25edo

Theory

25EDO divides the octave in 25 equal steps of exact size 48 cents each. It is a good way to tune the Blackwood temperament, which takes the very sharp fifths of 5EDO as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (5/4) and 7 (7/4). It also tunes sixix temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not consistent. It therefore makes sense to use it as a 2.5.7 subgroup tuning. Looking just at 2, 5, and 7, it equates five 8/7s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a 128/125 diesis and two septimal tritones of 7/5 with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is 50EDO. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the 2*25 subgroup 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

Intervals

Degrees	Cents	Approximate Ratios*	Armodue Notation	ups and downs notation
0	0	1/1	1	P1	perfect 1sn	D, Eb
1	48	33/32, 39/38, 34/33	1#	^1, ^m2	up 1sn, upminor 2nd	^D, ^Eb
2	96	17/16, 20/19, 18/17	2b	v~2	downmid 2nd	^^Eb
3	144	12/11, 38/35	2	^~2	upmid 2nd	vvE
4	192	9/8, 10/9, 19/17	2#	vM2	downmajor 2nd	vE
5·	240	8/7	3b	M2, m3	major 2nd, minor 3rd	E, F
6	288	19/16, 20/17	3	^m3	upminor 3rd	^F
7	336	39/32, 17/14, 40/33	3#	v~3	downmid 3rd	^^F
8·	384	5/4	4b	^~3	upmid 3rd	vvF#
9	432	9/7, 32/25, 50/39	4	vM3	downmajor	vF#
10	480	33/25, 25/19	4#/5b	M3, P4	major 3rd, perfect 4th	F#, G
11·	528	31/21, 34/25	5	^4	up 4th	^G
12	576	7/5, 39/28	5#	v~4,v~5	downmid 4th, downmid 5th	^^G, ^^Ab
13	624	10/7, 56/39	6b	^~4,^~5	upmid 4th, upmid 5th	vvG#, vvA
14·	672	42/31, 25/17	6	v5	down 5th	vA
15	720	50/33, 38/25	6#	P5, m6	perfect 5th, minor 6th	A, Bb
16	768	14/9, 25/16, 39/25	7b	^m6	upminor 6th	^Bb
17·	816	8/5	7	v~6	downmid 6th	^^Bb
18	864	64/39, 28/17, 33/20	7#	^~6	upmid 6th	vvB
19	912	32/19, 17/10	8b	vM6	downmajor 6th	vB
20·	960	7/4	8	M6, m7	major 6th, minor 7th	B, C
21	1008	16/9, 9/5, 34/19	8#	^m7	upminor 7th	^C
22	1056	11/6, 35/19	9b	v~7	downmid 7th	^^C
23	1104	32/17, 17/9, 19/10	9	^~7	upmid 7th	vvC#
24	1152	33/17, 64/33, 76/39	9#/1b	vM7	downmajor 7th	vC#
25	1200	2/1	1	P8	perfect 8ve	C#, D

• based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.

25-edo chords can be named with ups and downs, see Ups and Downs Notation - Chords and Chord Progressions.

25ed2-001.svg

Relationship to Armodue

Like 16-EDO and 23-EDO, 25-EDO contains the 9-note "Superdiatonic" scale of 7L2s (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9-EDO or 666.67 cents, to 4\7-EDO or 685.71 cents). The 25-EDO generator for this scale is the 672-cent interval. This allows 25-EDO to be used with the Armodue notation system in much the same way that 19-EDO is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25-EDO Armodue 6th is flatter than that of 16-EDO (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.

Commas

25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.)

Ratio	Monzo	Cents	Color Name	Name 1	Name 2	Name 3
256/243	| 8 -5 >	90.22	Sawa	Limma	Pythagorean Minor 2nd
3125/3072	| -10 -1 5 >	29.61	Laquinyo	Small Diesis	Magic Comma
	| 38 -2 -15 >	1.38	Sasa-quintrigu	Hemithirds Comma
16807/16384	| -14 0 0 5 >	44.13	Laquinzo
49/48	| -4 -1 0 2 >	35.70	Zozo	Slendro Diesis
64/63	| 6 -2 0 -1 >	27.26	Ru	Septimal Comma	Archytas' Comma	Leipziger Komma
3125/3087	| 0 -2 5 -3 >	21.18	Triru-aquinyo	Gariboh
50421/50000	| -4 1 -5 5 >	14.52	Quinzogu	Trimyna
1029/1024	| -10 1 0 3 >	8.43	Latrizo	Gamelisma
3136/3125	| 6 0 -5 2 >	6.08	Zozoquingu	Hemimean
65625/65536	| -16 1 5 1 >	2.35	Lazoquinyo	Horwell
100/99	| 2 -2 2 0 -1 >	17.40	Luyoyo	Ptolemisma
176/175	| 4 0 -2 -1 1 >	9.86	Lorugugu	Valinorsma
91/90	| -1 -2 -1 1 0 1 >	19.13	Thozogu	Superleap
676/675	| 2 -3 -2 0 0 2 >	2.56	Bithogu	Parizeksma

A 25edo keyboard

Music

Study in Fives by Paul Rapoport

Fantasy for Piano in 25 Note per Octave Tuning play by Chris Vaisvil

Flat fourth blues by Fabrizio Fulvio Fausto Fiale

File:25edochorale.mid 25edochorale.mid Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")

File:25 edo prelude largo.mid 25 edo prelude largo.mid Peter Kosmorsky (2011, Blackwood)