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 exactly, 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. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.

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, 17/15	10/9	E#	Lbb
8	240	[23/20], 55/48	38/33, 39/34	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, 68/39	7/4, 26/15	Cbb	Q#
33	990	[39/22], 23/13	16/9, 30/17	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

Tempering properties

Tempered commas

Important commas tempered out by 40et within the 2.5.7/3.11/3.13.19 subgroup include:

• 176/175, S8/S10 (valinorsmic), equating a stack of two 5/4s to 11/7
• 456/455 (abnobismic), equating (5/4)*(7/6) to 19/13
• 540/539, S12/S14 (swetismic), equating two 7/6s to 15/11
• 1573/1568 (lambeth), equating a stack of two 14/11s to 13/8
• 1728/1715, S6/S7 (orwellismic), equating three 7/6s to 8/5
• 3584/3575, S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant
• 48013/48000, S19/S20, splitting 7/6 into three 20/19's.

The dual-{3 7 11 17} interpretation additionally tempers out the following:

• 136/135 (diatismic), S16*S17, equating 9/8 with 17/15
• 289/288 (semitonismic), S17, splitting the octave into two 17/12~24/17 periods
• 361/360 (dudon), S19, and 400/399 (devichromic), S20, equating 20/19 with 19/18 and 21/20, splitting 7/6 in three
• 390625/388962 (dimcomp), equating 25/21 to the quarter-octave.

In its patent 13-limit, 40et tempers out the first set of commas alongside:

• 66/65, S11*S12 (winmeanmic), equating 6/5 and 13/11
• 99/98 (mothwellsmic), equating 9/7 and 14/11
• 105/104, S14*S15 (animist), equating 8/7 and 15/13
• 121/120, S11 (biyatismic), equating 11/8 and 15/11, and 12/11 to 11/10
• 225/224, S15 (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to 14/9
• 648/625 (diminished), setting 6/5 to the quarter-octave
• 1053/1024 (superflat), making 16/13 the diatonic major third
• 2187/2080, placing 5/4 an apotome above 16/13 (making it the augmented third)
• 16807/16384 (cloudy), setting 8/7 to a fifth of the octave.

Notable structural chains

40edo has eight distinct generator chains that span the EDO with a full-octave period: these being generated by intervals of 1, 3, 7, 9, 11, 13, 17, and 19 steps.

The most significant structural relation is that three intervals of ~7/6 (270¢) comprise ~8/5 (810¢), and furthermore that two intervals of 7/6 comprise ~15/11 (540¢). This is Guanyintet temperament, defined on the subgroup 2.5.7/3.11/3; if 40edo's flat fifth is considered acceptable, this continues into undecimal Orwell. Otherwise, Guanyintet approximates the 13th harmonic at 12 steps along the chain of 7/6s, and since ~15/11~48/35 approximates also 26/19, the 19th harmonic occurs at 10 generators, leaving only the 23rd harmonic difficult to approximate.

The positions of 13 and 19 in the chain can be made more accessible by dividing 7/6 into three intervals of 20/19. As a result, 8/5 is split into 9 parts, with 4 parts and 5 parts forming close approximations of 16/13 and 13/10, respectively; 10 parts form (8/5)(20/19) = 32/19. This makes several 10:13:16:19 tetrads available within the 13- and 14-note scales of this temperament.

Finally, 40edo's chain of fifths, generated by 23\40 (690¢) is of note. Interpreting the fifth as 3/2, the major third (formally ~81/64) is mapped to 16/13, and as the apotome is the narrow 30¢ difference between it and the minor third (~39/32), 5/4 occurs at the augmented third, or 11 fifths upward, which is the definition of Deeptone temperament. While the whole tone in Deeptone approximates 10/9 very well, it cannot be interpreted that way outside of the dual-prime interpretation.

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

As 40edo is composite, it incorporates the scales of all of its subset EDOs, including 8edo, 10edo, and 20edo. These will not be discussed here; what follows will be a sampling of structurally significant scales unique to 40edo.

Deeptone

40edo's native diatonic scale, generated by its flat 690¢ fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even meantone diatonics, especially as interval qualities converge significantly towards 7edo, with 16/13 submajor thirds in place of 5/4 or 81/64.

The 12-note chromatic generated by the fifth, 7L 5s, is somewhat perversely a very hard scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480¢ and 720¢ blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the clustering nature of deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

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/Guanyintet

The fundamental scale of 40edo's Orwell temperament is the 9-note scale, 4L 5s, with step pattern 4-5-4-5-4-5-4-5-4. The scale is generated by 7/6, and while two 690¢ fifths occur in the enneatonic, far more common are 660¢ 22/15~35/24~19/13 subfifths. It is also notable that the long step and short step very closely approximate the intervals 12/11 and 15/14, respectively (four 12/11s and five 15/14s differ from the octave by 246071287/246037500, about 0.24¢).

The 9-note scale is followed by a 13-note chromatic (9L 4s) with steps of 4\40 and 1\40. These cluster severely around 9edo, however, and leave much to be desired in terms of melody, the former having too little distinction between steps and the latter being too commatic for many purposes (though it can be considered an aberrismic scale). This can be remedied partially by spacing out Orwell chains by another interval (such as 3\40), or by taking a subset of either MOS.

One example of such a subset will be provided: 5-4-9-5-4-5-8 is a heptatonic subset of Orwell[9] which retains both perfect fifths while simulating the 2L 5s scale of 9edo and providing melodic contrast between steps. This can also be considered to be an approximation of pelog tunings.

Diminished and Blackwood

Two scale families of note are generated by the interval 5/4 (13\40) against a period of either 1/4 or 1/5 of the octave. Interestingly, in either case, 5/4 is a 90¢ semitone away from a period, and so both types of scales can be considered to be generated by this interval as well as 5/4. Useful mappings for this interval include 21/20, 20/19, and 19/18, and it should be noted that when stacked thrice, it forms 7/6. As 5/4 stacked twice, 11/7, also occurs aplenty in these scales, the implication is that these scales work well with the dual-3 dual-7 (dual-11) interpretation of 40edo as 2.9.5.21.(33.)19.

1/4 of the octave can be interpreted as 6/5 by 40edo's patent val, and the temperament this represents is called Diminished. More accurately, this interval represents 25/21~19/16. As 5/4 rests a 90¢ semitone above a quarter-octave, taken together, these imply that the flat fifth (690¢) is found at 5/4 plus a quarter-octave; while the Blackwood fourth (480¢, identified with 21/16) is found at a quarter-octave up two semitones. Scales of Diminished include an 8-note (3 - 7 in a period), 12-note (3 - 3 - 4 in a period), and a 16-note scale (3 - 3 - 3 - 1 in a period), corresponding to a depth of 1 (including the flat fifth and 5/4), 2 (including 10/9, 11/7 and the Blackwood fifth), and 3 semitones (including 7/6) respectively.

2/5 of the octave can be interpreted as 4/3 by the 40b val, and the temperament this represents is called Blackwood. In the dual interpretation, this interval instead represents 21/16. Noting that 5/4 rests a 90¢ semitone below the Blackwood fourth, and 1/5 of the octave represents 55/48, the interval 12/11 exists at a semitone below a single period. Scales of Blackwood include a 10-note (5 - 3 in a period), and a 15-note (2 - 3 - 3 in a period), corresponding to a depth of 1 (including 12/11 and 5/4), and 2 semitones (including 10/9 and 11/7) respectively.

13edo-derived muddles

A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The principal generated MOS is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a well-temperament of 13edo - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, MOS muddles.

As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing 3L 4s/3L 7s, and oneirotonic respectively. The former sequence is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

Multiples

As 40edo's primes 5, 13, and 19 are relatively accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 40edo which preserve elements of its structure. The supersets listed below also have the advantage of their step size being an integer number of cents.

80edo

Doubling 40edo is the obvious solution to the issue of its inaccurate dual fifths, with 80edo correcting the mapping of primes 3, 7, and 11 in accordance with the dual-fifth interpretation of 40edo, although it has a strong sharp tendency. 80edo is more notable for highly accurate representations of certain specific intervals, such as 6/5 (0.64¢ flat), 9/7 (0.087¢ flat), 17/16 (0.045¢ sharp), and most incredibly, 11/10 (0.004¢ flat), and as a tuning for temperaments such as Diaschismic and Echidna.

Approximation of prime harmonics in 80edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+3.0	+3.7	+6.2	+3.7	-0.5	+0.0	+2.5	+1.7	+5.4	-5.0
	Relative (%)	0.0	+20.3	+24.6	+41.2	+24.5	-3.5	+0.3	+16.6	+11.5	+36.2	-33.6
Steps (reduced)		80 (0)	127 (47)	186 (26)	225 (65)	277 (37)	296 (56)	327 (7)	340 (20)	362 (42)	389 (69)	396 (76)

120edo

120edo splits the octave into three, and includes the familiar 700¢ fifth of 12edo. As 40edo's prime 7 is close to 1/3 of a step off, 120edo tunes it near just. 40edo's 5 and 7/6 become high in relative error at this resolution, but 120edo supports all of these mappings. 120edo also serves as an optimized tuning of Myna.

Approximation of prime harmonics in 120edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-2.0	+3.7	+1.2	-1.3	-0.5	-5.0	+2.5	+1.7	+0.4	+5.0
	Relative (%)	0.0	-19.6	+36.9	+11.7	-13.2	-5.3	-49.6	+24.9	+17.3	+4.2	+49.6
Steps (reduced)		120 (0)	190 (70)	279 (39)	337 (97)	415 (55)	444 (84)	490 (10)	510 (30)	543 (63)	583 (103)	595 (115)

200edo

200edo's most notable feature is its highly accurate 3/2, being the smallest EDO with a better approximation thereof than 53edo. It, somewhat conveniently, splits this fifth into nine, allowing it to tune Carlos Alpha. In particular, with the 5/4 inherited from 40edo, it tunes 5-limit Valentine. However, the patent val chooses to inherit 5/4 and 7/6 from 50edo rather than 40.

Approximation of prime harmonics in 200edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+0.0	-2.3	-2.8	+0.7	-0.5	-3.0	+2.5	+1.7	+2.4	+1.0
	Relative (%)	0.0	+0.7	-38.6	-47.1	+11.4	-8.8	-49.3	+41.4	+28.8	+40.4	+16.1
Steps (reduced)		200 (0)	317 (117)	464 (64)	561 (161)	692 (92)	740 (140)	817 (17)	850 (50)	905 (105)	972 (172)	991 (191)