User:R-4981/Redbull
Redbull is a irregular temperament obtained by recursively dividing one octave on the logarithmic scale (1200 ¢) by 1:√3 into 16 tones. This is not a regular temperament, and it is impossible to approximate it with them, but on the other hand it has very systematic and unique properties that are completely different from them.
As we all know, 400¢ is the most commonly used approximation to 5/4. This interval is also expressed as 1\3, but its square root on the logarithmic axis, 692.82¢ (hereinafter expressed as 1\√3 for convenience), functions as an approximation of 3/2. Furthermore, the interval divided into √3 equal parts with 1\√3 as the center is 985.64¢, which works as an approximation of 7/4 or 9/5, and the tetrachord that combines these is 4:5:6:7, the so-called It will be the minor 7th. Applying this property, the temperament that is created as a result of recursively dividing those intervals futhermore twice is Redbull.
Most of the notes on this temperament are irrational numbers in both cent and frequency units, so redbull cannot be reproduced with EDO. Also, since there is no interval that can be called a generator, so redbull is also impossible to approximate it as a MOS scale. Therefore, as mentioned at the beginning, redbull is very special and irregular temperament. Also, since there is no interval that can be called a generator, and it varies even by one step, there are many intervals within Redbull that approximate Just intonation, just like AFDO.
Intervals
If you need exact cent values, please refer to the Scala file below.
| Degree | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 133.33 | 14/13, 13/12, 12/11 |
| 2 | 230.94 | 8/7 |
| 3 | 328.55 | 6/5 |
| 4 | 400 | 5/4, 24/19 |
| 5 | 497.6 | 4/3 |
| 6 | 569.06 | 25/18, 7/5 |
| 7 | 640.51 | 10/7, 13/9, 16/11 |
| 8 | 692.82 | 3/2 |
| 9 | 790.43 | 11/7 |
| 10 | 861.88 | 13/8, 18/11, 5/3 |
| 11 | 933.33 | 12/7 |
| 12 | 985.64 | 7/4, 9/5 |
| 13 | 1057.09 | 11/6, 24/13, 13/7 |
| 14 | 1109.40 | 15/8, 17/9, 28/15 |
| 15 | 1161.71 | 33/17, 64/33 |
| 16 | 1200 | 2/1 |
Propertys and Trivia
- As mentioned above, the 4-step chord in Redbull is similar to the seventh in 12EDO, but since 4 is a divisor of 16, there are only 4 types of these chords, and the others are only inversions.
- in Redbull, a chord obtained by estimating three steps five times can be approximated in only one way by Just intonation: 5:6:7:8:9.
- Furthermore, Redbull has a pentatonic, which is similar to 2L 3s, and the constituent notes of that scale can be approximated as 9:12:13:16:17 in pure ratio.
- Redbull is the (probably) first scale to have an article on the Xenharmonic Wiki, even though it is neither Regular nor Historical temperaments.
- The name "Redbull" comes from a famous energy drink from Australia. This drink is popular all over the world due to its fruity taste and reasonable price.
Scala file
! redbull.scl 16-tone logarithmic fractal scale with 1:√3 16 ! 133.33333333 230.94010768 328.54688202 400 497.60677434 569.05989232 640.51301031 692.82032303 790.42709737 861.88021535 933.33333333 985.64064606 1057.09376404 1109.40107676 1161.70838948 1200