87edo

Prime factorization

3 × 29

Step size

13.7931¢

Fifth

51\87 (703.448¢) (→17\29)

Semitones (A1:m2)

9:6 (124.1¢ : 82.76¢)

Consistency limit

15

Distinct consistency limit

13

Special properties

Zeta peak integer

87 equal divisions of the octave (abbreviated 87edo or 87ed2), also called 87-tone equal temperament (87tet) or 87 equal temperament (87et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 87 equal parts of about 13.8 ¢ each. Each step represents a frequency ratio of 2^1/87, or the 87th root of 2.

Theory

87edo is solid as both a 13-limit (or 15-odd-limit) and as a 5-limit system, and does well enough in any limit in between. It is the smallest edo that is distinctly consistent in the 13-odd-limit tonality diamond, and also the smallest edo with relative interval errors of no greater than 25% on all of the first 16 harmonics of the harmonic series. It is also a zeta peak integer edo.

87edo also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.

The equal temperament tempers out 15625/15552 (kleisma), [26 -12 -3⟩ (misty comma), and [46 -29⟩ (29-comma) in the 5-limit, in addition to 245/243, 1029/1024, 3136/3125, and 5120/5103 in the 7-limit. In the 13-limit, notably 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000.

87edo is a particularly good tuning for rodan, the 41 & 46 temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit POTE generator and is close to the 11-limit POTE generator also. Also, the 32\87 generator for clyde temperament is 0.04455 cents sharp of the 7-limit POTE generator.

Prime harmonics

In higher limits it excels as a subgroup temperament, especially as an incomplete 71-limit temperament with 128/127 and 129/128 (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of Ringer 87, thus tempering S116 through S137 by patent val and corresponding to the gravity of the fact that 87edo is a circle of 126/125's, meaning (126/125)⁸⁷ only very slightly exceeds the octave.

Approximation of prime harmonics in 87edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.00	+1.49	-0.11	-3.31	+0.41	+0.85	+5.39	+5.94	+6.21	+4.91	-0.21	-3.07
Error	Relative (%)	+0.0	+10.8	-0.8	-24.0	+2.9	+6.2	+39.1	+43.0	+45.0	+35.6	-1.5	-22.2
Steps (reduced)		87 (0)	138 (51)	202 (28)	244 (70)	301 (40)	322 (61)	356 (8)	370 (22)	394 (46)	423 (75)	431 (83)	453 (18)

Approximation of prime harmonics in 87edo (continued)
Harmonic		41	43	47	53	59	61	67	71	73	79	83	89
Error	Absolute (¢)	-1.48	-1.17	-3.44	-4.54	+2.90	+0.36	+3.45	-0.39	+6.69	-5.92	+5.13	-5.36
Error	Relative (%)	-10.7	-8.5	-24.9	-32.9	+21.0	+2.6	+25.0	-2.8	+48.5	-42.9	+37.2	-38.9
Steps (reduced)		466 (31)	472 (37)	483 (48)	498 (63)	512 (77)	516 (81)	528 (6)	535 (13)	539 (17)	548 (26)	555 (33)	563 (41)

Intervals

#	Cents	Approximated Ratios		Ups and Downs Notation
#	Cents	13-Limit	31-Limit No-7s Extension	Ups and Downs Notation
0	0.000	1/1		P1	D
1	13.793	91/90, 100/99, 126/125		^1	^D
2	27.586	49/48, 55/54, 64/63, 65/64, 81/80		^^1	^^D
3	41.379	40/39, 45/44, 50/49	39/38	^³1	^³D/v³Eb
4	55.172	28/27, 33/32, 36/35	30/29, 31/30, 32/31, 34/33	vvm2	vvEb
5	68.966	25/24, 26/25, 27/26	24/23	vm2	vEb
6	82.759	21/20, 22/21	20/19, 23/22	m2	Eb
7	96.552	35/33	18/17, 19/18	^m2	^Eb
8	110.345	16/15	17/16, 31/29, 33/31	^^m2	^^Eb
9	124.138	14/13, 15/14	29/27	vv~2	^³Eb
10	137.931	13/12, 27/25	25/23	v~2	^⁴Eb
11	151.724	12/11, 35/32		^~2	v⁴E
12	165.517	11/10	32/29, 34/31	^^~2	v³E
13	179.310	10/9		vvM2	vvE
14	193.103	28/25	19/17, 29/26	vM2	vE
15	206.897	9/8	26/23	M2	E
16	220.690	25/22	17/15, 33/29	^M2	^E
17	234.483	8/7	31/27	^^M2	^^E
18	248.276	15/13	22/19, 23/20, 38/33	^³M2/v³m3	^³E/v³F
19	262.089	7/6	29/25, 36/31	vvm3	vvF
20	275.862	75/64	20/17, 27/23, 34/29	vm3	vF
21	289.655	13/11, 32/27, 33/28		m3	F
22	303.448	25/21	19/16, 31/26	^m3	^F
23	317.241	6/5		^^m3	^^F
24	331.034	40/33	23/19, 29/24	vv~3	^³F
25	344.828	11/9, 39/32		v~3	^⁴F
26	358.621	16/13, 27/22	38/31	^~3	v⁴F#
27	372.414	26/21	31/25, 36/29	^^3	v³F#
28	386.207	5/4		vvM3	vvF#
29	400.000	44/35	24/19, 29/23, 34/27	vM3	vF#
30	413.793	14/11, 33/26, 81/64	19/15	M3	F#
31	427.586	32/25	23/18	^M3	^F#
32	441.379	9/7, 35/27	22/17, 31/24, 40/31	^^M3	^^F#
33	455.172	13/10	30/23	^³M3/v³4	^³F#/v³G
34	468.966	21/16	17/13, 25/19, 38/29	vv4	vvG
35	482.759	33/25		v4	vG
36	496.552	4/3		P4	G
37	510.345	35/26	31/23	^4	^G
38	524.138	27/20	23/17	^^4	^^G
39	537.931	15/11	26/19, 34/25	^³4	^³G
40	551.724	11/8, 48/35		^⁴4	^⁴G
41	565.517	18/13	32/23	v⁴A4, vd5	v⁴G#, vAb
42	579.310	7/5	46/33	v³A4, d5	v³G#, Ab
43	593.103	45/32	24/17, 31/22, 38/27	vvA4, ^d5	vvG#, ^Ab
…	…	…	…	…	…

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 87edo. Prime harmonics are in bold.

As 87edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 87edo
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
5/4, 8/5	0.107	0.8
11/8, 16/11	0.406	2.9
13/11, 22/13	0.445	3.2
11/10, 20/11	0.513	3.7
15/13, 26/15	0.535	3.9
13/12, 24/13	0.642	4.7
13/8, 16/13	0.852	6.2
13/10, 20/13	0.958	6.9
15/11, 22/15	0.980	7.1
11/6, 12/11	1.087	7.9
15/8, 16/15	1.386	10.1
3/2, 4/3	1.493	10.8
5/3, 6/5	1.600	11.6
13/9, 18/13	2.135	15.5
11/9, 18/11	2.580	18.7
9/8, 16/9	2.987	21.7
9/5, 10/9	3.093	22.4
7/5, 10/7	3.202	23.2
7/4, 8/7	3.309	24.0
11/7, 14/11	3.715	26.9
13/7, 14/13	4.160	30.2
15/14, 28/15	4.695	34.0
7/6, 12/7	4.802	34.8
9/7, 14/9	6.295	45.6

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3.5	15625/15552, 67108864/66430125	[⟨87 138 202]]	−0.299	0.455	3.30
2.3.5.7	245/243, 1029/1024, 3136/3125	[⟨87 138 202 244]]	+0.070	0.752	5.45
2.3.5.7.11	245/243, 385/384, 441/440, 3136/3125	[⟨87 138 202 244 301]]	+0.033	0.676	4.90
2.3.5.7.11.13	196/195, 245/243, 352/351, 364/363, 625/624	[⟨87 138 202 244 301 322]]	−0.011	0.625	4.53
2.3.5.7.11.13.17	154/153, 196/195, 245/243, 273/272, 364/363, 375/374	[⟨87 138 202 244 301 322 356]]	−0.198	0.738	5.35
2.3.5.7.11.13.17.19	154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363	[⟨87 138 202 244 301 322 356 370]]	−0.348	0.796	5.77

13-limit detempering

Main article: 87edo/13-limit detempering

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperament
1	2\87	27.586	64/63	Arch
1	4\87	55.172	33/32	Escapade / escaped / alphaquarter
1	10\87	137.931	13/12	Quartemka
1	14\87	193.103	28/25	Luna / didacus / hemithirds
1	17\87	234.483	8/7	Slendric / rodan
1	23\87	317.241	6/5	Hanson / countercata / metakleismic
1	26\87	358.621	16/13	Restles
1	32\87	441.379	9/7	Clyde
1	38\87	524.138	65/48	Widefourth
1	40\87	551.724	11/8	Emka / emkay
3	18\87 (11\87)	248.276 (151.724)	15/13 (12/11)	Hemimist
3	23\87 (6\87)	317.241 (82.759)	6/5 (21/20)	Tritikleismic
3	28\87 (1\87)	386.207 (13.793)	5/4 (126/125)	Mutt
3	36\87 (7\87)	496.552 (96.552)	4/3 (18/17~19/18)	Misty
29	28\87 (1\87)	386.207 (13.793)	5/4 (121/120)	Mystery

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

87 can serve as a MOS in these:

Avicenna (87&270) ⟨⟨ 24 -9 -66 12 27 … ]]
87&494 ⟨⟨ 51 -1 -133 11 32 … ]]

Scales

MOS scales

Main article: List of MOS scales in 87edo

Harmonic scale

87edo accurately approximates the mode 8 of harmonic series, and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.

Mode 8

Overtones	8	9	10	11	12	13	14	15	16
JI Ratios	1/1	9/8	5/4	11/8	3/2	13/8	7/4	15/8	2/1
… in cents	0.0	203.9	386.3	551.3	702.0	840.5	968.8	1088.3	1200.0
Degrees in 87edo	0	15	28	40	51	61	70	79	87
… in cents	0.0	206.9	386.2	551.7	703.5	841.4	965.5	1089.7	1200.0

The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.

Mode 12

Overtones	12	13	14	15	16	17	18	19	20	21	22	23	24
JI Ratios	1/1	13/12	7/6	5/4	4/3	17/12	3/2	19/12	5/3	7/4	11/6	23/12	2/1
… in cents	0.0	138.6	266.9	386.3	498.0	603.0	702.0	795.6	884.4	968.8	1049.4	1126.3	1200.0
Degrees in 87edo	0	10	19	28	36	44	51	58	64	70	76	82	87
… in cents	0.0	137.9	262.1	386.2	496.6	606.9	703.4	800.0	882.8	965.5	1048.3	1131.0	1200.0

The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.
13, 15, 16, 18, 20, and 22 are close matches.
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.

Music

Gene Ward Smith

Pianodactyl (archived 2010) – SoundCloud | detail | play – rodan[26] in 87edo tuning