87edo

The 87 equal temperament, often abbreviated 87-tET, 87-EDO, or 87-ET, is the scale derived by dividing the octave into 87 equally-sized steps, where each step is 13.79 cents.

Theory

87et is solid as both a 13-limit (or 15-odd-limit) and as a 5-limit system, and of course does well enough in any limit in between. It represents the 13-limit tonality diamond both uniquely and consistently (see 87edo/13-limit detempering), and is the smallest equal temperament to do so. It is a zeta peak integer edo.

87et 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 don't combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.

87et tempers out 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.

87et is a particularly good tuning for rodan 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.

Intervals

#	Cents	Approximated Ratios		Ups and Downs Notation
		13-Limit	31-Limit No-7s Extension
0	0.000	1/1		P1	D
1	13.793	126/125, 100/99, 91/90		^1	^D
2	27.586	81/80, 64/63, 49/48, 55/54, 65/64		^^1	^^D
3	41.379	50/49, 45/44, 40/39	39/38	^31	^3D/v3Eb
4	55.172	28/27, 36/35, 33/32	34/33, 30/29, 32/31, 31/30	vvm2	vvEb
5	68.966	25/24, 27/26, 26/25	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, 33/31, 31/29	^^m2	^^Eb
9	124.138	15/14, 14/13	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, 38/33, 23/20	^3M2/v3m3	^3E/v3F
19	262.089	7/6	29/25, 36/31	vvm3	vvF
20	275.862	75/64	27/23, 34/29	vm3	vF
21	289.655	32/27, 33/28, 13/11		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	27/22, 16/13	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	34/27, 24/19, 29/23	vM3	vF#
30	413.793	81/64, 14/11, 33/26	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, 38/27, 31/22	vvA4, ^d5	vvG#, ^Ab
…	…	…	…	…	…

Just approximation

Temperament Measures

The following table shows TE temperament measures (RMS normalized by the rank) of 87et.

		3-limit	5-limit	7-limit	11-limit	13-limit	17-limit	19-limit	23-limit	29-limit	31-limit
Octave stretch (¢)		-0.471	-0.299	+0.070	+0.033	-0.011	-0.198	-0.348	-0.462	-0.517	-0.466
Error	absolute (¢)	0.471	0.455	0.752	0.676	0.625	0.738	0.796	0.816	0.792	0.772
	relative (%)	3.42	3.30	5.45	4.90	4.53	5.35	5.77	5.92	5.74	5.60

13-limit detempering of 87et

See also: Detempering

Main article: 87edo/13-limit detempering

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (Reduced)	Cents (Reduced)	Associated Ratio (Reduced)	Temperament
1	2\87	27.586	64/63	Arch
1	4\87	55.172	33/32	Escapade / sensa / alphaquarter
1	10\87	137.931	13/12	Quartemka
1	14\87	193.103	28/25	Luna / hemithirds
1	17\87	234.483	8/7	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

87 can serve as a MOS in these:

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

Scales

Harmonic Scale

87edo accurately approximates the mode 8 of harmonic series, and the only intervals not distinct are 14/13 and 15/14. It does mode 16 fairly decent, with the only anomaly at 28/27 (4 steps) and 29/28 (5 steps).

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 16

Odd overtones	17	19	21	23	25	27	29	31
JI Ratios	17/16	19/16	21/16	23/16	25/16	27/16	29/16	31/16
… in cents	105.0	297.5	470.8	628.3	772.6	905.9	1029.6	1145.0
Degrees in 87edo	8	22	34	46	56	66	75	83
… in cents	110.3	303.4	469.0	634.5	772.4	910.3	1034.5	1144.8

• The scale in adjacent steps is 8, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 5, 4, 4, 4.
• 25 and 31 are close matches.
• 21 is a little bit flat, but still decent.
• The others (17, 19, 23, 27 and 29) are extremely sharp, but the intervals between them are close.

Music

• Pianodactyl play by Gene Ward Smith