45zpi: Difference between revisions

Revision as of 22:54, 11 August 2024

45 zeta peak index (abbreviated 45zpi), is the equal-step tuning system obtained from the 45th peak of the Riemann zeta function.

Tuning			Strength			Closest EDO		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	EDO	Octave (cents)	Consistent	Distinct
45zpi	14.5944346577250	82.2231232756126	2.097730	0.344839	10.594800	15edo	1233.34684913419	2	2

Theory

45zpi is characterized by a very broad octave error, yet it maintains a quite decent zeta strength. This combination makes it an ideal candidate for no-octave tuning applications.

No other zeta peak indexes exhibit both a larger octave error and greater zeta height than 45zpi.

45zpi supports a complex chord structure with ratios of 1:3:4:5:7:9:13:15:18:19:20:21:22:23:24:25, which further exemplifies its capabilities.

The closest zeta peak indexes to 45zpi that exceed its strength are 42zpi and 47zpi, though 43zpi is nearly as strong as 45zpi.

Harmonic series

Approximation of harmonics in 45zpi
Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+33.3	-10.8	-15.5	+9.3	+22.5	+2.3	+17.8	-21.6	-39.6	-40.2	-26.4	-0.5	+35.7	-1.6	-31.1
Error	Relative (%)	+40.6	-13.2	-18.9	+11.3	+27.4	+2.8	+21.7	-26.3	-48.2	-48.8	-32.1	-0.6	+43.4	-1.9	-37.8
Step		15	23	29	34	38	41	44	46	48	50	52	54	56	57	58

Approximation of harmonics in 45zpi
Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+28.4	+11.7	+0.3	-6.3	-8.5	-6.8	-1.5	+7.0	+18.5	+32.9	-32.5	-13.2	+8.3	+31.8	-25.0	+2.3
Error	Relative (%)	+34.6	+14.2	+0.4	-7.6	-10.3	-8.3	-1.9	+8.5	+22.6	+40.0	-39.5	-16.1	+10.1	+38.7	-30.4	+2.8
Step		60	61	62	63	64	65	66	67	68	69	69	70	71	72	72	73

Intervals

Intervals in 45zpi
Degree	Cents	Ratios	Ups and Downs Notation	Step
JI ratios are comprised of 32-integer limit ratios, and are stylized as follows to indicate their accuracy: Bold Underlined: relative error < 8.333 % Bold: relative error < 16.667 % Normal: relative error < 25 % Small: relative error < 33.333 % Small Small: relative error < 41.667 % Small Small Small: relative error < 50 %			⟨73 116] at every 5 steps Whole tone = 13 steps Limma = 4 steps Apotome = 9 steps
0	0.000		P1	0
1	82.223	32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24, 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17, 17/16, 16/15, 31/29, 15/14	^m2	5
2	164.446	29/27, 14/13, 27/25, 13/12, 25/23, 12/11, 23/21, 11/10, 32/29, 21/19, 31/28, 10/9, 29/26, 19/17, 28/25, 9/8	vvvM2	10
3	246.669	26/23, 17/15, 25/22, 8/7, 31/27, 23/20, 15/13, 22/19, 29/25, 7/6, 27/23, 20/17	^^M2, vvm3	15
4	328.892	13/11, 32/27, 19/16, 25/21, 31/26, 6/5, 29/24, 23/19, 17/14, 28/23, 11/9, 27/22, 16/13, 21/17, 26/21	^^^m3	20
5	411.116	31/25, 5/4, 29/23, 24/19, 19/15, 14/11, 23/18, 32/25, 9/7, 31/24, 22/17	vM3	25
6	493.339	13/10, 30/23, 17/13, 21/16, 25/19, 29/22, 4/3, 31/23, 27/20, 23/17, 19/14	P4	30
7	575.562	15/11, 26/19, 11/8, 29/21, 18/13, 25/18, 32/23, 7/5, 31/22, 24/17, 17/12, 27/19	v⁴A4	35
8	657.785	10/7, 23/16, 13/9, 29/20, 16/11, 19/13, 22/15, 25/17, 28/19, 31/21	vvv5	40
9	740.008	3/2, 32/21, 29/19, 26/17, 23/15, 20/13, 17/11, 31/20, 14/9, 25/16	^^5, vvm6	45
10	822.231	11/7, 30/19, 19/12, 27/17, 8/5, 29/18, 21/13, 13/8, 31/19, 18/11, 23/14	^^^m6	50
11	904.454	28/17, 5/3, 32/19, 27/16, 22/13, 17/10, 29/17, 12/7, 31/18	vM6	55
12	986.677	19/11, 26/15, 7/4, 30/17, 23/13, 16/9, 25/14, 9/5	m7	60
13	1068.901	29/16, 20/11, 31/17, 11/6, 24/13, 13/7, 28/15, 15/8, 32/17, 17/9	v⁴M7	65
14	1151.124	19/10, 21/11, 23/12, 25/13, 27/14, 29/15, 31/16	^M7	70
15	1233.347	2/1, 31/15, 29/14, 27/13, 25/12	^^1 +1 oct, vvm2 +1 oct	75
16	1315.570	23/11, 21/10, 19/9, 17/8, 32/15, 15/7, 28/13, 13/6, 24/11	^^^m2 +1 oct	80
17	1397.793	11/5, 31/14, 20/9, 29/13, 9/4, 25/11, 16/7	vM2 +1 oct	85
18	1480.016	23/10, 30/13, 7/3, 26/11, 19/8, 31/13, 12/5	m3 +1 oct	90
19	1562.239	29/12, 17/7, 22/9, 27/11, 32/13, 5/2	v⁴M3 +1 oct	95
20	1644.462	28/11, 23/9, 18/7, 31/12, 13/5, 21/8, 29/11	^M3 +1 oct	100
21	1726.686	8/3, 27/10, 19/7, 30/11, 11/4	^^4 +1 oct	105
22	1808.909	25/9, 14/5, 31/11, 17/6, 20/7, 23/8, 26/9, 29/10, 32/11	^^^d5 +1 oct	110
23	1891.132	3/1	v5 +1 oct	115
24	1973.355	31/10, 28/9, 25/8, 22/7, 19/6, 16/5	m6 +1 oct	120
25	2055.578	29/9, 13/4, 23/7, 10/3	v⁴M6 +1 oct	125
26	2137.801	27/8, 17/5, 24/7, 31/9, 7/2	^M6 +1 oct	130
27	2220.024	32/9, 25/7, 18/5, 29/8, 11/3	^^m7 +1 oct	135
28	2302.247	26/7, 15/4, 19/5, 23/6, 27/7	vvM7 +1 oct	140
29	2384.471	31/8, 4/1	v1 +2 oct	145
30	2466.694	29/7, 25/6, 21/5, 17/4	m2 +2 oct	150
31	2548.917	30/7, 13/3, 22/5, 31/7	v⁴M2 +2 oct	155
32	2631.140	9/2, 32/7, 23/5, 14/3	^M2 +2 oct	160
33	2713.363	19/4, 24/5, 29/6	^^m3 +2 oct	165
34	2795.586	5/1	vvM3 +2 oct	170
35	2877.809	31/6, 26/5, 21/4, 16/3	v4 +2 oct	175
36	2960.032	27/5, 11/2, 28/5	^⁴4 +2 oct	180
37	3042.256	17/3, 23/4, 29/5	v⁴5 +2 oct	185
38	3124.479	6/1, 31/5	^5 +2 oct	190
39	3206.702	25/4, 19/3, 32/5, 13/2	^^m6 +2 oct	195
40	3288.925	20/3, 27/4	vvM6 +2 oct	200
41	3371.148	7/1	vm7 +2 oct	205
42	3453.371	29/4, 22/3, 15/2	^⁴m7 +2 oct	210
43	3535.594	23/3, 31/4	M7 +2 oct	215
44	3617.817	8/1	^1 +3 oct	220
45	3700.041	25/3, 17/2, 26/3	^^m2 +3 oct	225
46	3782.264	9/1	vvM2 +3 oct	230
47	3864.487	28/3, 19/2	vm3 +3 oct	235
48	3946.710	29/3, 10/1	^⁴m3 +3 oct	240
49	4028.933	31/3	M3 +3 oct	245
50	4111.156	21/2, 32/3, 11/1	^4 +3 oct	250
51	4193.379	23/2	vvvA4 +3 oct	255
52	4275.602	12/1	vv5 +3 oct	260
53	4357.826	25/2	vm6 +3 oct	265
54	4440.049	13/1	^⁴m6 +3 oct	270
55	4522.272	27/2	M6 +3 oct	275
56	4604.495	14/1, 29/2	^m7 +3 oct	280
57	4686.718	15/1	vvvM7 +3 oct	285
58	4768.941	31/2, 16/1	^^M7 +3 oct, vv1 +4 oct	290
59	4851.164		vm2 +4 oct	295
60	4933.387	17/1	^⁴m2 +4 oct	300
61	5015.611	18/1	M2 +4 oct	305
62	5097.834	19/1	^m3 +4 oct	310
63	5180.057	20/1	vvvM3 +4 oct	315
64	5262.280	21/1	^^M3 +4 oct, vv4 +4 oct	320
65	5344.503	22/1	^^^4 +4 oct	325
66	5426.726	23/1	^⁴d5 +4 oct	330
67	5508.949	24/1	P5 +4 oct	335
68	5591.172	25/1	^m6 +4 oct	340
69	5673.396	26/1, 27/1	vvvM6 +4 oct	345
70	5755.619	28/1	^^M6 +4 oct, vvm7 +4 oct	350
71	5837.842	29/1	^^^m7 +4 oct	355
72	5920.065	30/1, 31/1	vM7 +4 oct	360
73	6002.288	32/1	P1 +5 oct	365

Approximation to JI

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45zpi: Difference between revisions

Revision as of 22:54, 11 August 2024

Contents

Theory

Harmonic series

Intervals

Approximation to JI

Navigation menu

@@ Line 46: / Line 46: @@
 == Intervals ==
+{| class="wikitable center-1 right-2 left-3 center-4 center-5"
+|-
+|+ style="white-space:nowrap" | Intervals in 45zpi
+| colspan="3" style="text-align:left;" | JI ratios are comprised of 32-integer limit ratios,<br>and are stylized as follows to indicate their accuracy:
+* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
+* '''Bold:''' relative error < 16.667 %
+* Normal: relative error < 25 %
+* <small>Small:</small> relative error < 33.333 %
+* <small><small>Small Small:</small></small> relative error < 41.667 %
+* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
+| colspan="2" style="text-align:right;" | <center>'''⟨73 116] at every 5 steps'''</center><br>[[9/8|Whole tone]] = 13 steps<br>[[256/243|Limma]] = 4 steps<br>[[2187/2048|Apotome]] = 9 steps
+|-
+! Degree
+! Cents
+! Ratios
+! Ups and Downs Notation
+! Step
+|-
+| 0
+| 0.000
+|
+| P1
+| 0
+|-
+| 1
+| 82.223
+| <small>[[32/31]]</small>, <small>[[31/30]]</small>, <small>[[30/29]]</small>, <small>[[29/28]]</small>, [[28/27]], [[27/26]], [[26/25]], '''[[25/24]]''', '''[[24/23]]''', '''<u>[[23/22]]'''</u>, '''<u>[[22/21]]'''</u>, '''<u>[[21/20]]'''</u>, '''<u>[[20/19]]'''</u>, '''[[19/18]]''', [[18/17]], <small>[[17/16]]</small>, <small><small>[[16/15]]</small></small>, <small><small>[[31/29]]</small></small>, <small><small><small>[[15/14]]</small></small></small>
+| ^m2
+| 5
+|-
+| 2
+| 164.446
+| <small><small><small>[[29/27]]</small></small></small>, <small><small><small>[[14/13]]</small></small></small>, <small><small>[[27/25]]</small></small>, <small>[[13/12]]</small>, [[25/23]], [[12/11]], '''[[23/21]]''', '''<u>[[11/10]]'''</u>, '''<u>[[32/29]]'''</u>, '''[[21/19]]''', '''[[31/28]]''', [[10/9]], <small>[[29/26]]</small>, <small><small>[[19/17]]</small></small>, <small><small>[[28/25]]</small></small>, <small><small><small>[[9/8]]</small></small></small>
+| vvvM2
+| 10
+|-
+| 3
+| 246.669
+| <small><small><small>[[26/23]]</small></small></small>, <small><small>[[17/15]]</small></small>, <small>[[25/22]]</small>, [[8/7]], '''[[31/27]]''', '''<u>[[23/20]]'''</u>, '''<u>[[15/13]]'''</u>, '''[[22/19]]''', '''[[29/25]]''', [[7/6]], <small><small>[[27/23]]</small></small>, <small><small><small>[[20/17]]</small></small></small>
+| ^^M2, vvm3
+| 15
+|-
+| 4
+| 328.892
+| <small><small><small>[[13/11]]</small></small></small>, <small><small><small>[[32/27]]</small></small></small>, <small><small>[[19/16]]</small></small>, <small>[[25/21]]</small>, <small>[[31/26]]</small>, '''[[6/5]]''', '''<u>[[29/24]]'''</u>, '''<u>[[23/19]]'''</u>, '''[[17/14]]''', '''[[28/23]]''', [[11/9]], <small>[[27/22]]</small>, <small><small>[[16/13]]</small></small>, <small><small><small>[[21/17]]</small></small></small>, <small><small><small>[[26/21]]</small></small></small>
+| ^^^m3
+| 20
+|-
+| 5
+| 411.116
+| <small><small><small>[[31/25]]</small></small></small>, <small>[[5/4]]</small>, '''[[29/23]]''', '''<u>[[24/19]]'''</u>, '''<u>[[19/15]]'''</u>, '''<u>[[14/11]]'''</u>, '''[[23/18]]''', [[32/25]], <small>[[9/7]]</small>, <small><small>[[31/24]]</small></small>, <small><small><small>[[22/17]]</small></small></small>
+| vM3
+| 25
+|-
+| 6
+| 493.339
+| <small><small><small>[[13/10]]</small></small></small>, <small><small>[[30/23]]</small></small>, <small><small>[[17/13]]</small></small>, <small>[[21/16]]</small>, [[25/19]], [[29/22]], '''<u>[[4/3]]'''</u>, <small>[[31/23]]</small>, <small>[[27/20]]</small>, <small><small>[[23/17]]</small></small>, <small><small><small>[[19/14]]</small></small></small>
+| P4
+| 30
+|-
+| 7
+| 575.562
+| <small><small><small>[[15/11]]</small></small></small>, <small><small>[[26/19]]</small></small>, <small>[[11/8]]</small>, [[29/21]], '''[[18/13]]''', '''<u>[[25/18]]'''</u>, '''<u>[[32/23]]'''</u>, '''[[7/5]]''', [[31/22]], <small>[[24/17]]</small>, <small><small>[[17/12]]</small></small>, <small><small>[[27/19]]</small></small>
+| v<sup>4</sup>A4
+| 35
+|-
+| 8
+| 657.785
+| <small><small><small>[[10/7]]</small></small></small>, <small><small>[[23/16]]</small></small>, <small>[[13/9]]</small>, [[29/20]], '''[[16/11]]''', '''<u>[[19/13]]'''</u>, '''<u>[[22/15]]'''</u>, '''[[25/17]]''', '''[[28/19]]''', [[31/21]]
+| vvv5
+| 40
+|-
+| 9
+| 740.008
+| <small><small><small>[[3/2]]</small></small></small>, '''[[32/21]]''', '''[[29/19]]''', '''<u>[[26/17]]'''</u>, '''<u>[[23/15]]'''</u>, '''<u>[[20/13]]'''</u>, '''[[17/11]]''', [[31/20]], <small>[[14/9]]</small>, <small><small>[[25/16]]</small></small>
+| ^^5, vvm6
+| 45
+|-
+| 10
+| 822.231
+| <small><small><small>[[11/7]]</small></small></small>, <small><small>[[30/19]]</small></small>, <small>[[19/12]]</small>, <small>[[27/17]]</small>, '''[[8/5]]''', '''<u>[[29/18]]'''</u>, '''[[21/13]]''', [[13/8]], <small>[[31/19]]</small>, <small><small>[[18/11]]</small></small>, <small><small><small>[[23/14]]</small></small></small>
+| ^^^m6
+| 50
+|-
+| 11
+| 904.454
+| <small><small><small>[[28/17]]</small></small></small>, [[5/3]], '''<u>[[32/19]]'''</u>, '''<u>[[27/16]]'''</u>, '''<u>[[22/13]]'''</u>, [[17/10]], [[29/17]], <small><small>[[12/7]]</small></small>, <small><small><small>[[31/18]]</small></small></small>
+| vM6
+| 55
+|-
+| 12
+| 986.677
+| <small><small><small>[[19/11]]</small></small></small>, <small><small><small>[[26/15]]</small></small></small>, [[7/4]], '''<u>[[30/17]]'''</u>, '''<u>[[23/13]]'''</u>, '''[[16/9]]''', [[25/14]], <small><small>[[9/5]]</small></small>
+| m7
+| 60
+|-
+| 13
+| 1068.901
+| <small><small><small>[[29/16]]</small></small></small>, <small><small>[[20/11]]</small></small>, <small><small>[[31/17]]</small></small>, [[11/6]], '''[[24/13]]''', '''<u>[[13/7]]'''</u>, '''[[28/15]]''', [[15/8]], <small>[[32/17]]</small>, <small><small>[[17/9]]</small></small>
+| v<sup>4</sup>M7
+| 65
+|-
+| 14
+| 1151.124
+| <small><small><small>[[19/10]]</small></small></small>, <small><small>[[21/11]]</small></small>, <small>[[23/12]]</small>, [[25/13]], [[27/14]], '''[[29/15]]''', '''<u>[[31/16]]'''</u>
+| ^M7
+| 70
+|-
+| 15
+| 1233.347
+| <small><small>[[2/1]]</small></small>, <small>[[31/15]]</small>, <small>[[29/14]]</small>, <small><small>[[27/13]]</small></small>, <small><small><small>[[25/12]]</small></small></small>
+| ^^1 +1 oct, vvm2 +1 oct
+| 75
+|-
+| 16
+| 1315.570
+| <small><small><small>[[23/11]]</small></small></small>, <small><small>[[21/10]]</small></small>, <small>[[19/9]]</small>, '''[[17/8]]''', '''<u>[[32/15]]'''</u>, '''<u>[[15/7]]'''</u>, '''[[28/13]]''', <small>[[13/6]]</small>, <small><small><small>[[24/11]]</small></small></small>
+| ^^^m2 +1 oct
+| 80
+|-
+| 17
+| 1397.793
+| <small><small>[[11/5]]</small></small>, <small>[[31/14]]</small>, [[20/9]], '''[[29/13]]''', '''<u>[[9/4]]'''</u>, <small>[[25/11]]</small>, <small><small>[[16/7]]</small></small>
+| vM2 +1 oct
+| 85
+|-
+| 18
+| 1480.016
+| <small><small><small>[[23/10]]</small></small></small>, <small><small>[[30/13]]</small></small>, '''[[7/3]]''', '''[[26/11]]''', [[19/8]], <small>[[31/13]]</small>, <small><small><small>[[12/5]]</small></small></small>
+| m3 +1 oct
+| 90
+|-
+| 19
+| 1562.239
+| <small><small><small>[[29/12]]</small></small></small>, <small>[[17/7]]</small>, [[22/9]], '''[[27/11]]''', '''<u>[[32/13]]'''</u>, <small>[[5/2]]</small>
+| v<sup>4</sup>M3 +1 oct
+| 95
+|-
+| 20
+| 1644.462
+| <small>[[28/11]]</small>, [[23/9]], '''[[18/7]]''', '''<u>[[31/12]]'''</u>, '''[[13/5]]''', <small>[[21/8]]</small>, <small><small>[[29/11]]</small></small>
+| ^M3 +1 oct
+| 100
+|-
+| 21
+| 1726.686
+| <small><small>[[8/3]]</small></small>, '''[[27/10]]''', '''<u>[[19/7]]'''</u>, '''[[30/11]]''', <small>[[11/4]]</small>
+| ^^4 +1 oct
+| 105
+|-
+| 22
+| 1808.909
+| <small><small><small>[[25/9]]</small></small></small>, <small>[[14/5]]</small>, [[31/11]], '''<u>[[17/6]]'''</u>, '''[[20/7]]''', [[23/8]], <small><small>[[26/9]]</small></small>, <small><small><small>[[29/10]]</small></small></small>, <small><small><small>[[32/11]]</small></small></small>
+| ^^^d5 +1 oct
+| 110
+|-
+| 23
+| 1891.132
+| '''[[3/1]]'''
+| v5 +1 oct
+| 115
+|-
+| 24
+| 1973.355
+| [[31/10]], '''[[28/9]]''', '''<u>[[25/8]]'''</u>, '''[[22/7]]''', <small>[[19/6]]</small>, <small><small><small>[[16/5]]</small></small></small>
+| m6 +1 oct
+| 120
+|-
+| 25
+| 2055.578
+| <small><small>[[29/9]]</small></small>, [[13/4]], '''<u>[[23/7]]'''</u>, <small><small>[[10/3]]</small></small>
+| v<sup>4</sup>M6 +1 oct
+| 125
+|-
+| 26
+| 2137.801
+| <small><small>[[27/8]]</small></small>, [[17/5]], '''<u>[[24/7]]'''</u>, '''<u>[[31/9]]'''</u>, <small><small>[[7/2]]</small></small>
+| ^M6 +1 oct
+| 130
+|-
+| 27
+| 2220.024
+| <small>[[32/9]]</small>, [[25/7]], '''<u>[[18/5]]'''</u>, '''[[29/8]]''', <small><small>[[11/3]]</small></small>
+| ^^m7 +1 oct
+| 135
+|-
+| 28
+| 2302.247
+| <small><small>[[26/7]]</small></small>, [[15/4]], '''[[19/5]]''', <small>[[23/6]]</small>, <small><small><small>[[27/7]]</small></small></small>
+| vvM7 +1 oct
+| 140
+|-
+| 29
+| 2384.471
+| <small><small><small>[[31/8]]</small></small></small>, [[4/1]]
+| v1 +2 oct
+| 145
+|-
+| 30
+| 2466.694
+| '''<u>[[29/7]]'''</u>, '''<u>[[25/6]]'''</u>, [[21/5]], <small><small><small>[[17/4]]</small></small></small>
+| m2 +2 oct
+| 150
+|-
+| 31
+| 2548.917
+| <small><small>[[30/7]]</small></small>, '''[[13/3]]''', [[22/5]], <small>[[31/7]]</small>
+| v<sup>4</sup>M2 +2 oct
+| 155
+|-
+| 32
+| 2631.140
+| <small>[[9/2]]</small>, '''<u>[[32/7]]'''</u>, '''[[23/5]]''', <small><small><small>[[14/3]]</small></small></small>
+| ^M2 +2 oct
+| 160
+|-
+| 33
+| 2713.363
+| [[19/4]], '''<u>[[24/5]]'''</u>, [[29/6]]
+| ^^m3 +2 oct
+| 165
+|-
+| 34
+| 2795.586
+| '''[[5/1]]'''
+| vvM3 +2 oct
+| 170
+|-
+| 35
+| 2877.809
+| <small><small><small>[[31/6]]</small></small></small>, <small>[[26/5]]</small>, '''[[21/4]]''', [[16/3]]
+| v4 +2 oct
+| 175
+|-
+| 36
+| 2960.032
+| <small><small><small>[[27/5]]</small></small></small>, '''[[11/2]]''', <small>[[28/5]]</small>
+| ^<sup>4</sup>4 +2 oct
+| 180
+|-
+| 37
+| 3042.256
+| <small><small><small>[[17/3]]</small></small></small>, [[23/4]], '''<u>[[29/5]]'''</u>
+| v<sup>4</sup>5 +2 oct
+| 185
+|-
+| 38
+| 3124.479
+| <small>[[6/1]]</small>, <small><small>[[31/5]]</small></small>
+| ^5 +2 oct
+| 190
+|-
+| 39
+| 3206.702
+| <small><small>[[25/4]]</small></small>, '''[[19/3]]''', '''[[32/5]]''', <small><small>[[13/2]]</small></small>
+| ^^m6 +2 oct
+| 195
+|-
+| 40
+| 3288.925
+| '''<u>[[20/3]]'''</u>, [[27/4]]
+| vvM6 +2 oct
+| 200
+|-
+| 41
+| 3371.148
+| '''<u>[[7/1]]'''</u>
+| vm7 +2 oct
+| 205
+|-
+| 42
+| 3453.371
+| <small>[[29/4]]</small>, '''<u>[[22/3]]'''</u>, <small><small><small>[[15/2]]</small></small></small>
+| ^<sup>4</sup>m7 +2 oct
+| 210
+|-
+| 43
+| 3535.594
+| '''[[23/3]]''', '''[[31/4]]'''
+| M7 +2 oct
+| 215
+|-
+| 44
+| 3617.817
+| [[8/1]]
+| ^1 +3 oct
+| 220
+|-
+| 45
+| 3700.041
+| <small><small>[[25/3]]</small></small>, '''<u>[[17/2]]'''</u>, <small><small><small>[[26/3]]</small></small></small>
+| ^^m2 +3 oct
+| 225
+|-
+| 46
+| 3782.264
+| <small>[[9/1]]</small>
+| vvM2 +3 oct
+| 230
+|-
+| 47
+| 3864.487
+| '''<u>[[28/3]]'''</u>, <small><small>[[19/2]]</small></small>
+| vm3 +3 oct
+| 235
+|-
+| 48
+| 3946.710
+| [[29/3]], <small><small><small>[[10/1]]</small></small></small>
+| ^<sup>4</sup>m3 +3 oct
+| 240
+|-
+| 49
+| 4028.933
+| [[31/3]]
+| M3 +3 oct
+| 245
+|-
+| 50
+| 4111.156
+| <small><small><small>[[21/2]]</small></small></small>, '''[[32/3]]''', <small><small><small>[[11/1]]</small></small></small>
+| ^4 +3 oct
+| 250
+|-
+| 51
+| 4193.379
+| <small><small><small>[[23/2]]</small></small></small>
+| vvvA4 +3 oct
+| 255
+|-
+| 52
+| 4275.602
+| <small>[[12/1]]</small>
+| vv5 +3 oct
+| 260
+|-
+| 53
+| 4357.826
+| [[25/2]]
+| vm6 +3 oct
+| 265
+|-
+| 54
+| 4440.049
+| '''<u>[[13/1]]'''</u>
+| ^<sup>4</sup>m6 +3 oct
+| 270
+|-
+| 55
+| 4522.272
+| [[27/2]]
+| M6 +3 oct
+| 275
+|-
+| 56
+| 4604.495
+| <small><small><small>[[14/1]]</small></small></small>, <small>[[29/2]]</small>
+| ^m7 +3 oct
+| 280
+|-
+| 57
+| 4686.718
+| '''<u>[[15/1]]'''</u>
+| vvvM7 +3 oct
+| 285
+|-
+| 58
+| 4768.941
+| <small>[[31/2]]</small>, <small><small>[[16/1]]</small></small>
+| ^^M7 +3 oct, vv1 +4 oct
+| 290
+|-
+| 59
+| 4851.164
+|
+| vm2 +4 oct
+| 295
+|-
+| 60
+| 4933.387
+| <small><small>[[17/1]]</small></small>
+| ^<sup>4</sup>m2 +4 oct
+| 300
+|-
+| 61
+| 5015.611
+| '''[[18/1]]'''
+| M2 +4 oct
+| 305
+|-
+| 62
+| 5097.834
+| '''<u>[[19/1]]'''</u>
+| ^m3 +4 oct
+| 310
+|-
+| 63
+| 5180.057
+| '''<u>[[20/1]]'''</u>
+| vvvM3 +4 oct
+| 315
+|-
+| 64
+| 5262.280
+| '''[[21/1]]'''
+| ^^M3 +4 oct, vv4 +4 oct
+| 320
+|-
+| 65
+| 5344.503
+| '''<u>[[22/1]]'''</u>
+| ^^^4 +4 oct
+| 325
+|-
+| 66
+| 5426.726
+| '''<u>[[23/1]]'''</u>
+| ^<sup>4</sup>d5 +4 oct
+| 330
+|-
+| 67
+| 5508.949
+| '''[[24/1]]'''
+| P5 +4 oct
+| 335
+|-
+| 68
+| 5591.172
+| [[25/1]]
+| ^m6 +4 oct
+| 340
+|-
+| 69
+| 5673.396
+| <small><small>[[26/1]]</small></small>, <small><small>[[27/1]]</small></small>
+| vvvM6 +4 oct
+| 345
+|-
+| 70
+| 5755.619
+| '''[[28/1]]'''
+| ^^M6 +4 oct, vvm7 +4 oct
+| 350
+|-
+| 71
+| 5837.842
+| '''[[29/1]]'''
+| ^^^m7 +4 oct
+| 355
+|-
+| 72
+| 5920.065
+| <small><small>[[30/1]]</small></small>, <small>[[31/1]]</small>
+| vM7 +4 oct
+| 360
+|-
+| 73
+| 6002.288
+| '''<u>[[32/1]]'''</u>
+| P1 +5 oct
+| 365
+|}
 == Approximation to JI ==

45zpi: Difference between revisions

Revision as of 22:54, 11 August 2024

Theory

Harmonic series

Intervals

Approximation to JI

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