A#	sharp 7th	s7	G#	minor 6th	m6	Bb
15	818	p-minor sixth	pm6	minor 6th	m6	Bb	double-aug 5thoid, double-dim sub7th	AA5d, dds7	Ax, Cb3	flat 8th	f8	αb	upminor 6th	^m6	^Bb
16	873	p-Major sixth	pM6	major 6th	M6	B	dim sub7th	ds7	Cbb	natural 8th	N8	α	downmajor 6th	vM6	vB
17	927	s-Major sixth	sM6	aug 6th	A6	B#	minor sub7th	ms7	Cb	sharp 8th, flat 9th	s8, f9	α#, Ab	major 6th	M6	B
18	982	(s/p) minor seventh	m7	dim 7th	d7	Cb	major sub7th	Ms7	C	natural 9th	N9	A	minor 7th	m7	C
19	1036	p-Major seventh	pM7	perfect 7th	P7	C	aug sub7th	As7	C#	sharp 9th, flat 10th	s9, f10	A#, βb	upminor 7th	^m7	^C
20	1091	p-Augmented seventh	pA7	aug 7th	A7	C#	double-aug sub7th, double-dim octave	AAs7, dd8	Cx, Dbb	natural 10th	N10	β	downmajor 7th	vM7	vC#
21	1145	s-Major seventh	sM7	dim 8ve	d8	Db	dim octave	d8	Db	sharp 10th	s10	β#, Cb	major 7th	M7	C#
22	1200	Octave	8	perfect octave	P8	D	perfect octave	P8	D	natural 11th	N11	C	perfect octave	P8	D

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Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality	color name	monzo format	examples
minor	zo	[a b 0 1>	7/6, 7/4
minor	fourthward wa	[a b> where b < -1	32/27, 16/9
upminor	gu	[a b -1>	6/5, 9/5
downmajor	yo	[a b 1>	5/4, 5/3
major	fifthward wa	[a b> where b > 1	9/8, 27/16
major	ru	[a b 0 -1>	9/7, 12/7

All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:

color of the 3rd	JI chord	notes as edosteps	notes of C chord	written name	spoken name
zo	6:7:9	0-5-13	C Eb G	Cm	C minor
gu	10:12:15	0-6-13	C ^Eb G	C^m	C upminor
yo	4:5:6	0-7-13	C vE G	Cv	C downmajor or C down
ru	14:18:21	0-8-13	C E G	C	C major or C

Examples:

0-4-13 = C D G = C2
0-9-13 = C F G = C4
0-10-13 = C ^F G = C^4 or C(^4)
0-5-10 = C Eb Gb = Cd = Cdim
0-5-11 = C Eb ^Gb = Cd(^5)
0-5-12 = C Eb vG = Cm(v5)

Further discussion of 22edo chord naming:

音楽

Main article: 22edo/Music

See also: Category:22edo tracks

外部リンク

参考文献

Barbour, James Murray, Tuning and temperament, a historical survey, East Lansing, Michigan State College Press, 1953 [c1951]
Bosanquet, R.H.M. On the Hindoo division of the octave, with additions to the theory of higher orders, Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, Hindu Music from Various Authors, Chowkhamba Sanskrit Series, Varanasi, India, 1965

[1] Ratios longer than 10 digits are presented by placeholders with informative hints

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