1ed148.5c
| ← 7ed1869/941 | 8ed1869/941 | 9ed1869/941 → |
(semiconvergent)
1 equal division of 148.5¢ (1ed148.5c), also known as arithmetic pitch sequence of 148.5¢ (APS148.5¢), is an equal and nonoctave scale generated by making a continuous chain of intervals of exactly 148.5 cents.
It is closely related to 8edo. 1ed148.5c can be seen as a compressed-octave version of 8edo.
It can be treated as an equalized variant of the octatonic scale from mainstream 12edo music theory, a.k.a. a variant of the scale diminished[8].
Harmonics
Prime
Of all prime harmonics up to 31, pure-octave 8edo only manages to approximate 2/1 and 19/1 within 15 cents, completely missing all the others.
By contrast, 1ed148.5c approximates 2/1, 11/1, 13/1, 17/1 and 31/1 all within 15 cents.
In other words, 1ed148.5c upgrades 8edo into a strong low-complexity tuning for the 2.11.13.17.31 subgroup.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.0 | +28.5 | +35.2 | +46.7 | +6.7 | +14.5 | -4.5 | -48.5 | +66.2 | -38.1 | -5.0 |
| Relative (%) | -8.1 | +19.2 | +23.7 | +31.4 | +4.5 | +9.7 | -3.0 | -32.7 | +44.6 | -25.6 | -3.4 | |
| Steps | 8 | 13 | 19 | 23 | 28 | 30 | 33 | 34 | 37 | 39 | 40 | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +48.0 | +63.7 | -68.8 | +48.7 | +59.5 | +45.0 | +2.5 | -28.3 | +20.4 | +55.0 |
| Relative (%) | +0.0 | +32.0 | +42.5 | -45.9 | +32.5 | +39.6 | +30.0 | +1.7 | -18.8 | +13.6 | +36.6 | |
| Steps (reduced) |
8 (0) |
13 (5) |
19 (3) |
22 (6) |
28 (4) |
30 (6) |
33 (1) |
34 (2) |
36 (4) |
39 (7) |
40 (0) | |
Integer
Of all integer harmonics up to 27, pure-octave 8edo approximates the following within 20 cents:
2, 4, 8, 16, 19, 27.
Of all integer harmonics up to 27, 1ed148.5c approximates the following within 20 cents:
2, 6, 11, 12, 13, 17, 20, 22, 25, 26.
So while the composite subgroup of 8edo could be described as 2.19.27, the composite subgroup of 1ed148.5c could be described as 2.6.11.13.17.20.25. This provides 1ed148.5c with a comparatively larger, more diverse array of consonances.
Integer harmonics in 1ed148.5c
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.0 | +28.5 | -24.0 | +35.2 | +16.5 | +46.7 | -36.0 | +57.1 |
| Relative (%) | -8.1 | +19.2 | -16.2 | +23.7 | +11.1 | +31.4 | -24.2 | +38.4 | |
| Steps | 8 | 13 | 16 | 19 | 21 | 23 | 24 | 26 | |
| Harmonic | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +23.2 | +6.7 | +4.5 | +14.5 | +34.7 | +63.7 | -48.0 | -4.5 | +45.1 |
| Relative (%) | +15.6 | +4.5 | +3.1 | +9.7 | +23.3 | +42.9 | -32.3 | -3.0 | +30.4 | |
| Steps | 27 | 28 | 29 | 30 | 31 | 32 | 32 | 33 | 34 | |
| Harmonic | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -48.5 | +11.2 | -73.3 | -5.3 | +66.2 | -7.5 | +70.4 | +2.5 | -62.9 |
| Relative (%) | -32.7 | +7.5 | -49.3 | -3.6 | +44.6 | -5.0 | +47.4 | +1.7 | -42.3 | |
| Steps | 34 | 35 | 35 | 36 | 37 | 37 | 38 | 38 | 38 | |
Integer harmonics in 8edo
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +48.0 | +0.0 | +63.7 | +48.0 | -68.8 | +0.0 | -53.9 |
| Relative (%) | +0.0 | +32.0 | +0.0 | +42.5 | +32.0 | -45.9 | +0.0 | -35.9 | |
| Steps (reduced) |
8 (0) |
13 (5) |
16 (0) |
19 (3) |
21 (5) |
22 (6) |
24 (0) |
25 (1) | |
| Harmonic | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +63.7 | +48.7 | +48.0 | +59.5 | -68.8 | -38.3 | +0.0 | +45.0 | -53.9 |
| Relative (%) | +42.5 | +32.5 | +32.0 | +39.6 | -45.9 | -25.5 | +0.0 | +30.0 | -35.9 | |
| Steps (reduced) |
27 (3) |
28 (4) |
29 (5) |
30 (6) |
30 (6) |
31 (7) |
32 (0) |
33 (1) |
33 (1) | |
| Harmonic | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.5 | +63.7 | -20.8 | +48.7 | -28.3 | +48.0 | -22.6 | +59.5 | -5.9 |
| Relative (%) | +1.7 | +42.5 | -13.9 | +32.5 | -18.8 | +32.0 | -15.1 | +39.6 | -3.9 | |
| Steps (reduced) |
34 (2) |
35 (3) |
35 (3) |
36 (4) |
36 (4) |
37 (5) |
37 (5) |
38 (6) |
38 (6) | |
Intervals
| Step | Interval (¢) | JI approximated | Comments |
|---|---|---|---|
| 1 | 148.5 | 12/11 | |
| 2 | 297.0 | 13/11, 24/20 | Reduces to 6/5 |
| 3 | 445.5 | 22/17 | |
| 4 | 594.0 | 17/12, 24/17 | 17/6 in the next octave |
| 5 | 742.5 | 17/11, 20/13 | |
| 6 | 891.0 | 20/12 | Reduces to 5/3 |
| 7 | 1039.5 | 11/6, 20/11 | 11/3 in the next octave |
| 8 | 1188.0 | 2/1 |
Chords
Not an exhaustive list at all:
Scale degrees 0, 1, 2, 5, 7, 8 create the hexad 11:12:13:17:20:22. Any subset of this chord can be a consonance in its own right too.
Scale degrees 0, 4, 6, 7, 8 create the pentad 12:17:20:22:24. Any subset of this chord can be a consonance in its own right too.
Scale degrees 0, 3, 4, 8 create the tetrad 17:22:24:34. Any subset of this chord can be a consonance in its own right too.
Notation
1ed148.5c can be notated using most systems that work for 8edo. It can also be notated using most systems that work for the octatonic scale/diminished[8].