'''Redbull''' is a "irregular" [[temperament]] obtained by recursively dividing one octave on the logarithmic scale (1200 [[cent|¢]]) 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.
[[File:Redbull Cromatic Scale.mp3|thumb|A chromatic Redbull scale on C.]]
[[File:Redbull Scale's Theory.png|thumb|An illustration of the structure of the Redbull scale.]]
[[File:Redbull Scale's Theory.png|thumb|An easy-to-understand (not accurate) illustration of the theory of the Redbull Scale.]]
The '''Redbull scale'''{{idiosyncratic}} is a 16-tone logarithmic [[fractal scale]] obtained by recursively dividing one [[octave]] on the logarithmic scale (1200{{cent}}) with a 1:√3 ratio.
As we all know, 400¢ is the most commonly used approximation to [[5/4]]. This interval is also expressed as [[3edo|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.
== Theory ==
400{{cent}} is the most commonly used approximation to [[5/4]], mainly due to its use in [[12edo]]. This interval is also expressed as [[3edo|1\3]], and its square root on the logarithmic scale, ≈692.82{{cent}} (hereinafter expressed as 1\√3 for convenience), functions as an approximation of [[3/2]]. (The comma between these two intervals is called as [[Caffeinterval]].) Furthermore, the interval divided into √3 equal parts with 1\√3 as the center{{clarify}} is ≈985.641¢, which works as an approximation of [[7/4]] or [[9/5]], and the [[tetrad]] that combines these is 4:5:6:7, the so-called It will be the C7. Applying this property, the scale that is created as a result of recursively dividing those intervals furthermore 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]].
Most of the notes on this scale are irrational numbers in both cent and frequency units, so Redbull cannot be reproduced with an [[edo]]. Also, since there is no interval that can be called a generator, it is also impossible to approximate Redbull with a [[mos scale]]. Also, since there is no interval that can be called a generator, and it varies even by one step{{clarify}}, there are many intervals within Redbull that approximate [[just intonation]], just like [[afdo]]s.
== Intervals ==
== Intervals ==
For more precise cent values, refer to the [[#Scala file|Scala file]] below.
If you need exact cent values, please refer to the Scala file below.
{| class="wikitable center-all"
{| class="wikitable center-all"
Line 15:
Line 16:
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
Line 22:
Line 23:
|-
|-
| 1
| 1
| 133.33
| 133.333
| 14/13, 13/12, 12/11
| 14/13, 13/12, 12/11
|-
|-
| 2
| 2
| 230.94
| 230.940
| 8/7
| 8/7, 9/8
|-
|-
| 3
| 3
| 328.55
| 328.547
| 6/5
| 6/5
|-
|-
Line 38:
Line 39:
|-
|-
| 5
| 5
| 497.6
| 497.607
| 4/3
| 4/3
|-
|-
| 6
| 6
| 569.06
| 569.060
| 25/18, 7/5
| 25/18, 7/5, 11/8
|-
|-
| 7
| 7
| 640.51
| 640.513
| 10/7, 13/9, 16/11
| 10/7, 13/9, 16/11
|-
|-
| 8
| 8
| 692.82
| 692.820
| 3/2
| 3/2
|-
|-
| 9
| 9
| 790.43
| 790.427
| 11/7
| 11/7
|-
|-
| 10
| 10
| 861.88
| 861.880
| 13/8, 18/11, 5/3
| 13/8, 18/11, 5/3
|-
|-
| 11
| 11
| 933.33
| 933.333
| 12/7
| 12/7
|-
|-
| 12
| 12
| 985.64
| 985.641
| 7/4, 9/5
| 7/4, 9/5
|-
|-
| 13
| 13
| 1057.09
| 1057.094
| 11/6, 24/13, 13/7
| 11/6, 24/13, 13/7
|-
|-
| 14
| 14
| 1109.40
| 1109.401
| 15/8, 17/9, 28/15
| 15/8, 17/9, 28/15
|-
|-
| 15
| 15
| 1161.71
| 1161.708
| 33/17, 64/33
| 33/17, 64/33
|-
|-
Line 86:
Line 87:
|}
|}
== Propertys and Trivia ==
== Properties and trivia ==
[[File:Redbull Pentic.mp3|thumb|A pentatonic Redbull scale on C.]]
[[File:Redbull Do-Re-Mi.mp3|thumb|A "Do-Re-Mi" song in wholetone Redbull scale.]]
* 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.
* As mentioned above, the [[tetrad]] obtained by stacking 4-steps from the tonic (i.e. starting on degree 0) is similar to the seventh tetrad in [[12edo]], approximating 4:5:6:7. Since 4 is a divisor of 16, there are only 4 types of 4-step tetrads, the others being only inversions, and the other types of 4-step tetrads do not approximate 4:5:6:7. (in example, chords stacked 4 steps from 2 steps above the tonic is approximating 9:11:13:17.)
* 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.
* The [[pentad]] obtained by stacking 3-steps from the tonic approximates 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.
* Furthermore, Redbull has a pentatonic subset which is similar to [[2L 3s]]{{clarify}}, and the constituent notes of that scale can be approximated as 9:12:13:16:17 in just intonation.
* 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'', proposed by [[User:R-4981|R-4981]], comes from the {{w|Red Bull|energy drink brand from Austria}}.
* The name "Redbull" comes from a [https://en.wikipedia.org/wiki/Red_Bull famous energy drink from Australia]. This drink is popular all over the world due to its fruity taste and reasonable price.
== Scala file ==
== Scala file ==
<pre>
<pre>
! redbull.scl
! redbull.scl
!
16-tone logarithmic fractal scale with 1:√3
16-tone logarithmic fractal scale with 1:√3
16
16
Line 104:
Line 106:
230.94010768
230.94010768
328.54688202
328.54688202
400
400.
497.60677434
497.60677434
569.05989232
569.05989232
Line 116:
Line 118:
1109.40107676
1109.40107676
1161.70838948
1161.70838948
1200
1200.
</pre>
</pre>
[[Category:Temperaments]]
== See also ==
* [[Caffeinterval]]
* [[Fractal scale]]
* [[Pepsi]]
[[Category:Tempered scales]]
[[Category:Tempered scales]]
[[Category:16-tone scales]]
[[Category:16-tone scales]]
[[Category:Pages with Scala files]]
[[Category:Pages with Scala files]]
Latest revision as of 23:13, 31 March 2025
A chromatic Redbull scale on C.An illustration of the structure of the Redbull scale.
The Redbull scale[idiosyncratic term] is a 16-tone logarithmic fractal scale obtained by recursively dividing one octave on the logarithmic scale (1200 ¢) with a 1:√3 ratio.
400 ¢ is the most commonly used approximation to 5/4, mainly due to its use in 12edo. This interval is also expressed as 1\3, and its square root on the logarithmic scale, ≈692.82 ¢ (hereinafter expressed as 1\√3 for convenience), functions as an approximation of 3/2. (The comma between these two intervals is called as Caffeinterval.) Furthermore, the interval divided into √3 equal parts with 1\√3 as the center[clarification needed] is ≈985.641¢, which works as an approximation of 7/4 or 9/5, and the tetrad that combines these is 4:5:6:7, the so-called It will be the C7. Applying this property, the scale that is created as a result of recursively dividing those intervals furthermore twice is Redbull.
Most of the notes on this scale are irrational numbers in both cent and frequency units, so Redbull cannot be reproduced with an edo. Also, since there is no interval that can be called a generator, it is also impossible to approximate Redbull with a mos scale. Also, since there is no interval that can be called a generator, and it varies even by one step[clarification needed], there are many intervals within Redbull that approximate just intonation, just like afdos.
Intervals
For more precise cent values, refer to the Scala file below.
Degree
Cents
Approximate ratios
0
0
1/1
1
133.333
14/13, 13/12, 12/11
2
230.940
8/7, 9/8
3
328.547
6/5
4
400
5/4, 24/19
5
497.607
4/3
6
569.060
25/18, 7/5, 11/8
7
640.513
10/7, 13/9, 16/11
8
692.820
3/2
9
790.427
11/7
10
861.880
13/8, 18/11, 5/3
11
933.333
12/7
12
985.641
7/4, 9/5
13
1057.094
11/6, 24/13, 13/7
14
1109.401
15/8, 17/9, 28/15
15
1161.708
33/17, 64/33
16
1200
2/1
Properties and trivia
A pentatonic Redbull scale on C.A "Do-Re-Mi" song in wholetone Redbull scale.
As mentioned above, the tetrad obtained by stacking 4-steps from the tonic (i.e. starting on degree 0) is similar to the seventh tetrad in 12edo, approximating 4:5:6:7. Since 4 is a divisor of 16, there are only 4 types of 4-step tetrads, the others being only inversions, and the other types of 4-step tetrads do not approximate 4:5:6:7. (in example, chords stacked 4 steps from 2 steps above the tonic is approximating 9:11:13:17.)
The pentad obtained by stacking 3-steps from the tonic approximates 5:6:7:8:9.
Furthermore, Redbull has a pentatonic subset which is similar to 2L 3s[clarification needed], and the constituent notes of that scale can be approximated as 9:12:13:16:17 in just intonation.
