Ed5/2

The equal division of 5/2 (ed5/2) is a tuning obtained by dividing the classic major tenth (5/2) in a certain number of equal steps.

Properties

Division of 5/2 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed5/2 scales have a perceptually important false octave, with various degrees of accuracy.

The structural utility of 5/2 (or another tenth) is apparent by its being the base of so much common practice tonal harmony^{[clarification needed]}, and by 5/2 being the best option for “no-threes” harmony excluding the octave^{[clarification needed]}.

One way to approach ed5/2 tunings is the use of the 2:3:4:(5) chord as the fundamental complete sonority in a very similar way to the 3:4:5:(6) chord in meantone. Whereas in meantone it takes three 4/3 to get to 6/5, here it takes three 3/2 to get to 6/5 (tempering out the comma 3125/3048). So, doing this yields 5-, 7-, and 12-note mos, just like meantone. While the notes are rather closer together, the scheme shares the scale shape of meantone.

Joseph Ruhf proposes the term "Macrodiatonic"^{[idiosyncratic term]} for the above approach because it uses a scheme that turns out exactly identical to meantone, though severely stretched. These are also the MOS scales formerly known as Middletown^{[idiosyncratic term]} because a tenth base stretches the meantone scheme to the point where it tempers out 64/63.

Another option is to treat ed5/2's as "no-threes" systems (like how edts are usually treated as no-twos), using the 4:5:7:(10) chord as the fundamental complete sonority instead of 4:5:6:(8). Whereas in meantone it takes four 4/3 to get to 6/5, here it takes one 10/7 to get to 7/5 (tempering out the comma 50/49 in the no-threes 7-limit), producing a nonoctave version of jubilic temperament. Doing this yields 5-, 8-, 13-, and 21-note mos.

Individual pages for ed5/2's

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

Todo: cleanup , explain edonoi Most people do not think 5/2 sounds like an equivalence, so there must be some other reason why people are dividing it — some property other than equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.