25ed4

25ed4 is the equal division of the double octave into 25 parts of exactly 96 cents each (every second step of 25edo). It corresponds to 12.5edo and is notable as a type of compressed 12edo.

← 23ed4

25ed4

27ed4 →

Prime factorization	52
Step size	96 ¢
Octave	13\25ed4 (1248 ¢)
Twelfth	20\25ed4 (1920 ¢) (→ 4\5ed4)
Consistency limit	1
Distinct consistency limit	1

Theory

On the surface, 25ed4 seems fairly similar to 12edo. Its step is 96¢, its perfect 5th is actually quite flat at 672¢ (but still 7 steps), but it has an excellent 5/4 at 4 steps (384¢). However, try to map this to a 12edo keyboard and you will run into multiple issues. First, the octave is WAY out. 12\25ed4 is 1152¢. Second, the best fifth of 25edo is only found up an octave. Third, 24≠25, so double octaves don't work unless you use some isomorphic keyboard.

Here are some important ratios and their relative errors in 25ed4.

3/1: 18.8%

3/2: -31.2%

4/3: -18.8%

5/4: -2.4%

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	96
2	192	10/9, 19/17
3	288	13/11
4	384
5	480	25/19
6	576	7/5
7	672	22/15, 25/17
8	768	14/9
9	864	23/14
10	960	26/15
11	1056
12	1152
13	1248
14	1344
15	1440	23/10
16	1536	17/7, 22/9
17	1632	23/9
18	1728	19/7
19	1824
20	1920
21	2016
22	2112	17/5
23	2208	25/7
24	2304	19/5
25	2400

Harmonics

Approximation of harmonics in 25ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+48.0	+18.0	+0.0	-2.3	-30.0	-8.8	+48.0	+36.1	+45.7	-23.3	+18.0
	Relative (%)	+50.0	+18.8	+0.0	-2.4	-31.2	-9.2	+50.0	+37.6	+47.6	-24.3	+18.8
Steps (reduced)		13 (13)	20 (20)	25 (0)	29 (4)	32 (7)	35 (10)	38 (13)	40 (15)	42 (17)	43 (18)	45 (20)

Approximation of harmonics in 25ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-24.5	+39.2	+15.7	+0.0	-9.0	-11.9	-9.5	-2.3	+9.2	+24.7	+43.7
	Relative (%)	-25.5	+40.8	+16.4	+0.0	-9.3	-12.4	-9.9	-2.4	+9.6	+25.7	+45.5
Steps (reduced)		46 (21)	48 (23)	49 (24)	50 (0)	51 (1)	52 (2)	53 (3)	54 (4)	55 (5)	56 (6)	57 (7)

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