User:BudjarnLambeth/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".

A recreation of the disc from the dream.

Breuddwyd scales

This list is not exhaustive. There are many other possible breuddwyd scales.

The 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

Just sonhar tunings

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.

Tempered sonhar tunings

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.

The 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