Breuddwyd scale
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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 scale generated by 5\31, a scale generated by 11\31, and a scale generated by 13\31. (Where n\31 is n steps of 31edo or another 31-tone equal tuning.) Often but not always the three scales are MOS scales.
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 (very weak), 5/3, 7/4, 2/1, 7/3, 5/2, 3/1 (very weak), 7/2, 4/1, 5/1, 6/1 (very weak), 7/1 |
| 5&11&13&31ifdo (Breuddwyd inverse) Main article: 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 (very 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) Main article: Breuddwyd6 |
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 (warts in brackets) |
|---|---|---|---|---|---|---|---|---|---|---|
| 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) |
Wijzerplaat scales
All of these scales are tempered by definition. Sometimes multiple scales may generate the same tone, which is why when one combines an x-tone, y-tone and z-tone scale, 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.
| Systematic name (& idiosyncratic common name) |
Parent tuning used | Tones per period used | Scale pattern | Tones generated by 5\31, 11\31, 13\31 (mos or no?) | 5\31 generators up:down, 11\31 up:down, 13\31 up:down |
7 integer limit intervals approximated within 15¢ |
|---|---|---|---|---|---|---|
| PolyMOS <5\31(u2d0), <11\31(u0d4), 13\31(u1d3)> (Oclock) Main article: Oclock |
31edo | 9 tones per 31\31 | 5 4 1 3 5 2 3 6 2 | 3(mos), 5(mos), 5(mos) | 2:0, 0:4, 1:3 |
5/4, 4/3, 3/2, 5/3, 2/1, 5/2, 3/1, 4/1, 5/1, 6/1 |