Ed12

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Ed12 means division of the twelfth harmonic (12/1) into n equal parts.

Properties

Improving primes

Ed12 is like ed6 in that it is a compromise between primes 2 and 3 — but ed12 leans more towards prime 2 than ed6 does. An ed12 sometimes gives you the right amount of stretch for equal temperaments whose 3 is more inaccurate than its higher primes.

Here for example, you can choose how much you wish to stretch 31edo depending on your harmonic style: 80ed6 vs 111ed12.

(See also Stretched and compressed tuning.)

Width of its period

The twelfth harmonic is particularly wide as far as periods go. There are (at absolute most) ~3.1 dodecataves within the human hearing range; imagine if that were the case with octaves.

This width means that the listener probably won't hear the period (12/1) as an equivalence, but instead will hear the pseudo-octave or pseudo-tritave or similar as one — this disconnect between period versus equivalence could be used by a composer to surprise their listener, in a similar way that 13edo can be used to make melodies that sound like 12edo, until they suddenly don't.

NTSC and PAL devices

The ed12-edo correspondences fall particularly close to the harmonic series of the NTSC or PAL-M color subcarrier. Perhaps a multimedia artist could find some way to put this to use.

Ed12-edo correspondences

edo	ed12	NTSC*n	PAL-M*n
1	3.5849625	3.579545 MHz	3.575611 MHz
2	7.169925	7.158909	7.151222
3	10.7548875	10.7383635	10.726833
4	14.33985	14.317818	14.302444
5	17.9248125	17.8972725	17.878055
6	21.509775	21.476727	21.453666
7	25.0947375	25.0561815	25.029277
8	28.6797	28.635636	28.604888
9	32.2646625	32.2150905	32.180299
10	35.849625	35.79545	35.75611
11	39.4345875	39.374	39.331521
12	43.01955	42.953454	42.907332
13	46.6045125	46.5329085	46.482743
14	50.189475	50.112363	50.058554
15	53.7744375	53.6918175	53.634265
16	57.3594	57.271272	57.209776
17	60.9443625	60.8507265	60.785487
18	64.529325	64.430181	64.360598
19	68.1142875	68.0096355	67.936709
20	71.69925	71.58909	71.51222
21	75.2842125	75.1685445	75.087931
22	78.869175	78.747999	78.663442
23	82.4541375	82.3274535	82.239153
24	86.0391	85.906908	85.814664
25	89.6240625	89.4863625	89.390375
26	93.209025	93.065817	92.965886
27	96.7939875	96.6452715	96.541597
28	100.37895	100.224726	100.117108
29	103.9639125	103.8041805	103.692819
30	107.548875	107.38365	107.28633
31	111.1338375	110.9630895	110.894041
32	114.7188	114.542544	114.437552
33	118.3037625	118.1219985	118.045263
34	121.888725	121.701453	121.588774
35	125.4736875	125.2809075	125.096485
36	129.05865	128.860362	128.739296
37	132.6436125	132.4398165	132.247707
38	136.228575	136.019271	135.860518
39	139.8135375	139.5987255	135.398929
40	143.3985	143.17818	143.02444
41	146.41815	146.7576345	146.600151
42	150.568425	150.337089	150.175862
43	154.0533875	153.9165435	153.751373
44	157.73835	157.495998	157.326884
45	161.3233125	161.0754525	160.902595
46	164.908275	164.654907	164.478306
47	168.4932375	168.2343615	168.053817
48	172.0782	171.813816	171.629328
49	175.6631625	175.3932705	175.205039
50	179.248125	178.972725	178.78075
51	182.8330875	182.5521795	182.356261
52	186.41805	186.131634	185.931772
53	190.003125	189.7110885	189.507483
54	193.597975	193.290543	193.083194
55	197.1729375	196.869975	196.658705
56	200.7579	200.449452	200.234216

Table of Ed12s

0…299

0…99

0	1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99