61edt: Difference between revisions

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{{ED intro}}
{{ED intro}}


61edt provides a good tuning of [[mintaka]] temperament in the 3.7.11 subgroup, and contains an intersection of it with [[Bohlen-Pierce-Stearns]], despite the 5th harmonic being rather far from accurate. Notably, the octave is almost halfway in between steps, and therefore this system minimizes the prospect of a shimmering octave appearing, although it has a good 4th harmonic.
61edt provides a good tuning of [[mintaka]] temperament in the 3.7.11 subgroup, and contains an intersection of it with [[Bohlen–Pierce–Stearns]], despite lacking an accurate approximation to the 5th harmonic. Notably, the octave is almost halfway in between steps, and therefore this system minimizes the prospect of a shimmering octave appearing, although it has a good fourth harmonic.


== Intervals ==
== Intervals ==
Line 8: Line 8:


== Harmonics ==
== Harmonics ==
{{Harmonics in equal
{{Harmonics in equal|61|3|1|intervals = prime|columns = 9}}
| steps = 61
{{Harmonics in equal|61|3|1|start = 12|collapsed = 1|intervals = odd}}
| num = 3
 
| denom = 1
== Music ==
| intervals = integer
; [[Francium]]
}}
* "Ideal Spiritual Enrichment Cat Toys" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/4yGmTekIUn1Ealke6PDJwG Spotify] | [https://francium223.bandcamp.com/track/ideal-spiritual-enrichment-cat-toys Bandcamp] | [https://www.youtube.com/watch?v=0MyIKlXRVS0 YouTube]
{{Harmonics in equal
| steps = 61
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}


{{Stub}}
{{Stub}}

Latest revision as of 11:05, 10 June 2025

← 60edt 61edt 62edt →
Prime factorization 61 (prime)
Step size 31.1796 ¢ 
Octave 38\61edt (1184.82 ¢)
Consistency limit 3
Distinct consistency limit 3

61 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 61edt or 61ed3), is a nonoctave tuning system that divides the interval of 3/1 into 61 equal parts of about 31.2 ¢ each. Each step represents a frequency ratio of 31/61, or the 61st root of 3.

61edt provides a good tuning of mintaka temperament in the 3.7.11 subgroup, and contains an intersection of it with Bohlen–Pierce–Stearns, despite lacking an accurate approximation to the 5th harmonic. Notably, the octave is almost halfway in between steps, and therefore this system minimizes the prospect of a shimmering octave appearing, although it has a good fourth harmonic.

Intervals

Steps Cents Hekts Approximate ratios
0 0 0 1/1
1 31.2 21.3
2 62.4 42.6
3 93.5 63.9 19/18
4 124.7 85.2 14/13, 29/27
5 155.9 106.6 23/21
6 187.1 127.9
7 218.3 149.2 17/15, 25/22
8 249.4 170.5 15/13, 22/19
9 280.6 191.8 27/23
10 311.8 213.1 6/5
11 343 234.4
12 374.2 255.7
13 405.3 277 19/15, 29/23
14 436.5 298.4 9/7
15 467.7 319.7 17/13
16 498.9 341
17 530.1 362.3 19/14, 34/25
18 561.2 383.6 18/13, 29/21
19 592.4 404.9
20 623.6 426.2 33/23
21 654.8 447.5 19/13
22 686 468.9
23 717.1 490.2
24 748.3 511.5
25 779.5 532.8 11/7
26 810.7 554.1
27 841.8 575.4
28 873 596.7
29 904.2 618
30 935.4 639.3
31 966.6 660.7
32 997.7 682
33 1028.9 703.3
34 1060.1 724.6 35/19
35 1091.3 745.9
36 1122.5 767.2 21/11
37 1153.6 788.5 35/18
38 1184.8 809.8
39 1216 831.1
40 1247.2 852.5 35/17
41 1278.4 873.8 23/11
42 1309.5 895.1
43 1340.7 916.4 13/6
44 1371.9 937.7
45 1403.1 959
46 1434.3 980.3
47 1465.4 1001.6 7/3
48 1496.6 1023
49 1527.8 1044.3
50 1559 1065.6
51 1590.2 1086.9 5/2
52 1621.3 1108.2 23/9
53 1652.5 1129.5 13/5
54 1683.7 1150.8
55 1714.9 1172.1 35/13
56 1746.1 1193.4
57 1777.2 1214.8
58 1808.4 1236.1
59 1839.6 1257.4
60 1870.8 1278.7
61 1902 1300 3/1

Harmonics

Approximation of prime harmonics in 61edt
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) -15.2 +0.0 -11.3 -1.4 -4.4 -13.0 -9.8 -15.2 -3.0
Relative (%) -48.7 +0.0 -36.3 -4.6 -14.2 -41.8 -31.3 -48.9 -9.7
Steps
(reduced) 		38
(38) 		61
(0) 		89
(28) 		108
(47) 		133
(11) 		142
(20) 		157
(35) 		163
(41) 		174
(52)
Approximation of odd harmonics in 61edt
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) +8.5 +0.0 +1.0 +10.3 -4.4 -12.8 -15.4 -13.0 -6.1 +5.0 -11.3
Relative (%) +27.3 +0.0 +3.2 +32.9 -14.2 -40.9 -49.5 -41.8 -19.5 +16.1 -36.3
Steps
(reduced) 		179
(57) 		183
(0) 		187
(4) 		191
(8) 		194
(11) 		197
(14) 		200
(17) 		203
(20) 		206
(23) 		209
(26) 		211
(28)

Music

Francium
  • "Ideal Spiritual Enrichment Cat Toys" from Microtonal Six-Dimensional Cats (2025) – Spotify | Bandcamp | YouTube
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