Kite's thoughts on 31edo brass instruments

See Kite's thoughts on 41edo brass instruments for the reasoning behind these calculations.

Approximating 2\31 as 43/41 gives us these valve lengths:

valve	edosteps	ratio	cents	as 31edo
1st	2	43/41	82.5	2\31 + 5.0¢
2nd	3	44/41	122.3	3\31 + 6.1¢
3rd	4	45/41	161.2	4\31 + 6.3¢
4th	8	49/41	308.6	8\31 - 1.1¢

These valves make the subharmonic series 15-note fragment 58::41, with 57 and 42 missing. In the chart below, ○ is a closed/released valve, and ● is an open/depressed valve. The 3 largest errors are bolded. Notes are for a Bb trumpet.

The intervals are descending so 82.5¢ means 82.5¢ below the unvalved note. Thus the error is the opposite sign from what one expects. For example, the 1st valve lowers by 82.5¢, 5¢ wider than 2\31, thus the note is 5¢ flat of 31edo.

Note that both 52/41 and 53/41 map to 11\31. Thus 4+8 = 11, 2+3+8 = 12, etc.

edosteps to lower by		valves	subhar- monics	as a ratio	cents	error from 31		harmonics
								3	4	5	6	7	8	9
0	0	○○○○	41	1/1	0.0	0.0		F	^A# Bb	D	F	G# vAb	^A# Bb	C
2	2	●○○○	43	43/41	82.5	-5.0		^E	^A	^C# Db	^E	G	^A	^B
3	3	○●○○	44	44/41	122.3	-6.1		E	A	C# vDb	E	vG	A	B
4	4	○○●○	45	45/41	161.2	-6.3		vE	vA	^C	vE	^F# Gb	vA	vB
5	2+3	●●○○	46	46/41	199.2	-5.7		^D# Eb	^G# Ab	C	^D# Eb	F# vGb	^G# Ab	^A# Bb
6	2+4	●○●○	47	47/41	236.4	-4.2		D# vEb	G# vAb	vC	D# vEb	^F	G# vAb	A# vBb
7	3+4	○●●○	48	48/41	272.9	-1.9		^D	^G	^B	^D	F	^G	^A
8	8	○○○●	49	49/41	308.6	1.1		D	G	B	D	vF	G	A
9	2+3+4	●●●○	50	50/41	343.6	4.8		vD	vG	vB	vD	^E	vG	vA
10	2+8	●○○●	51	51/41	377.8	9.2		^C# Db	^F# Gb	^A# Bb	^C# Db	E	^F# Gb	^G# Ab
11	3+8	○●○●	52	52/41	411.5	14.3		C# vDb	F# vGb	A# vBb	C# vDb	vE	F# vGb	G# vAb
	4+8	○○●●	53	53/41	444.4	-18.6
12	2+3+8	●●○●	54	54/41	476.8	-12.3		^C	^F	^A	^C	^D# Eb	^F	^G
13	2+4+8	●○●●	55	55/41	508.6	-5.3		C	F	A	C	D# vEb	F	G
14	3+4+8	○●●●	56	56/41	539.8	2.2		vC	vF	vA	vC	^D	vF	vG
16	2+3+4+8	●●●●	58	58/41	600.5	18.8		B	E	G# vAb	B	vD	E	F#

Here are the full 2 octaves produced by a Bb trumpet. Note that the bolded notes for the three largest errors often have alternative notes.

Full 2 octave range (low note B below middle-C)

31-edo gamut	harmonics
	3	4	5	6	7	8	9
C							C
							(missing)
							^B
B							B
							vB
						^A# Bb	^A# Bb
							A# vBb
						^A	^A
A						A	A
						vA	vA
						^G# Ab	^G# Ab
					G# vAb	G# vAb	G# vAb
						^G	^G
G					G	G	G
					vG	vG	vG
					^F# Gb	^F# Gb
					F# vGb	F# vGb	F#
					^F	^F
F				F	F	F
					vF	vF
				^E	^E
E				E	E	E
				vE	vE
				^D# Eb	^D# Eb
				D# vEb	D# vEb
				^D	^D
D			D	D
				vD	vD
			^C# Db	^C# Db
			C# vDb	C# vDb
			^C	^C
C			C	C
			vC	vC
			^B
B			B	B
			vB
		^A# Bb	^A# Bb
			A# vBb
		^A	^A
A		A	A
		vA	vA
		^G# Ab
		G# vAb	G# vAb
		^G
G		G
		vG
		^F# Gb
		F# vGb
		^F
F	F	F
		vF
	^E
E	E	E
	vE
	^D# Eb
	D# vEb
	^D
D	D
	vD
	^C# Db
	C# vDb
	^C
C	C
	vC
	(missing)
B	B