270edo

← 269edo

270edo

271edo →

Prime factorization	2 × 33 × 5
Step size	4.44444 ¢
Fifth	158\270 (702.222 ¢) (→ 79\135)
Semitones (A1:m2)	26:20 (115.6 ¢ : 88.89 ¢)
Consistency limit	15
Distinct consistency limit	15

270 equal divisions of the octave (abbreviated 270edo or 270ed2), also called 270-tone equal temperament (270tet) or 270 equal temperament (270et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 270 equal parts of about 4.44 ¢ each. Each step represents a frequency ratio of 2^1/270, or the 270th root of 2. 270edo's step size is called a tredek when used as an interval size unit.

Theory

270edo is an extremely strong 13-limit system, distinctly consistent through the 15-odd-limit and almost consistent to distance 2 in it, missing 15/13 and 26/15 as they have 25.8% error (tempering out 676/675). It is the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo, making it a strict zeta edo.

In the 5-limit it tempers out the ennealimma, [1 -27 18⟩, the vulture comma, [24 -21 4⟩, and the vishnuzma, [23 6 -14⟩.

In the 7-limit it tempers out the breedsma (2401/2400), the ragisma (4375/4374), and by extension the wizma (420175/419904), and the landscape comma (250047/250000) so that it supports ennealimmal temperament. It also tempers out the quasiorwellisma (29360128/29296875) and the garischisma (33554432/33480783).

In the 11-limit, it tempers out the lehmerisma (3025/3024), the vishdel comma (5632/5625), the kalisma (9801/9800), the symbiotic comma (19712/19683), the nexus comma (1771561/1769472), and the quartisma (117440512/117406179). Notably, it is consistent to distance 3 in the 11-odd-limit, and almost to distance 4 ((11/10)⁴ and (20/11)⁴ are a hair off, 50.4%).

Finally, in the 13-limit it is slightly worse but still excellent. It tempers out 676/675, 1001/1000, 1716/1715, and 2080/2079, making it an archipelago tuning, and the optimal patent val for some of the archipelago temperaments such as hemiennealimmal, vulture, eagle, and avicenna.

The excellent tuning accuracy does not bar it from the utility of essentially tempered chords, including sinbadmic chords in the 13-odd-limit, and island chords in the 15-odd-limit.

Beyond the 13-limit, the approximated harmonic 17 is more than 1/3-edostep, but the harmonic 19 is very accurately tuned. 17/13 and its octave complement 26/17 are the only inconsistently approximated 21-odd-limit intervals, each barely missing the mark (50.4% relative error). The harmonic 23 is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full 23-limit. It tempers out 715/714, 936/935, 1089/1088, 1225/1224, 1701/1700, 2025/2023, 2058/2057, and 2431/2430 in the 17-limit; 1216/1215, 1331/1330, 1521/1520, 1540/1539, and 1729/1728 in the 19-limit. If the full 23-limit is desired, then 460/459, 529/528, 736/735, 897/896, 1288/1287, 1311/1309, and 1771/1768 are further tempered out.

The harmonics 29 and 31 are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency (29/26 and 31/26 are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 35-odd-limit. Notably, it tempers out 784/783, 900/899, and 1024/1023, while inflating 841/840 and 961/960.

On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance (beyond the 13-limit on which it is spot on), as the maximum error (assuming consistency) is only 2.2 ¢, yet having a step size that can be discernible.

If, however, you want a edo for more rounded, consistent very high-limit use, the obvious alternative choice is 311edo, which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a prime edo as opposed to a very composite one. While 270edo approximates the first 16 harmonics with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.

Prime harmonics

Approximation of prime harmonics in 270edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.267	+0.353	+0.063	-0.207	-0.528	+1.711	+0.265	-1.608	+1.534	+1.631
	Relative (%)	+0.0	+6.0	+7.9	+1.4	-4.7	-11.9	+38.5	+6.0	-36.2	+34.5	+36.7
Steps (reduced)		270 (0)	428 (158)	627 (87)	758 (218)	934 (124)	999 (189)	1104 (24)	1147 (67)	1221 (141)	1312 (232)	1338 (258)

Approximation of prime harmonics in 270edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+1.989	+2.049	-0.407	+1.160	+2.051	-1.394	-1.329	+0.693	-1.919	-1.123	-0.092
	Relative (%)	+44.8	+46.1	-9.1	+26.1	+46.1	-31.4	-29.9	+15.6	-43.2	-25.3	-2.1
Steps (reduced)		1407 (57)	1447 (97)	1465 (115)	1500 (150)	1547 (197)	1588 (238)	1601 (251)	1638 (18)	1660 (40)	1671 (51)	1702 (82)

Subsets and supersets

270 is a very composite number. The prime factorization is 270 = 2 × 3³ × 5, with divisors 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, and 135. This means that 270edo can be conceptualised as the superset of, for example, 10edo and 27edo, which are both interesting and somewhat peculiar in their own right.

540edo, which divides the edostep in two, and 810edo, which divides the edostep in three, provide good correction for harmonics 17, 23, and beyond.

Intervals

As 270edo is a large edo, its intervals can be found on a separate page: Table of 270edo intervals.

Notation

Ups and downs notation

270edo can be notated using ups and downs with Stein-Zimmerman quarter-tone accidentals representing half-sharps and half-flats. These can be spoken as sha and fla. For example, the note 12\270 above C is C downsha, and the note 39\270 above C is C shasharp.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53
Sharp symbol
Flat symbol

Sagittal notation

The Sagittal notation for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and half-flat may be used.

+ edosteps		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26
Symbol	Evo-SZ																										
	Evo															
	Revo																										

Note that the Revo notation has matching flag sequences between the double-shaft symbols and a subsequence of the single-shaft symbols.

Alternate spellings in the Promethean set (comma tempered out):

•  =  (2621440/2617839)
•  =  (1949696/1948617)
•  =  (1216/1215)
•  =  (22528/22491)
•  =  (1540/1539)
•  =  (19712/19683)
•  =  (20493/20480)
•  =  =  (729/728) (1540/1539)
•  =  (131072/130977) (3969/3968)

See apotome complements for equivalent accidental pairs.

Approximation to JI

23-odd-limit interval mappings

The following tables show how 23-odd-limit intervals are represented in 270edo. Prime harmonics are in bold; inconsistent intervals are in italics.

23-odd-limit intervals in 270edo (direct approximation, even if inconsistent)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
19/12, 24/19	0.002	0.1
21/20, 40/21	0.023	0.5
7/4, 8/7	0.063	1.4
21/19, 38/21	0.065	1.5
5/3, 6/5	0.086	1.9
19/10, 20/19	0.088	2.0
9/5, 10/9	0.181	4.1
19/14, 28/19	0.202	4.5
7/6, 12/7	0.204	4.6
11/8, 16/11	0.207	4.7
19/16, 32/19	0.265	6.0
3/2, 4/3	0.267	6.0
19/18, 36/19	0.270	6.1
11/7, 14/11	0.270	6.1
7/5, 10/7	0.290	6.5
13/11, 22/13	0.321	7.2
21/16, 32/21	0.330	7.4
5/4, 8/5	0.353	7.9
19/15, 30/19	0.355	8.0
9/7, 14/9	0.471	10.6
19/11, 22/19	0.472	10.6
11/6, 12/11	0.474	10.7
13/8, 16/13	0.528	11.9
9/8, 16/9	0.534	12.0
21/11, 22/21	0.537	12.1
15/14, 28/15	0.557	12.5
11/10, 20/11	0.560	12.6
13/7, 14/13	0.591	13.3
15/8, 16/15	0.620	14.0
11/9, 18/11	0.741	16.7
19/13, 26/19	0.792	17.8
13/12, 24/13	0.795	17.9
15/11, 22/15	0.827	18.6
21/13, 26/21	0.858	19.3
13/10, 20/13	0.881	19.8
13/9, 18/13	1.062	23.9
23/13, 26/23	1.080	24.3
17/15, 30/17	1.091	24.5
23/17, 34/23	1.126	25.3
15/13, 26/15	1.148	25.8
17/9, 18/17	1.177	26.5
17/10, 20/17	1.358	30.6
21/17, 34/21	1.381	31.1
23/22, 44/23	1.401	31.5
17/12, 24/17	1.444	32.5
19/17, 34/19	1.446	32.5
23/16, 32/23	1.608	36.2
17/14, 28/17	1.648	37.1
23/14, 28/23	1.671	37.6
17/16, 32/17	1.711	38.5
23/19, 38/23	1.872	42.1
23/12, 24/23	1.875	42.2
17/11, 22/17	1.918	43.2
23/21, 42/23	1.938	43.6
23/20, 40/23	1.961	44.1
23/18, 36/23	2.142	48.2
17/13, 26/17	2.206	49.6
23/15, 30/23	2.217	49.9

23-odd-limit intervals in 270edo (patent val mapping)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
19/12, 24/19	0.002	0.1
21/20, 40/21	0.023	0.5
7/4, 8/7	0.063	1.4
21/19, 38/21	0.065	1.5
5/3, 6/5	0.086	1.9
19/10, 20/19	0.088	2.0
9/5, 10/9	0.181	4.1
19/14, 28/19	0.202	4.5
7/6, 12/7	0.204	4.6
11/8, 16/11	0.207	4.7
19/16, 32/19	0.265	6.0
3/2, 4/3	0.267	6.0
19/18, 36/19	0.270	6.1
11/7, 14/11	0.270	6.1
7/5, 10/7	0.290	6.5
13/11, 22/13	0.321	7.2
21/16, 32/21	0.330	7.4
5/4, 8/5	0.353	7.9
19/15, 30/19	0.355	8.0
9/7, 14/9	0.471	10.6
19/11, 22/19	0.472	10.6
11/6, 12/11	0.474	10.7
13/8, 16/13	0.528	11.9
9/8, 16/9	0.534	12.0
21/11, 22/21	0.537	12.1
15/14, 28/15	0.557	12.5
11/10, 20/11	0.560	12.6
13/7, 14/13	0.591	13.3
15/8, 16/15	0.620	14.0
11/9, 18/11	0.741	16.7
19/13, 26/19	0.792	17.8
13/12, 24/13	0.795	17.9
15/11, 22/15	0.827	18.6
21/13, 26/21	0.858	19.3
13/10, 20/13	0.881	19.8
13/9, 18/13	1.062	23.9
23/13, 26/23	1.080	24.3
17/15, 30/17	1.091	24.5
15/13, 26/15	1.148	25.8
17/9, 18/17	1.177	26.5
17/10, 20/17	1.358	30.6
21/17, 34/21	1.381	31.1
23/22, 44/23	1.401	31.5
17/12, 24/17	1.444	32.5
19/17, 34/19	1.446	32.5
23/16, 32/23	1.608	36.2
17/14, 28/17	1.648	37.1
23/14, 28/23	1.671	37.6
17/16, 32/17	1.711	38.5
23/19, 38/23	1.872	42.1
23/12, 24/23	1.875	42.2
17/11, 22/17	1.918	43.2
23/21, 42/23	1.938	43.6
23/20, 40/23	1.961	44.1
23/18, 36/23	2.142	48.2
23/15, 30/23	2.228	50.1
17/13, 26/17	2.239	50.4
23/17, 34/23	3.319	74.7

Higher-limit JI

270edo's approximation to higher harmonics, starting from 29, demonstrates a somewhat monotonous sharp tendency. This allows it to be considered as a temperament of very high limits – specifically the 53-limit. In fact, 270edo is the first edo to be monotonic in the 47- through 51-odd-limit, using the 270i val with the sharp mapping of 23.

For primes 37 and 41, this means the pairs 37/36 and 38/37, and the pairs 41/40 and 42/41, are distinct, observing 1369/1368 (S37) and 1681/1680 (S41). In fact 38/37, 39/38, 40/39, and 41/40 are tempered together. The sharp mapping for prime 23 is required here so that 37/33 (198.071 ¢ just) is not tuned wider 46/41 (199.212 ¢ just). Prime 43 then fits naturally with 42/41, 43/42, 44/43, and 45/44 all tempered together, while 47 may be added such that 48/47 is tempered together with 49/48, 50/49, and 51/50. Again the sharp mapping for prime 23 is required so that 46/45 is tempered together with 45/44 and that 47/46 is tempered together with 48/47. Prime 53, if desired, is tuned with 51/50~53/52 and 52/51~54/53, so monotonicity is unavoidably lost in the 53-odd-limit.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[23 6 -14⟩, [24 -21 4⟩	[⟨270 428 627]]	−0.1069	0.0759	1.71
2.3.5.7	2401/2400, 4375/4374, 29360128/29296875	[⟨270 428 627 758]]	−0.0858	0.0752	1.69
2.3.5.7.11	2401/2400, 3025/3024, 4375/4374, 5632/5625	[⟨270 428 627 758 934]]	−0.0567	0.0889	2.00
2.3.5.7.11.13	676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095	[⟨270 428 627 758 934 999]]	−0.0235	0.1100	2.48
2.3.5.7.11.13.19	676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728	[⟨270 428 627 758 934 999 1147]]	−0.0290	0.1028	2.31
2.3.5.7.11.13.17	676/675, 715/714, 936/935, 1001/1000, 1225/1224, 4096/4095	[⟨270 428 627 758 934 999 1104]]	−0.0799	0.1718	3.86
2.3.5.7.11.13.17.19	676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1225/1224, 1331/1330	[⟨270 428 627 758 934 999 1104 1147]]	−0.0777	0.1608	3.62
2.3.5.7.11.13.17.19.23	460/459, 529/528, 676/675, 715/714, 736/735, 936/935, 1001/1000, 1216/1215	[⟨270 428 627 758 934 999 1104 1147 1221]]	−0.0296	0.2037	4.58

• 270et has lower relative errors than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat 72 in the 11-limit, 224 in the 13-limit, and 217 in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is 342, in the 13-limit 494, in the 23-limit 282; and in the 19-limit, 311 for absolute error and 581 for relative error. It is also a record edo for Pepper ambiguity in the 11-, 13- and 15-odd-limit, and the edo with the lowest TE logflat badness in the 11-limit, 13-limit and 19-limit up until 342edo, 96478edo and 3395edo respectively.
• 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
• It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until 552, and the least relative error until 2190.
• It is also notable in the 17-limit, with lower absolute errors than smaller equal temperaments despite inconsistency in the corresponding odd limit.

Commas

Prime limit	Ratio	Cents	Monzo	Color name		Name(s)
5	(22 digits)	4.20	[24 -21 4⟩	Sasaquadyo	ssy4	Vulture comma
5	(20 digits)	3.34	[23 6 -14⟩	Sasepbigu	sg14	Vishnuzma
7	(16 digits)	3.80	[25 -14 0 -1⟩	Sasaru	ssr	Garischisma
7	2401/2400	0.72	[-5 -1 -2 4⟩	Bizozogu	z4gg	Breedsma
7	4375/4374	0.40	[-1 -7 4 1⟩	Zoquadyo	zy4	Ragisma
7	(16 digits)	3.73	[22 -1 -10 1⟩	Sazoquinbigu	szg10	Quasiorwellisma
11	(22 digits)	6.35	[33 -23 0 0 1⟩	Trisalo	s1o3	Pythrabian comma
11	5632/5625	2.15	[9 -2 -4 0 1⟩	Saloquagu	s1og4	Vishdel comma
11	(12 digits)	2.04	[-16 -3 0 0 6⟩	Tribilo	1o3	Nexus comma
11	3025/3024	0.57	[-4 -3 2 -1 2⟩	Loloruyoyo	1ooryy	Lehmerisma
11	9801/9800	0.18	[-3 4 -2 -2 2⟩	Bilorugu	(1org)2	Kalisma
13	676/675	2.56	[2 -3 -2 0 0 2⟩	Bithogu	3oogg	Island comma, parizeksma
13	1001/1000	1.73	[-3 0 -3 1 1 1⟩	Tholozotrigu	3o1ozg3	Fairytale comma, sinbadma
13	2080/2079	0.83	[5 -3 1 -1 -1 1⟩	Tholuruyo	3o1ury	Ibnsinma, sinaisma
13	4096/4095	0.42	[12 -2 -1 -1 0 -1⟩	Sathurugu	s3urg	Minisma
17	12376/12375	0.14	[3 -2 -3 1 -1 1 1⟩	Sotholuzotrigu	7o3o1uzg3	Flashma
19	1216/1215	1.42	2.3.5.19 [6 -5 -1 1⟩	Sanogu	s9og	Password, Eratosthenes' comma
19	(16 digits)	0.00	[-1 -4 -1 1 -4 1 0 4⟩	Quadno-athoquadlu-azogu	9o43o1u4zg	Tredekisma
23	529/528	3.24	2.3.11.23 [-4 -1 -1 2⟩	Bitwetho-alu	23oo1u	Preziosisma
29	784/783	2.20	2.3.7.29 [4 -3 2 -1⟩	Twenuzozo	23uzz	Biminorisma
31	621/620	2.79	2.3.5.23.31 [-2 3 -1 1 -1⟩	Thiwutwethogu	31u23og	Owowhatsthisma

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	1\270	4.4	385/384	Keenanose
1	29\270	128.8	14/13	Tertiathirds
1	61\270	271.1	90/77	Quasiorwell
1	71\270	315.5	6/5	Acrokleismic / counteracro
1	79\270	351.1	49/40	Newt
1	97\270	431.1	77/60	Lockerbie
1	107\270	475.5	25/19	Vulture
2	14\270	62.2	28/27	Eagle
2	16\270	71.1	25/24	Vishnu / ananta / acyuta
2	112\270 (23\270)	497.7 (102.2)	4/3 (35/33)	Gariwizmic
2	28\270	124.4	275/256	Semivulture
2	47\270	208.8	44/39	Abigail
2	52\270	231.1	8/7	Orga
2	131\270 (4\270)	582.2 (17.7)	7/5 (99/98)	Quarvish
3	17\270	75.5	24/23	Terture
3	31\270	137.7	13/12	Avicenna
5	83\270 (25\270)	368.8 (111.1)	1024/891 (16/15)	Quintosec
6	112\270 (4\270)	497.7 (97.7)	4/3 (128/121)	Sextile
9	71\270 (11\270)	315.5 (48.8)	6/5 (36/35)	Ennealimmal / enneabiotic / ennealympic
10	16\270 (11\270)	71.1 (48.8)	25/24 (36/35)	Decavish
10	56\270 (2\270)	248.8 (8.8)	15/13 (176/175)	Decoid
10	71\270 (10\270)	315.5 (44.4)	6/5 (40/39)	Deca
18	71\270 (4\270)	248.8 (17.7)	15/13 (99/98)	Hemiennealimmal
18	71\270 (2\270)	475.5 (8.8)	1053/800 (1287/1280)	Semihemiennealimmal
27	61\270 (1\270)	271.1 (4.4)	1375/1176 (385/384)	Trinealimmal
30	82\270 (1\270)	364.4 (4.4)	216/175 (385/384)	Zinc
45	59\270 (1\270)	262.2 (4.4)	64/55 (385/384)	Rhodium

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

Scales

Mos scales

• Ennealimmal[45]: 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2
• Vishnu[34]: 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7

Harmonic scales

270edo very accurately approximates the mode 16 of harmonic series. The scale in adjacent steps is 24, 22, 21, 20, 19, 18, 17, 17, 16, 15, 15, 14, 14, 13, 13, 12. Four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28, and 30/29~31/30.

It further does decently in the mode 24. The scale in adjacent steps is 16, 15, 15, 14, 14, 13, 13, 12, 12, 12, 11, 11, 11, 10, 10, 10, 10, 9, 9, 9, 9, 9, 8, 8.

Other scales

• Gutierrez-Lambeth quasi-subharmonic pentatonic (octave reduced: 37 23 93 65 52)
• Gutierrez Moonglade scale (24 tones): 3 17 22 3 20 5 17 25 5 14 22 5 3 16 18 4 19 5 5 12 2 6 17 5

External links

• tredek, 270-edo on Tonalsoft Encyclopedia