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. 40edo's native diatonic scale is nearly equiheptatonic, with a hardness of 6:5; major and minor intervals of the scale differ by only 30¢. In particular, the major third of the diatonic scale is 360¢ (essentially 16/13), generally considered a high neutral or submajor third, and 5/4 is mapped not to the major third, but the augmented third, which implies that the syntonic comma is mapped negatively in 40edo.

Despite the impurity of its approximations to 3/2 (if this does not deny them usability as bounding intervals for chords), 40edo has a range of more accurate concordances to draw from. 40edo's 11-limit is a tuning for undecimal Orwell, and while at first glance it appears like a rather poor one, it is in fact essentially optimized for a subset of the 11-limit, that being the 2.5.7/3.11/3 subgroup, which Orwell connects together remarkably well, and most of whose important intervals are available within the 9-note MOS, 4L 5s - most notably 40edo's approximations to 7/6 and 5/4, each just over 3¢ sharp.

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} tuning system, 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 2.5.7/3.11/3.13.19 subgroup, 40edo's step size represents:

• 56/55 (the difference between 5/4 and 14/11)
• 57/56 (the difference between 7/6 and 19/16)
• 65/64 (the difference between 16/13 and 5/4)
• 128/125 (the residue between three stacked 5/4s and the octave).

With the dual-prime interpretation (i.e. 2.9.5.21.33.13), it can additionally be taken to be, amongst other things:

• 50/49 (the difference between 49/40 and 5/4, or 7/6 and 25/21)
• 55/54 (the difference between 12/11 and 10/9)
• 81/80 (the difference between 10/9 and 9/8)
• 105/104 (the difference between 13/10 and 21/16);

alongside the first set of representations.

If the patent mapping of the 13-limit is taken instead, it represents:

• 27/26 (the difference between 10/9 and 15/13)
• 33/32 (the difference between 4/3 and 11/8)
• 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
• 45/44 (the difference between 11/9 and 5/4)
• 49/48 (the difference between 8/7 and 7/6)
• 80/81 (the negative difference between 9/8 and 10/9);

alongside the first set of representations.

Intervals and notation

As 40edo's diatonic fifth is so flat, its native diatonic scale has 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; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the 13-limit according to 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 13-limit val	Native-fifths	Orwell
0	0	1/1			D	J
1	30	[56/55], [57/56], 65/64	50/49, 51/50, 52/51	33/32, 36/35, 45/44, 49/48	D#	J#
2	60	26/25	33/32, 35/34	21/20	Dx	Jx, Kbb
3	90	[20/19]	52/49, 19/18, 21/20	22/21, 25/24	D#x, Ebbb	Kb
4	120	[15/14]	49/46	14/13, 16/15	Ebb	K
5	150	[12/11], 25/23	23/21	11/10, 13/12	Eb	K#
6	180	39/35	10/9, [51/46], 21/19	9/8	E	Kx
7	210	26/23, [44/39]	9/8	10/9	E#	Lbb
8	240	[23/20], 55/48	38/33	15/13, 8/7	Ex	Lb
9	270	7/6			Fbb	L
10	300	19/16	[25/21]	6/5, 13/11	Fb	L#
11	330	23/19	17/14, 40/33		F	Lx, Mbb
12	360	[16/13]	26/21, 49/40	11/9	F#	Mb
13	390	44/35, 5/4	64/51		Fx	M
14	420	32/25, 14/11	23/18, [51/40], 33/26	9/7	F#x, Gbbb	M#
15	450	13/10	64/49, 22/17, 49/38		Gbb	Mx
16	480	25/19	[33/25], 21/16		Gb	Nbb
17	510	66/49	51/38	4/3	G	Nb
18	540	48/35, 26/19, 15/11		11/8	G#	N
19	570	39/28, [32/23]	46/33, [25/18], 18/13	7/5	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	10/7	Abb	Ob
22	660	22/15, 19/13, 35/24		16/11	Ab	O
23	690	49/33	76/51	3/2	A	O#
24	720	38/25	32/21, [50/33]		A#	Ox
25	750	20/13	76/49, 17/11, 49/32		Ax	Pbb
26	780	11/7, 25/16	52/33, 80/51, 36/23	14/9	A#x, Bbbb	Pb
27	810	8/5, 35/22	51/32		Bbb	P
28	840	[13/8]	80/49, 21/13	18/11	Bb	P#
29	870	38/23	33/20, 28/17		B	Px, Qbb
30	900	32/19	[42/25]	22/13, 5/3	B#	Qb
31	930	12/7			Bx	Q
32	960	96/55, [40/23]	33/19	7/4, 26/15	Cbb	Q#
33	990	[39/22], 23/13	16/9	9/5	Cb	Qx
34	1020	70/39	38/21, [92/51], 9/5	16/9	C	Rbb
35	1050	46/25, [11/6]	42/23	24/13, 20/11	C#	Rb
36	1080	[28/15]	92/49	15/8, 13/7	Cx	R
37	1110	[19/10]	40/21, 36/19, 49/26		C#x, Dbbb	R#
38	1140	25/13	68/35, 64/33		Dbb	Rx, Jbb
39	1170	128/65, [112/57], [55/28]	51/26, 100/51, 49/25		Db	Jb
40	1200	2/1			D	J

Compositional theory

Tertian structure

Six intervals in 40edo can be considered functional "thirds" with the 690¢ diatonic fifth taken as the bounding interval; a seventh (450¢) can be included with the acknowledgement of the 720¢ blackwood fifth as competing. As neither fifth is very close to 3/2, it is best to treat the approximations of 40edo's thirds asymmetrically; in doing so, it can be seen that most of them are a couple of cents sharp of reasonably simple JI intervals. This, somewhat intriguingly, allows for treating 14/11 and 7/6 as a pair of fifth complements while maintaining the dyadic integrity of each third, and similarly 5/4 and 19/16 as a pair of fifth complements, if the diatonic fifth is used.

Thirds in 40edo

Quality (ADIN)	Subminor	Nearminor	Supraminor	Submajor	Nearmajor	Supermajor	Ultramajor
Cents	270	300	330	360	390	420	450
Just interpretation	7/6 (+3.1¢)	19/16 (+2.5¢)	23/19 (-0.8¢)	16/13 (+0.5¢)	5/4 (+3.7¢)	14/11 (+2.5¢)	13/10 (-4.2¢)
Steps	9	10	11	12	13	14	15

Diatonic thirds are bolded.

Chords

In addition to triads bounded by a perfect fifth, in 40edo one finds that 810¢ (8/5) and 660¢ (35/24~22/15~19/13) serve as important bounding intervals for chords.

In particular, we have the no-threes isoharmonic segment 10:13:16:19, mapped to [0 15 27 37]\40, which can serve as an equivalent to the classic 7-limit tetrad 4:5:6:7. This can be split into the triads 10:13:16, within 8/5, and 13:16:19, within 19/13. Another pair of triads that fit within 8/5 are those formed by 5/4 and 14/11: [0 13 27] and [0 14 27]\40.

The latter 660¢ interval also represents 35/24, a stack of 7/6 and 5/4, and hence within it are the chords 24:28:35 ([0 9 22]) and 24:30:35 ([0 13 22]). In between them, a stack of two 330¢ supraminor thirds ([0 11 22]) can be represented as 24:29:35.

Within 8/5 is formed the orwell tetrad, formed from the first three generators of Orwell stacked, [0 9 18 27]\40, i.e. 1/1 - 7/6 - 15/11 - 8/5, an interesting otonal representation of which is 30:35:41:48~35:41:48:56. Reducing the stack to two generators forms a chthonic triad, of which Orwell[9] provides two additional variants stacked within the perfect fourth (17\40).

Lastly, 40edo contains a nearly-isoharmonic diminished triad, similarly to 22edo, at [0 11 20]\40, approximating 24:29:34.

Scales

Deeptone diatonic

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Omnidiatonic/Diasem

Consider the Zarlino diatonic scale, representing the series of intervals 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1, with step sizes representing 9/8, 10/9, and 16/15. While normally, this results in a step pattern LMsLMLs, with L > M > s, because the syntonic comma is mapped negatively in 40edo, it results in the pattern MLsMLMs (known as "omnidiatonic") instead: 6 - 7 - 4 - 6 - 7 - 6 - 4 in steps of 40edo. A different variety of omnidiatonic uses supermajor thirds instead of major, and has the step pattern 6 - 8 - 3 - 6 - 8 - 6 - 3. The large steps of these scales can then be split further into a commatic interval and a wholetone (6\40), forming step patterns of the form LsLmLsLLm, known as diasem.

Additionally, these scales are chiral, so that they can be both rotated into different modes, and reflected between "left-handed" and "right-handed" variants.

Orwell

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Diminished

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Blackwood

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