User:BudjarnLambeth/79ed5

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← 78ed5

79ed5

80ed5 →

Prime factorization	79 (prime)
Step size	35.2698 ¢
Octave	34\79ed5 (1199.17 ¢)
Twelfth	54\79ed5 (1904.57 ¢)
Consistency limit	6
Distinct consistency limit	6

79 equal divisions of the 5th harmonic (abbreviated 79ed5) is a nonoctave tuning system that divides the interval of 5/1 into 79 equal parts of about 35.3 ¢ each. Each step represents a frequency ratio of 5^1/79, or the 79th root of 5.

I am documenting this tuning for sentimental reasons. I shared a page I was working on that had a table of a bunch of ETs, with a queerplatonic online partner, who isn't involved in microtonality, and they spontaneously without prompting declared 79ed5 their favourite tuning, so now it’s special to me.

Practically speaking, 79ed5 is a slightly octave-compressed 34edo.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	35.3
2	70.5	24/23, 25/24, 26/25
3	105.8	17/16, 33/31
4	141.1	13/12, 38/35
5	176.3	21/19, 31/28
6	211.6	26/23, 35/31
7	246.9	15/13, 38/33
8	282.2	20/17, 33/28
9	317.4	6/5
10	352.7	27/22, 38/31
11	388	5/4
12	423.2	23/18, 37/29
13	458.5	30/23
14	493.8
15	529	19/14
16	564.3	18/13
17	599.6	24/17
18	634.9	13/9
19	670.1	25/17, 28/19
20	705.4
21	740.7	23/15
22	775.9	36/23
23	811.2	8/5
24	846.5	31/19
25	881.7	5/3
26	917	17/10, 39/23
27	952.3	26/15
28	987.6	23/13
29	1022.8
30	1058.1	35/19
31	1093.4	32/17
32	1128.6	23/12
33	1163.9
34	1199.2	2/1
35	1234.4
36	1269.7	25/12
37	1305	17/8
38	1340.3	13/6
39	1375.5	31/14
40	1410.8
41	1446.1	30/13
42	1481.3	33/14
43	1516.6	12/5
44	1551.9	27/11
45	1587.1	5/2
46	1622.4	23/9
47	1657.7
48	1693
49	1728.2	19/7
50	1763.5	36/13
51	1798.8
52	1834	26/9
53	1869.3
54	1904.6	3/1
55	1939.8
56	1975.1	25/8
57	2010.4
58	2045.6
59	2080.9
60	2116.2	17/5
61	2151.5
62	2186.7
63	2222
64	2257.3
65	2292.5
66	2327.8	23/6
67	2363.1
68	2398.3	4/1
69	2433.6
70	2468.9	25/6
71	2504.2	17/4
72	2539.4	13/3
73	2574.7	31/7
74	2610
75	2645.2
76	2680.5
77	2715.8	24/5
78	2751
79	2786.3	5/1

Harmonics

Compared to 34edo, 79ed5 improves upon harmonics 3, 5, 11 and 13 at the expense of 2 and 7. Though if one treats it as a dual-7 tuning (which is very reasonable because of how close 34’s relative error on 7 is to 50%), then it actually improves the dual 7s by making them more balanced.

Approximation of prime harmonics in 79ed5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-0.8	+2.6	+0.0	+17.1	+10.5	+3.5	-2.5	+16.6	+3.3	-10.1	+15.6
	Relative (%)	-2.3	+7.4	+0.0	+48.4	+29.8	+9.8	-7.0	+47.1	+9.3	-28.5	+44.1
Steps (reduced)		34 (34)	54 (54)	79 (0)	96 (17)	118 (39)	126 (47)	139 (60)	145 (66)	154 (75)	165 (7)	169 (11)

34edo for comparison

Approximation of prime harmonics in 34edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+3.9	+1.9	-15.9	+13.4	+6.5	+0.9	-15.2	+7.0	-6.0	-15.6
	Relative (%)	+0.0	+11.1	+5.4	-45.0	+37.9	+18.5	+2.6	-43.0	+19.9	-17.1	-44.3
Steps (reduced)		34 (0)	54 (20)	79 (11)	95 (27)	118 (16)	126 (24)	139 (3)	144 (8)	154 (18)	165 (29)	168 (32)

Scales

See 34edo#Scales.