User:Moremajorthanmajor/Ed9/4: Difference between revisions
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Every other step of [[5edf]]. | Every other step of [[5edf]]. | ||
{{Harmonics in equal|5|9|4|intervals=integer}} | {{Harmonics in equal|5|9|4|intervals=integer}} | ||
Revision as of 02:04, 22 May 2025
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The equal division of 9/4 (ed9/4) is a tuning obtained by dividing the Pythagorean ninth (9/4) in a certain number of equal steps.
Properties
Relation to edfs
An ed9/4 can be generated by taking every other tone of an edf, so even-numbered ed9/4's are integer edfs.
This is the primary use for ed9/4s — to get the same benefits of a particular edf, without having to juggle such a large number of notes per period. This is a similar principle to using an ed4 in place of a very large edo.
Perhaps a composer wanting to explore Nedf but daunted by the number of notes, could instead simply use Ned9/4. Otherwise, they could also compose for two instruments, both tuned to Ned9/4, but each tuned one step of Nedf apart, making the piece overall in Nedf, but each individual instrument Ned9/4. This is a similar strategy to how some composers have approached 24edo — using two 12edo instruments tuned a 24edo-step apart.
Relation to common practice
9/4 or another major ninth is a standard replacement for the root in jazz piano voicings. Perhaps, then, a composer could approach the period of an ed9/4 not as an equivalence, but as a skeleton for chords to be built out of — potentially encouraging an approach that focuses more on individual chords than an overall scale.
Equivalence
Few would argue that 9/4 itself could be heard as an equivalence. Some might argue that some degree of 3/2 equivalence may be possible in a scale which has no 2/1, 3/1, or 4/1, though this is quite controversial. If that is the case, then perhaps in a similar scale that also has no 3/2, 9/4 may have some form of faint equivalence as it might sound like two periods of 3/2.
Important ed9/4s
5ed9/4
Completely misses 2/1 and 3/1, but aproximates 5/1 maybe passably, and 7/1 extremely well.
This makes it still posess some useful consonances, but with neither an octave nor tritave to pull the listener's ear — making it a prime candidate for perceptual 9/4-equivalence, if such a thing is even possible.
Every other step of 5edf.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -77 | +64 | +127 | +22 | -13 | +1 | +50 | +127 | -55 | +60 | -90 |
| Relative (%) | -27.4 | +22.6 | +45.2 | +7.7 | -4.8 | +0.2 | +17.9 | +45.2 | -19.7 | +21.5 | -32.1 | |
| Steps (reduced) |
4 (4) |
7 (2) |
9 (4) |
10 (0) |
11 (1) |
12 (2) |
13 (3) |
14 (4) |
14 (4) |
15 (0) |
15 (0) | |
9ed9/4
Every other step of 9edf (almost exactly every other step of Carlos Alpha).
11ed9/4
Every other step of 11edf (almost exactly every other step of Carlos Beta).
20ed9/4
Every other step of 20edf (almost exactly every other step of Carlos Gamma).
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -51.3 | +12.5 | +24.9 | +21.5 | -38.9 | -50.5 | -26.4 | +24.9 | -29.8 | +60.4 | +37.4 |
| Relative (%) | -40.2 | +9.8 | +19.5 | +16.9 | -30.5 | -39.6 | -20.7 | +19.5 | -23.4 | +47.3 | +29.3 | |
| Steps (reduced) |
9 (9) |
15 (4) |
19 (8) |
22 (0) |
24 (2) |
26 (4) |
28 (6) |
30 (8) |
31 (9) |
33 (0) |
34 (1) | |
29ed9/4
A compromise between 39edt (triple Bohlen-Pierce) and 25edo, combining the benefits and drawbacks of both systems while being audibly different from either.
Every other step of 29edf.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | -13.9 | +21.5 | +19.9 | +12.0 | +13.3 | -15.5 | -14.4 | -6.3 | -20.3 | +9.5 |
| Relative (%) | +21.2 | -28.8 | +44.4 | +41.2 | +24.8 | +27.4 | -32.0 | -29.7 | -13.0 | -41.9 | +19.6 | |
| Steps (reduced) |
25 (25) |
39 (10) |
58 (0) |
70 (12) |
86 (28) |
92 (5) |
101 (14) |
105 (18) |
112 (25) |
120 (4) |
123 (7) | |
43ed9/4
A compromise between 58edt and 37edo, combining the benefits and drawbacks of both systems while being audibly different from either.
Every other step of 43edf.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.0 | -8.3 | -11.1 | -6.0 | -4.9 | -0.3 | -7.6 | -4.3 | -8.5 | +14.6 | -2.9 |
| Relative (%) | +24.6 | -25.4 | -34.1 | -18.3 | -15.0 | -0.8 | -23.3 | -13.0 | -26.1 | +44.7 | -8.9 | |
| Steps (reduced) |
37 (37) |
58 (15) |
85 (42) |
103 (17) |
127 (41) |
136 (7) |
150 (21) |
156 (27) |
166 (37) |
179 (7) |
182 (10) | |
45ed9/4
Very similar to 61edt but improves on its approximations of JI.
Every other step of 45edf.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -14.5 | +1.1 | -9.7 | +0.6 | -2.0 | -10.4 | -6.9 | -12.2 | +0.2 | +4.4 | +13.8 |
| Relative (%) | -46.4 | +3.6 | -31.1 | +1.8 | -6.4 | -33.4 | -22.0 | -39.2 | +0.6 | +14.3 | +44.2 | |
| Steps (reduced) |
38 (38) |
61 (16) |
89 (44) |
108 (18) |
133 (43) |
142 (7) |
157 (22) |
163 (28) |
174 (39) |
187 (7) |
191 (11) | |
47ed9/4
A compromise between 64edt and 40edo, combining the benefits and drawbacks of both systems while being audibly different from either.
Every other step of 47edf.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.2 | +9.8 | -8.4 | +6.5 | +0.7 | +10.2 | -6.2 | +10.3 | +8.1 | -4.8 | -0.8 |
| Relative (%) | -17.4 | +32.6 | -28.0 | +21.9 | +2.2 | +34.0 | -20.8 | +34.6 | +27.3 | -16.2 | -2.7 | |
| Steps (reduced) |
40 (40) |
64 (17) |
93 (46) |
113 (19) |
139 (45) |
149 (8) |
164 (23) |
171 (30) |
182 (41) |
195 (7) |
199 (11) | |
57ed9/4
A compromise between 77edt and 49edo, combining the benefits and drawbacks of both systems while being audibly different from either.
Every other step of 57edf.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.9 | -5.4 | -3.1 | +5.5 | +11.2 | -7.1 | -3.6 | +0.9 | -9.7 | +7.7 | -9.2 |
| Relative (%) | +27.9 | -22.1 | -12.7 | +22.3 | +45.3 | -28.9 | -14.6 | +3.6 | -39.3 | +31.4 | -37.4 | |
| Steps (reduced) |
49 (49) |
77 (20) |
113 (56) |
137 (23) |
169 (55) |
180 (9) |
199 (28) |
207 (36) |
220 (49) |
237 (9) |
241 (13) | |
Individual pages for ed9/4's
| 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 |
| 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 |
| 41 | 43 | 45 | 47 | 49 | 51 | 53 | 55 | 57 | 59 |
| 61 | 63 | 65 | 67 | 69 | 71 | 73 | 75 | 77 | 79 |
| 81 | 83 | 85 | 87 | 89 | 91 | 93 | 95 | 97 | 99 |