40edo: Difference between revisions

From Xenharmonic Reference
← Older editNewer edit →
VisualWikitext
started interval table
added notation
Line 47: Line 47:
|
|
|
|
|
|D
|
|J
|-
|-
|1
|1
Line 55: Line 55:
|
|
|
|
|
|D#
|
|J#
|-
|-
|2
|2
Line 63: Line 63:
|
|
|
|
|
|Dx
|
|Jx, Kbb
|-
|-
|3
|3
Line 71: Line 71:
|
|
|
|
|
|D#x, Ebbb
|
|Kb
|-
|-
|4
|4
Line 79: Line 79:
|
|
|
|
|
|Ebb
|
|K
|-
|-
|5
|5
Line 87: Line 87:
|
|
|
|
|
|Eb
|
|K#
|-
|-
|6
|6
Line 95: Line 95:
|
|
|
|
|
|E
|
|Kx
|-
|7
|210
|210
|26/23, [44/39]
|26/23, [44/39]
|
|
|
|
|
|E#
|
|Lbb
|-
|-
|8
|8
Line 111: Line 109:
|
|
|
|
|
|Ex
|
|Lb
|-
|-
|9
|9
Line 119: Line 117:
|
|
|
|
|
|Fbb
|
|L
|-
|-
|10
|10
Line 127: Line 125:
|
|
|
|
|
|Fb
|
|L#
|-
|-
|11
|11
Line 135: Line 133:
|
|
|
|
|
|F
|
|Lx, Mbb
|-
|-
|12
|12
Line 143: Line 141:
|
|
|
|
|
|F#
|
|Mb
|-
|-
|13
|13
Line 151: Line 149:
|
|
|
|
|
|Fx
|
|M
|-
|-
|14
|14
Line 159: Line 157:
|
|
|
|
|
|F#x, Gbbb
|
|M#
|-
|-
|15
|15
Line 167: Line 165:
|
|
|
|
|
|Gbb
|
|Mx
|-
|-
|16
|16
Line 175: Line 173:
|
|
|
|
|
|Gb
|
|Nbb
|-
|-
|17
|17
Line 183: Line 181:
|
|
|
|
|
|G
|
|Nb
|-
|-
|18
|18
Line 191: Line 189:
|
|
|
|
|
|G#
|
|N
|-
|-
|19
|19
Line 199: Line 197:
|
|
|
|
|
|Gx
|
|N#
|-
|-
|20
|20
Line 207: Line 205:
|
|
|
|
|
|G#x, Abbb
|
|Nx, Obb
|-
|-
|21
|21
Line 215: Line 213:
|
|
|
|
|
|Abb
|
|Ob
|-
|-
|22
|22
Line 223: Line 221:
|
|
|
|
|
|Ab
|
|O
|-
|-
|23
|23
Line 231: Line 229:
|
|
|
|
|
|A
|
|O#
|-
|-
|24
|24
Line 239: Line 237:
|
|
|
|
|
|A#
|
|Ox
|-
|-
|25
|25
Line 247: Line 245:
|
|
|
|
|
|Ax
|
|Pbb
|-
|-
|26
|26
Line 255: Line 253:
|
|
|
|
|
|A#x, Bbbb
|
|Pb
|-
|-
|27
|27
Line 263: Line 261:
|
|
|
|
|
|Bbb
|
|P
|-
|-
|28
|28
Line 271: Line 269:
|
|
|
|
|
|Bb
|
|P#
|-
|-
|29
|29
Line 279: Line 277:
|
|
|
|
|
|B
|
|Px, Qbb
|-
|-
|30
|30
Line 287: Line 285:
|
|
|
|
|
|B#
|
|Qb
|-
|-
|31
|31
Line 295: Line 293:
|
|
|
|
|
|Bx
|
|Q
|-
|-
|32
|32
Line 303: Line 301:
|
|
|
|
|
|Cbb
|
|Q#
|-
|-
|33
|33
Line 311: Line 309:
|
|
|
|
|
|Cb
|
|Qx
|-
|-
|34
|34
Line 319: Line 317:
|
|
|
|
|
|C
|
|Rbb
|-
|-
|35
|35
Line 327: Line 325:
|
|
|
|
|
|C#
|
|Rb
|-
|-
|36
|36
Line 335: Line 333:
|
|
|
|
|
|Cx
|
|R
|-
|-
|37
|37
Line 343: Line 341:
|
|
|
|
|
|C#x, Dbbb
|
|R#
|-
|-
|38
|38
Line 351: Line 349:
|
|
|
|
|
|Dbb
|
|Rx, Jbb
|-
|-
|39
|39
Line 359: Line 357:
|
|
|
|
|
|Db
|
|Jb
|-
|-
|40
|40
Line 367: Line 365:
|
|
|
|
|
|D
|
|J
|}
|}
{{WIP}}
{{WIP}}

Revision as of 04:43, 5 March 2026

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)
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 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 D# J#
2 60 26/25 Dx Jx, Kbb
3 90 [20/19] D#x, Ebbb Kb
4 120 [15/14] Ebb K
5 150 [12/11], 25/23 Eb K#
6 180 39/35 E Kx 210 26/23, [44/39] E# Lbb
8 240 [23/20], 55/48 Ex Lb
9 270 7/6 Fbb L
10 300 19/16 Fb L#
11 330 23/19 F Lx, Mbb
12 360 [16/13] F# Mb
13 390 5/4 Fx M
14 420 32/25, 14/11 F#x, Gbbb M#
15 450 13/10 Gbb Mx
16 480 25/19 Gb Nbb
17 510 66/49 G Nb
18 540 26/19, 15/11 G# N
19 570 39/28, [32/23] Gx N#
20 600 55/39 G#x, Abbb Nx, Obb
21 630 [23/16], 56/39 Abb Ob
22 660 22/15, 19/13 Ab O
23 690 49/33 A O#
24 720 38/25 A# Ox
25 750 20/13 Ax Pbb
26 780 11/7, 25/16 A#x, Bbbb Pb
27 810 8/5 Bbb P
28 840 [13/8] Bb P#
29 870 38/23 B Px, Qbb
30 900 32/19 B# Qb
31 930 12/7 Bx Q
32 960 96/55, [40/23] Cbb Q#
33 990 [39/22], 23/13 Cb Qx
34 1020 70/39 C Rbb
35 1050 46/25, [11/6] C# Rb
36 1080 [28/15] Cx R
37 1110 [19/10] C#x, Dbbb R#
38 1140 25/13 Dbb Rx, Jbb
39 1170 128/65, [112/57], [55/28] Db Jb
40 1200 2/1 D J
This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

Compositional theory

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.


View • Talk • EditEqual temperaments
EDOs
Macrotonal 5 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 68 • 70
72-83 72 • 77 • 80 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9 • 11 • 20
Other
Retrieved from "https://xenreference.com/wiki/index.php?title=40edo&oldid=4536"