40edo: Difference between revisions
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'''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''', 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 primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. | 40edo can be considered a [[straddle primes|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (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, most notably its approximations to [[7/6]] and [[5/4]], each just over 3{{c}} sharp. 40edo's [[11-limit]] is a tuning for [[Orwell|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]]. | |||
== General theory == | == General theory == | ||
=== JI approximation === | === 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{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates | 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{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} 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. | 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. | ||
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=== Intervals and notation === | === Intervals and notation === | ||
As 40edo's diatonic fifth is so flat, its | 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 | 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. | 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. | ||
Revision as of 09:22, 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. 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, most notably its approximations to 7/6 and 5/4, each just over 3¢ sharp. 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.
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).
| 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)
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 | D# | J# | |
| 2 | 60 | 26/25 | 33/32 | Dx | Jx, Kbb | |
| 3 | 90 | [20/19] | 52/49, 19/18, 21/20 | 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 | 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 | 26/19, 15/11, 49/36 | 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 | 72/49, 22/15, 19/13 | 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 | 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 | 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
Chords
Scales
| 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 • 106 • 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 | |
