User:BudjarnLambeth/Breuddwyd scale
|
This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
A breuddwyd scale[idiosyncratic term] (pronounced "braid wood") is any polymicrotonal scale which combines four scales, the first scale with 5 tones per equave, the second with 11, the third with 13 and the fourth with 31.
A sonhar tuning[idiosyncratic term] (pronounced "sonyar") is any scale or temperament which uses or approximates the JI subgroup 5.11.13.31.
A wijzerplaat scale[idiosyncratic term] (pronounced "why, ser as in deserve, plat as in platypus") is any scale which is built by combining a MOS scale generated by 5\31, a MOS scale generated by 11\31, and a MOS scale generated by 13\31. (Where n\31 is n steps of 31edo or another 31-tone equal tuning.)
History and etymology
These three categories of scales were devised by Budjarn Lambeth in January 2025, after he had a dream featuring a disc inscribed with numbers - 31 in the middle, and 5, 11 and 13 around the outside.
Intending to make music based on these numbers, Lambeth started brainstorming scales and recorded what he found as the breuddwyd, sonhar and wijzerplaat scales.
"Breuddwyd" is Welsh for "dream". "Sonhar" is Brazilian Portugese for "dream". "Wijzerplaat" is Dutch for "clock face".
Breuddwyd scales
This list is not exhaustive. There are many other possible breuddwyd scales.
| Systematic name (& idiosyncratic common name) |
Just or tempered? | Equave | Tones per equave | Tones per octave | Definition | Additional valid definitions | Is a subset of | 7 integer limit intervals approximated within 15¢ |
|---|---|---|---|---|---|---|---|---|
| 5&11&13&31afdo (Breuddwyd arithmetic) |
Just | 2/1 | 57 tones | 57/octave | Polymicrotonal scale of 5afdo, 11afdo, 13afdo and 31afdo | The scale of all rational intervals with 5, 11, 13 or 31 in the denominator | 22165afdo | 7/6, 6/5, 5/4, 4/3 (weak), 7/5, 3/2 (weak), 5/3, 7/4, 2/1, 7/3, 5/2, 3/1 (weak), 7/2, 4/1, 5/1, 6/1 (weak), 7/1 |
| 5&11&13&31ifdo (Breuddwyd inverse) |
Just | 2/1 | 57 tones | 57/octave | Polymicrotonal scale of 5ifdo, 11ifdo, 13ifdo and 31ifdo | The scale of all rational intervals with 5, 11, 13 or 31, or any of their octave multiples (e.g. 10, 22, 26, 62 or 20, 44, 52, 124 or so on) in the numerator | 22165ifdo | 7/6, 6/5, 5/4, 4/3 (weak), 7/5, 3/2 (weak), 5/3, 7/4, 2/1, 7/3, 5/2, 3/1 (weak), 7/2, 4/1, 5/1, 6/1 (weak), 7/1 |
| 5&11&13&31edo (Breuddwyd-2) |
Tempered | 2/1 | 57 tones | 57/octave | Polymicrotonal scale of 5edo, 11edo, 13edo and 31edo | 22165edo | 7/6, 6/5, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4, 2/1, 7/3, 5/2, 3/1, 7/2, 4/1, 5/1, 6/1, 7/1 | |
| 5&11&13&31edt (Breuddwyd-3) |
Tempered | 3/1 | 57 tones | ~36/octave | Polymicrotonal scale of 5edt, 11edt, 13edt and 31edt | 22165edt | 7/6, 6/5, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4, 2/1, 7/3, 5/2, 3/1, 7/2, 4/1, 5/1, 6/1, 7/1 | |
| 5&11&13&31ed4 (Breuddwyd-4) |
Tempered | 4/1 | 57 tones | ~29/octave | Polymicrotonal scale of 5ed4, 11ed4, 13ed4 and 31ed4 | 22165ed4 | 6/5, 5/4, 4/3 (weak), 3/2, 5/3, 7/4, 7/3, 3/1 (weak), 7/2, 4/1, 5/1, 6/1, 7/1 | |
| 5&11&13&31ed5 (Breuddwyd-5) |
Tempered | 5/1 | 57 tones | ~25/octave | Polymicrotonal scale of 5ed5, 11ed5, 13ed5 and 31ed5 | 22165ed5 | 7/6, 4/3, 3/2 (weak), 5/3, 7/4, 7/2, 5/1, 7/1 (weak) | |
| 5&11&13&31ed6 (Breuddwyd-6) |
Tempered | 6/1 | 57 tones | ~19/octave | Polymicrotonal scale of 5ed6, 11ed6, 13ed6 and 31ed6 | 22165ed6 | 7/6, 6/5, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4, 2/1, 5/2, 3/1, 4/1, 5/1, 6/1, 7/1 | |
| 5&11&13&31ed14/3 (Breuddwyd-14/3) |
Tempered | 14/3 | 57 tones | ~26/octave | Polymicrotonal scale of 5ed14/3, 11ed14/3, 13ed14/3 and 31ed14/3 | 22165ed14/3 | 7/6, 4/3 (weak), 7/5, 3/2 (weak), 5/3 (weak), 7/4, 2/1, 7/3, 5/2, 3/1 (weak), 7/2, 4/1, 6/1, 7/1 |
Sonhar tunings
This list is not exhaustive. There are many other possible sonhar scales.
Just
| Systematic name (& idiosyncratic common name) |
Just or tempered? | Equave | Tones per equave | Tones per octave | Definition | Additional valid definitions | Is a subset of | 7 integer limit intervals approximated within 15¢ |
|---|---|---|---|---|---|---|---|---|
| CPS(2of5,11,13,31) (Breuddwyd hexany) |
Just | 2/1 | 6 tones | 6/octave | The hexany generated by 5/1, 11/1, 13/1 and 31/1 | The octave-repeating harmonic series subset 220:260:286:310:341:403:440 | 220afdo | (allowing rotations) 7/6, 6/5, 7/5, 5/3, 2/1, 7/3, 4/1 |
Tempered
You can find all necessary information to add a temperament to this table by using x31eq.com.
| Systematic name (& idiosyncratic common name) |
Equave | Equal temp mapping | Reduced mapping | TE generator tunings (¢) | TE step tunings (¢) | TE tuning map (¢) | TE mistunings (¢) | Complexity, adjusted error TE error |
Unison vectors | Recommended ETs | (x31 notation) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| c2 & c37 (Sonhar A) |
5/1 | 5,11,13,31 [<2,3,3,4] <37,55,59,79]> |
5,11,13,31 [<1,2,-2,-3] <0,-1,7,10]> |
2789.3304, 1431.2645 | 40.49033, 73.19864 | 2789.330, 4147.396, 4440.191, 5944.654 | 3.017, -3.922, -0.337, -0.382 | 0.454182, 4.281341, 0.864185 |
[-2,1,3,-2>, [-5,3,-1,1>, [-7,4,2,-1>, [-3,2,-4,3> | 39ed5, 37ed5, 41ed5, 35ed5, 76ed5, 78ed5, 74ed5, 43ed5(fk) | c39, c37, c41, c35, c76, c78, c74, c43fk |
Wijzerplaat scales
All of these scales are octave-repeating subsets of 31edo. They are tempered by definition. Sometimes multiple MOSes may generate the same tone, which is why when you combine an x-tone, y-tone and z-tone MOS, the total number of tones/octave may still be less than (x+y+z).
This list is not exhaustive. There are many other possible wijzerplaat scales.
The scale names are idiosyncratic.
| Name | Parent tuning used | Tones per period used | Scale pattern | Tones generated by 5\31, 11\31, 13\31 | 5\31 generators up:down, 11\31 up:down, 13\31 up:down |
7 integer limit intervals approximated within 15¢ |
|---|---|---|---|---|---|---|
| PolyMOS 5\31(up2down2) 11\31(up0down2) 13\31(up1down1) | 31edo | 9 tones per 31\31 | 5 4 1 3 5 2 1 5 5 | 5, 3, 3 | 2:2, 0:2, 1:1 |
5/4, 4/3, 3/2, 2/1, 5/2, 3/1, 4/1, 5/1, 6/1 |