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 2.3.5.7.13 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)

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

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		Interval names
		2.5.7/3.11/3.13.19.23 subgroup	Dual-{3 7 11 17}	Patent 13-limit val	Native-fifths	Orwell	Diatonic	ADIN
0	0	1/1			D	J	perfect unison	perfect unison
1	30	[56/55], [57/56], 65/64	50/49, 51/50, 52/51		D#	J#	augmented unison	near(major) unison
2	60	26/25	33/32, 35/34		Dx	Jx, Kbb	2x augmented unison	superunison
3	90	[20/19]	52/49, 19/18, 21/20		D#x, Ebbb	Kb	2x diminished 2nd	subminor second
4	120	[15/14]	49/46	14/13, 16/15	Ebb	K	diminished 2nd	nearminor second
5	150	[12/11], 25/23	23/21	11/10, 13/12	Eb	K#	minor 2nd	supraminor second
6	180	39/35	10/9, [51/46], 21/19	9/8	E	Kx	major 2nd	submajor second
7	210	26/23, [44/39]	9/8	10/9	E#	Lbb	augmented 2nd	nearmajor second
8	240	[23/20], 55/48	38/33	15/13, 8/7	Ex	Lb	2x augmented 2nd	supermajor second
9	270	7/6			Fbb	L	2x diminished 3rd	subminor third
10	300	19/16	[25/21]	6/5, 13/11	Fb	L#	diminished 3rd	nearminor third
11	330	23/19	17/14, 40/33		F	Lx, Mbb	minor 3rd	supraminor third
12	360	[16/13]	26/21, 49/40	11/9	F#	Mb	major 3rd	submajor third
13	390	44/35, 5/4	64/51		Fx	M	augmented 3rd	nearmajor third
14	420	32/25, 14/11	23/18, [51/40], 33/26	9/7	F#x, Gbbb	M#	2x augmented 3rd	supermajor third
15	450	13/10	64/49, 22/17, 49/38		Gbb	Mx	2x diminished 4th	subfourth
16	480	25/19	[33/25], 21/16		Gb	Nbb	diminished 4th	nearfourth
17	510	66/49	51/38	4/3	G	Nb	perfect 4th	perfect fourth
18	540	48/35, 26/19, 15/11		11/8	G#	N	augmented 4th	subaugmented fourth
19	570	39/28, [32/23]	46/33, [25/18], 18/13	7/5	Gx	N#	2x augmented 4th	nearaugmented fourth
20	600	55/39, 78/55	17/12, 24/17		G#x, Abbb	Nx, Obb	3x aug 4th, 3x dim 5th	subdim fifth, superaug fourth
21	630	[23/16], 56/39	13/9, [36/25], 33/23	10/7	Abb	Ob	2x diminished 5th	neardiminished fifth
22	660	22/15, 19/13, 35/24		16/11	Ab	O	diminished 5th	supradiminished fifth
23	690	49/33	76/51	3/2	A	O#	perfect 5th	perfect fifth
24	720	38/25	32/21, [50/33]		A#	Ox	augmented 5th	nearfifth
25	750	20/13	76/49, 17/11, 49/32		Ax	Pbb	2x augmented 5th	superfifth
26	780	11/7, 25/16	52/33, 80/51, 36/23	14/9	A#x, Bbbb	Pb	2x diminished 6th	subminor sixth
27	810	8/5, 35/22	51/32		Bbb	P	diminished 6th	nearminor sixth
28	840	[13/8]	80/49, 21/13	18/11	Bb	P#	minor 6th	supraminor sixth
29	870	38/23	33/20, 28/17		B	Px, Qbb	major 6th	submajor sixth
30	900	32/19	[42/25]	22/13, 5/3	B#	Qb	augmented 6th	nearmajor sixth
31	930	12/7			Bx	Q	2x augmented 6th	supermajor sixth
32	960	96/55, [40/23]	33/19	7/4, 26/15	Cbb	Q#	2x diminished 7th	subminor seventh
33	990	[39/22], 23/13	16/9	9/5	Cb	Qx	diminished 7th	nearminor seventh
34	1020	70/39	38/21, [92/51], 9/5	16/9	C	Rbb	minor 7th	supraminor seventh
35	1050	46/25, [11/6]	42/23	24/13, 20/11	C#	Rb	major 7th	submajor seventh
36	1080	[28/15]	92/49	15/8, 13/7	Cx	R	augmented 7th	nearmajor seventh
37	1110	[19/10]	40/21, 36/19, 49/26		C#x, Dbbb	R#	2x augmented 7th	supermajor seventh
38	1140	25/13	68/35, 64/33		Dbb	Rx, Jbb	2x diminished 8ve	suboctave
39	1170	128/65, [112/57], [55/28]	51/26, 100/51, 49/25		Db	Jb	diminished 8ve	near(minor) octave
40	1200	2/1			D	J	perfect 8ve	perfect octave

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

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

Scales

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.