Ed9/4
An 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 ed9/4's – 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 N-edf but daunted by the number of notes, could instead simply use N-ed9/4. Otherwise, they could also compose for two instruments, both tuned to N-ed9/4, but each tuned one step of N-edf apart, making the piece overall in N-edf, but each individual instrument N-ed9/4. 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 that claim is controversial. If that is the case though, then perhaps in a similar situation, 9/4 may have some form of faint equivalence as it might sound like two periods of 3/2. This is usually not really the point of using ed9/4 though as discussed above.
Important ed9/4's
5ed9/4
Every other step of 5edf. Completely misses 3/2, 2/1, 3/1 and 4/1, but aproximates 5/1, 6/1 and 7/1 well for its size. This makes it still posess useful consonances, but with no strong equivalence interval to pull the listener's ear – making it a prime candidate for perceptual 9/4-equivalence, if such a thing is even possible.
Its intervals are:
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -77 | +64 | +22 | +1 | +60 | +52 | -132 | -43 | -93 | +67 | -49 |
| Relative (%) | -27.4 | +22.6 | +7.7 | +0.2 | +21.5 | +18.5 | -46.9 | -15.5 | -33.3 | +23.8 | -17.3 | |
| Steps (reduced) |
4 (4) |
7 (2) |
10 (0) |
12 (2) |
15 (0) |
16 (1) |
17 (2) |
18 (3) |
19 (4) |
21 (1) |
21 (1) | |
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).
29ed9/4
Every other step of 29edf. A compromise between 39edt (triple Bohlen-Pierce) and 25edo, combining the benefits and drawbacks of both systems while being audibly different from either.
| 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
Every other step of 43edf. The smallest ed9/4 with a truly great approximation of full 11-limit JI (let alone all the way up to the full 23-limit).
A compromise between 58edt and 37edo, combining the benefits and drawbacks of both systems while being audibly different from either.
| 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
Every other step of 45edf. Very similar to 61edt but improves on its approximations of JI, with slightly better approximations of primes 2, 5, 7, 11, 13, 17, 19 and 23 compared to 61edt.
Compared to the nearest edo (38edo), it has a much worse prime 2 or course, but it has dramatically better primes 3, 7 and 11, and slightly better 13, 17, 19, 23 and even 29 compared to 38edo. It does however have a worse 5/1, but only slightly.
| 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
Every other step of 47edf. A compromise between 64edt and 40edo, combining the benefits and drawbacks of both systems while being audibly different from either.
| 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
Every other step of 57edf. A compromise between 77edt and 49edo, combining the benefits and drawbacks of both systems while being audibly different from either.
| 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 |
See also
|
Todo: explain edonoi
Most people do not think 9/4 sounds like an equivalence, so there must be some other reason why people are dividing it — some property other than equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is. |