40edo

40edo, or 40 equal divisions of the octave (sometimes called 40-TET or 40-tone equal temperament), is the equal tuning featuring steps of (1200/40) ~= 30 cents, 40 of which stack to the perfect octave 2/1.

40edo can be considered a straddle-3, or dual-3, system, as it has both the 5edo fifth of 720¢, and a very flat diatonic fifth at 690¢, being the smallest 5n EDO to have a diatonic perfect fifth.

General theory

JI approximation

While 40edo has two intervals that can be considered a perfect fifth, its patent 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the 7/4 inherited from 5edo (960¢) being a closer approximation compared to a very sharp mapping at 990¢; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390¢ interval, and due to being a multiple of 10edo and 4edo, it represents the 13th and 19th harmonics through those EDOs' respective approximations.

Therefore, the case is not dissimilar to 29edo's treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad subgroup of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.

As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17}, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of 80edo. Of course, the patent approximations can still be used, an interesting consequence of which is that 6/5 is mapped to the quarter-octave (300¢), like it is in 12edo (though note that this is not the best 6/5, the 330¢ interval being slightly closer).

Approximation of prime harmonics in 40edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-12.0	+3.7	-8.8	-11.3	-0.5	-15.0	+2.5	+1.7	-9.6	-5.0
	Relative (%)	0.0	-39.9	+12.3	-29.4	-37.7	-1.8	-49.9	+8.3	+5.8	-31.9	-16.8
Steps (reduced)		40 (0)	63 (23)	93 (13)	112 (32)	138 (18)	148 (28)	163 (3)	170 (10)	181 (21)	194 (34)	198 (38)

Edostep interpretations

In the 2.3.5.7.13 subgroup, 40edo's step has the following interpretations:

• 80/81, the negative syntonic comma (between 10/9 and 9/8)
• 512/507, the intertridecimal comma (between 16/13 and 39/32)
• 2187/2048, the chromatic semitone (between 256/243 and 9/8)
• 65/64, the wilsorma (between 5/4 and 16/13)
• 36/35, the mint comma (between 5/4 and 9/7)
• 49/48, the interseptimal comma (between 8/7 and 7/6)

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Intervals and notation

As 40edo's diatonic fifth is so flat, its native diatonic is nearly equiheptatonic, with a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as ups and downs therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple JI. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis.

40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23 and two supersets: one including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; the other including intervals of 3, 7, and 11 following the patent val. Inconsistent intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.

Edostep	Cents	JI approximations			Notation
		2.5.7/3.11/3.13.19.23 subgroup	Dual-{3 7 11 17}	Patent L13.19.23 val	Native-fifths	Orwell
0	0	1/1			D	J
1	30	[56/55], [57/56], 65/64	51/50		D#	J#
2	60	26/25	33/32		Dx	Jx, Kbb
3	90	[20/19]	19/18, 21/20		D#x, Ebbb	Kb
4	120	[15/14]			Ebb	K
5	150	[12/11], 25/23	23/21		Eb	K#
6	180	39/35	10/9, 21/19		E	Kx
7	210	26/23, [44/39]	9/8		E#	Lbb
8	240	[23/20], 55/48	38/33		Ex	Lb
9	270	7/6			Fbb	L
10	300	19/16	[25/21]		Fb	L#
11	330	23/19	40/33		F	Lx, Mbb
12	360	[16/13]	26/21		F#	Mb
13	390	5/4			Fx	M
14	420	32/25, 14/11	23/18, 33/26		F#x, Gbbb	M#
15	450	13/10			Gbb	Mx
16	480	25/19	[33/25], 21/16		Gb	Nbb
17	510	66/49			G	Nb
18	540	26/19, 15/11			G#	N
19	570	39/28, [32/23]	46/33, [25/18], 18/13		Gx	N#
20	600	55/39, 78/55	17/12, 24/17		G#x, Abbb	Nx, Obb
21	630	[23/16], 56/39	13/9, [36/25], 33/23		Abb	Ob
22	660	22/15, 19/13			Ab	O
23	690	49/33			A	O#
24	720	38/25	32/21, [50/33]		A#	Ox
25	750	20/13			Ax	Pbb
26	780	11/7, 25/16	52/33, 36/23		A#x, Bbbb	Pb
27	810	8/5			Bbb	P
28	840	[13/8]	21/13		Bb	P#
29	870	38/23	33/20		B	Px, Qbb
30	900	32/19	[42/25]		B#	Qb
31	930	12/7			Bx	Q
32	960	96/55, [40/23]	33/19		Cbb	Q#
33	990	[39/22], 23/13	16/9		Cb	Qx
34	1020	70/39	38/21, 9/5		C	Rbb
35	1050	46/25, [11/6]	40/21, 36/19		C#	Rb
36	1080	[28/15]			Cx	R
37	1110	[19/10]			C#x, Dbbb	R#
38	1140	25/13	64/33		Dbb	Rx, Jbb
39	1170	128/65, [112/57], [55/28]	100/51		Db	Jb
40	1200	2/1			D	J

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Compositional theory

Chords

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Scales

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