25ed4: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''25ed4''' is the [[Ed4|equal division of the double octave]] into 25 parts of exactly 96 [[cent|cents]] each (every second step of [[25edo]]). | '''25ed4''' is the [[Ed4|equal division of the double octave]] into 25 parts of exactly 96 [[cent|cents]] each (every second step of [[25edo]]). It corresponds to 12.5edo and is notable as a type of compressed [[12edo]]. | ||
== Theory == | == 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 == | |||
{{Interval table}} | |||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 25 | |||
| num = 4 | |||
| denom = 1 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 25 | |||
| num = 4 | |||
| denom = 1 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
{{stub}} | |||
Latest revision as of 16:22, 26 May 2026
| ← 23ed4 | 25ed4 | 27ed4 → |
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.
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
| 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) | |
| 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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