EDT
The equal division of the tritave or twelfth (EDT) or 3rd harmonic (ED3) is a tuning obtained by dividing the 3rd harmonic in a certain number of equal steps.
Introduction
Western music generally revolves around the principle of octave equivalence: notes an octave apart are often perceived in western music as being the same chroma but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "tritave".
It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the tonotopic representation of the mammalian auditory system^{[dead link]} is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics such as clarinets. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.
The Bohlen-Pierce (BP) scale, most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the second such arrangement to be seriously studied and made into music, the first being the Obikhod pitch set of the Russian Orthodox Church which seems to have been by extension of the diatonic scale. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.
Rank two temperaments
If factors of two are eliminated, the search for consonant intervals begins with the odd harmonic series, 1:3:5:7:9:.... We can take the second tritave of the series, 3:5:7:9, and find within it the two isoharmonic triads 3:5:7 and 5:7:9; the analogy here is with the third octave of the full harmonic series, 4:5:6:7:8, and the isoharmonic triad 4:5:6, the foundation of triadic harmony in 5-limit theory. Hence, 3:5:7 or 5:7:9 can be viewed as the fundamental consonant triad of no-twos music, and if we then apply the 5-limit analogy one more time, these triads are bounded by the intervals 7/3 or 9/5 respectively, either of them filling the role of the "fifth" in diatonicism.
The standard Bohlen-Pierce theory takes 3:5:7 to be the fundamental triad, and therefore naturally goes together with scales generated by 7/3, or equivalently 9/7 (the latter being convention), against the tritave. 7/3 generates pentatonic (4L 1s) and enneatonic (4L 5s) MOS scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from 7\9 to 3\4, 13edt is the smallest equal temperament supporting it, and can be seen as an equivalent of 12edo. However, 13edt's accuracy in the 3.5.7 subgroup is much better than 12edo's in the 5-limit, more comparable to that of 31edo. Therefore, higher multiples of 13edt remain excellent 3.5.7 subgroup tunings as well, and can be used to introduce higher harmonics (39edt is especially notable in this regard, with a good representation of both the 11th and 13th harmonics).
The linear temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is Bohlen-Pierce-Stearns, which tempers out the comma 245/243 and thereby equates the interval 5/3 to two generators down (81/49 considering tritave-reduction) - therefore flattening 7/3 by a fraction of this comma. It is also the 4 & 9 temperament in the 3.5.7 subgroup, and for these reasons serves a function very analogous to that of meantone in the 5-limit.
If we instead take 9/5, or more simply 5/3, as a generator, the temperament supported by 13edt is Arcturus, which equates 7/3, two tritaves up (i.e. 21/1) to six steps of 5/3. Naively, 5/3 as generator would be the most natural application of the Pythagorean principle of using the next higher prime harmonic (5) as a generator against the tritave. However, a larger MOS scale is needed to get full use out of the 7th harmonic, and due to the proximity of 5/3 to half the tritave, most simple MOS scales of Arcturus are quite hard. It is advisable to use (2L 9s) or (2L 11s) scales - and therefore, higher EDTs such as 28edt or 41edt.
The named but not necessarily no twos rank two temperament which 13EDT "supports" is Sirius, which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.
At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called Canopus begins to predominate. This has a mapping [⟨1 3 3], ⟨0 -5 -4]] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.
The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the 5L 3s unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.
Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale.
Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDT supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.
For example, CompactStar suggested the alternative fundamental chord 11:13:15 to avoid the highly-dissonant 7/5 tritone present in the simpler 3:5:7 chord, with the best temperament for this being Electra temperament. 15EDT very well approximates the 5th and 13th harmonics, and 12EDT, the 13th and 17th. 39EDT makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the [⟨13 19 23 0 2], ⟨0 0 0 1 1]] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc.
One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2. For example making the fifth harmonic 5:3 a "major sixth" by conventional pitch class terminology.
There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.
Below is a large list of EDTs; additionally, some equal divisions of the tritave are known by alternate names or have special interest:
- 3edt (Liese generator)
- 4edt (Vulture generator)
- 5edt (Tritave counterpart of Magic)
- 6edt (Tritave counterpart of Hanson)
- 7edt (Tritave counterpart of Orwell)
- 8edt (Tritave counterpart of Vulture)
- 11edt "Euler Temperament"
- "Bohlen-Pierce" or "BP"
- 15edt (Mowgli generator)
- "Bernhard Stopper"
- 39edt Triple Bohlen-Pierce (Erlich)
Individual pages for EDTs
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |
50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 |
60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 |
70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 |
80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 |
90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 |
100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 |
110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 |
120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129 |
130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 |
140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 |
150 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 |
160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 |
170 | 171 | 172 | 173 | 174 | 175 | 176 | 177 | 178 | 179 |
180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 |
190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 |
200 | 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | 209 |
210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 |
220 | 221 | 222 | 223 | 224 | 225 | 226 | 227 | 228 | 229 |
230 | 231 | 232 | 233 | 234 | 235 | 236 | 237 | 238 | 239 |
240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 |
250 | 251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 |
260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 |
270 | 271 | 272 | 273 | 274 | 275 | 276 | 277 | 278 | 279 |
280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 |
290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 |
- 300 and beyond
- A list of tritave reduced harmonics for easy comparison of JI and temperaments in tritave-based systems.
- Also may be found convenient: Nonoctave.com | Tuning | Equal Divisions of the Twelfth
EDT-EDO correspondences
It is useful to consider EDTs that both closely and poorly approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).
Otherwise, one can speak of EDTs that correspond to a diatonic val (i.e. the EDT's size is some EDO added to an approximation of 3/2 in that EDO that is a diatonic generator), which is equivalent to the EDT's approximation of 2/1 generating the 8L 3s scale against the tritave, therefore being between 5\8edt and 7\11edt.
EDTs with this property include 19, 27, 30, 35, 38, 41, 43, 46, 49, 51, 52, 54, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73 to 76, 78, 79, 81 to 87, and all greater than 88.
EDTs without a diatonic val are 1 to 7, 9, 10, 12 to 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 34, 36, 37, 39, 42, 45, 47, 50, 53, 58, 61, and 69.
Borderline cases (i.e. EDTs corresponding to a heptatonic or pentatonic fifth) are 8, 11, 16, 22, 24, 32, 33, 40, 44, 48, 55, 56, 64, 66, 72, 77, 80, and 88.
Correspondences are explained in more detail in the table below.
Multiples of 13EDT which approximate EDO
On the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly 33EDO and 78EDT is very nearly 49EDO, while 65EDT is practically identical to 41EDO.
Table of correspondences
EDT | EDO | Comments |
---|---|---|
8edt | 5edo | 8edt is equivalent to 5edo with ~11 cent octave compression. Equivalently, 5edo is 8edt with ~18 cent stretched tritaves. Patent vals match through the 13-limit. |
9edt | Neither 9edt nor 10edt is equivalent to 6edo. | |
10edt | ||
11edt | 7edo | 11edt is equivalent to 7edo with ~10 cent stretched octaves. Patent vals differ in the 7-limit, but neither can really be said to represent the 7th harmonic with a straight face. |
12edt | 12edt entirely misses 2/1, falling halfway between 7 and 8 edos. | |
13edt | The equal-tempered BP scale cannot be considered equivalent to 8edo. | |
14edt | 9edo | There is a lot of mismatch between the pure-octave and pure-tritave tunings, but the patent vals match through the 13-limit. Great for stretched-octave pelog! |
15edt | 15edt entirely misses 2/1, falling halfway between 9 and 10 edos. | |
16edt | 10edo | Similar situation to 8edt~5edo. Patent vals match through the 17-limit. |
17edt | Neither 17edt nor 18edt is equivalent to 11edo. | |
18edt | ||
19edt | 12edo | 19edt is 12edo with ~1.2 cent octave stretch. Patent vals match through the 31-limit, with the exception of 11. |
20edt | Neither 20edt nor 21edt is equivalent to 13edo. | |
21edt | ||
22edt | 14edo | Similar situation to 11edt~7edo, but the equivalence is rough. Patent vals match through the 11-limit, with the exception of 5 (which neither represents well). |
23edt | 23edt entirely misses 2/1, falling halfway between 14 and 15 edos. | |
24edt | 15edo | This is only a rough correspondence, as the (8n)edt ~ (5n)edo sequence begins to break down. Patent vals match through the 13-limit, with the exception of 7. |
25edt | 16edo | Also only a rough correspondence; 25edt corresponds to 16edo with ~17 cent octave stretch, and patent vals match through the 5-limit. |
26edt | Double BP scale entirely misses 2/1, falling halfway between 16 and 17 edos. | |
27edt | 17edo | 27edt is 17edo with ~2.5 cent compressed octaves. With the exception of 5 (which neither represents well), patent vals match through the 13-limit. |
28edt | Neither 28edt nor 29edt is equivalent to 18edo. | |
29edt | ||
30edt | 19edo | 30edt is 19edo with ~4.6 cent stretched octaves. Patent vals match through the 7-limit. |
31edt | 31edt entirely misses 2/1, falling halfway between 19 and 20 edos. | |
32edt | 32edt cannot be considered equivalent to 20edo. | |
33edt | 33edt cannot be considered equivalent to 21edo. | |
34edt | 34edt entirely misses 2/1, falling halfway between 21 and 22 edos. | |
35edt | 22edo | 35edt is 22edo with ~4.5 cent compressed octaves. Patent vals match through the 11-limit. |
36edt | Neither 36edt nor 37edt is equivalent to 23edo, although step of 36edt is close to step recommended for 23edo and octave stretching. | |
37edt | ||
38edt | 24edo | Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 19-limit. |
39edt | Neither 39edt (Triple BP scale) nor 40edt is equivalent to 25edo. | |
40edt | ||
41edt | 26edo | 41edt is 26edo with ~6.1 cent stretched octaves. Patent vals match through the 7-limit. |
42edt | 42edt falls exactly halfway between 26 and 27 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\53 of an octave. | |
43edt | 27edo | 43edt is 27edo with ~5.7 cent compressed octaves. Patent vals match through the 7-limit. |
44edt | Neither 44edt nor 45edt is equivalent to 28edo. | |
45edt | ||
46edt | 29edo | 46edt is 29edo with ~0.94 cent compressed octaves. Patent vals match through the 89-limit. |
47edt | Neither 47edt nor 48edt is equivalent to 30edo. | |
48edt | ||
49edt | 31edo | 49edt is 31edo with ~3.3 cent stretched octaves. Patent vals match through the 11-limit. |
50edt | 50edt falls exactly halfway between 31 and 32 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\63 of an octave. | |
51edt | 32edo | 51edt is 32edo with ~6.6 cent octave compression. Patent vals match through the 11-limit, with the exception of 5. |
52edt | 33edo | 52edt is 33edo with ~7 cent octave stretch (rough correspondence). Patent vals differ in the 5-limit. |
53edt | 53edt falls exactly halfway between 33 and 34 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\67 of an octave. | |
54edt | 34edo | Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 17-limit, with the exception of 7. |
55edt | Neither 55edt nor 56edt is equivalent to 35edo. | |
56edt | ||
57edt | 36edo | Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 31-limit, with the exception of 11. |
58edt | 58edt falls exactly halfway between 36 and 37 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it resembles the scale with generator 2\73 of an octave. | |
59edt | 37edo | 59edt is 37edo with ~7.2 cent octave compression (rough correspondence). Patent vals match through the 5-limit. |
60edt | 38edo | Same ~4.6 cent octave stretch as 30edt~19edo. Patent vals match through the 5-limit. |
61edt | 61edt falls exactly halfway between 38 and 39 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\77 of an octave. | |
62edt | 39edo | 62edt is 39edo with ~3.6 cent compressed octaves. Patent vals match through the 5-limit. |
63edt | 40edo | 63edt is 40edo with ~7.6 cent stretched octaves (rough correspondence). Patent vals match through the 11-limit, with the exception of 5. |
64edt | 64edt falls exactly halfway between 40 and 41 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it resembles the scale with generator 2\81 of an octave. | |
65edt | 41edo | 65edt is 41edo with ~0.31 cent compressed octaves. Patent vals match through the 19-limit. |
66edt | 66edt falls exactly halfway between 41 and 42 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it resembles the scale with generator 2\83 of an octave. | |
67edt | 42edo | 67edt is 42edo with ~7.3 cent compressed octaves (rough correspondence). Patent vals match through the 5-limit, though the 5s cannot be said to match with a straight face. |
68edt | 43edo | 68edt is 43edo with ~2.7 cent stretched octaves. Patent vals match through the 5-limit. |
69edt | 69edt falls exactly halfway between 43 and 44 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\87 of an octave. | |
70edt | 44edo | Same ~4.5 cent octave compression as 35edt~22edo. Patent vals match through the 7-limit, with the exception of 5, though the 7s cannot be said to match with a straight face. |
71edt | 45edo | 71edt is is 45edo with ~4.5 cent stretched octaves (rough correspondence). Patent vals match through the 7-limit, with the exception of 5. |
72edt | 72edt falls exactly halfway between 45 and 46 edos. It is the last edt which entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\91 of an octave. | |
73edt | 46edo | 73edt is 46edo with ~1.5 cent compressed octaves. Patent vals match through the 17-limit. |
74edt | Neither 74edt nor 75edt is equivalent to 47edo. | |
75edt | ||
76edt | 48edo | Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 11-limit. |
77edt | 77edt falls exactly halfway between 48 and 49 edos, so it strongly resembles the scale with generator 2\97 of an octave, but technically does not entirely miss 2/1 due to having a step of ~24.7¢. | |
78edt | 49edo | 78edt is 49edo with ~5.2 cent compressed octaves (rough correspondence). Patent vals match through the 11-limit, though the 77s rather than either the 7s or 11s individually can be said to match with a straight face. |
79edt | 50edo | 79edt is 50edo with ~3.8 cent stretched octaves. Patent vals match through the 7-limit. |
80edt | 80edt falls exactly halfway between 50 and 51 edos, so it strongly resembles the scale with generator 2\101 of an octave, but technically does not entirely miss 2/1 due to having a step of ~23.8¢. | |
81edt | 51edo | Same ~2.5 cent octave compression as 27edt~17edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
82edt | 52edo | Same ~6.1 cent octave stretch as 41edt~26edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
83edt | 83edt falls exactly halfway between 52 and 53 edos. so it resembles the scale with generator 2\105 of an octave, but technically does not entirely miss 2/1 due to having a step of ~22.9¢. | |
84edt | 53edo | 84edt is 53edo with ~0.04 cent stretched octaves. Patent vals match through the 61-limit. |
85edt | 85edt falls exactly halfway between 53 and 54 edos, so it resembles the scale with generator 2\107 of an octave, but technically does not entirely miss 2/1 due to having a step of ~22.4¢. | |
86edt | 54edo | Same ~5.7 cent octave compression as 43edt~27edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
87edt | 55edo | 87edt is 55edo with ~2.4 cent stretched octaves. Patent vals match through the 11-limit, with the exception of 5. |
88edt | 88edt falls exactly halfway between 55 and 56 edos, so it strongly resembles the scale with generator 2\111 of an octave, but technically does not entirely miss 2/1 due to having a step of ~21.6¢. | |
89edt | 56edo | 89edt is 56edo with ~3.3 cent compressed octaves. Patent vals match through the 5-limit. |
90edt | 57edo | Same ~4.6 cent octave stretch as 30edt~19edo (rough correspondence). Patent vals match through the 5-limit. |
91edt | 91edt falls exactly halfway between 57 and 58 edos, so it strongly resembles the scale with generator 2\115 of an octave, but technically does not entirely miss 2/1 due to having a step of ~20.9¢. | |
92edt | 58edo | Same ~0.94 cent octave compression as 46edt~29edo. Patent vals match through the 17-limit. |
93edt | Neither 93edt nor 94edt is equivalent to 59edo. | |
94edt | ||
95edt | 60edo | Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 7-limit. |
96edt | 96edt falls exactly halfway between 60 and 61 edos, so it strongly resembles the scale with generator 2\121 of an octave, but technically does not entirely miss 2/1 due to having a step of ~19.8¢. | |
97edt | 61edo | 97edt is 61edo with ~3.9 cent compressed octaves (rough correspondence). Patent vals match through the 5-limit. |
98edt | 62edo | Same ~3.3 cent octave compression as 49edt~31edo. Patent vals match through the 23-limit. |
99edt | 99edt falls exactly halfway between 62 and 63 edos, so it strongly resembles the scale with generator 2\125 of an octave, but technically does not entirely miss 2/1 due to having a step of ~19.2¢. | |
100edt | 63edo | 100edt is 63edo with ~1.8 cent compressed octaves. Patent vals match through the 23-limit. |
101edt | Neither 101edt nor 102edt is equivalent to 64edo. | |
102edt | ||
103edt | 65edo | 103edt is 65edo with ~2.4 cent stretched octaves. Patent vals match through the 181-limit, with the exception of 13. |
104edt | 104edt falls exactly halfway between 65 and 66 edos, so it resembles the scale with generator 2\131 of an octave, but technically does not entirely miss 2/1 due to having a step of ~18.3¢. | |
105edt | 66edo | Same ~4.5 cent octave compression as 35edt~22edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
106edt | 67edo | 106edt is 67edo with ~2.2 cent stretched octaves, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
107edt | 107edt falls exactly halfway between 67 and 68 edos, so it strongly resembles the scale with generator 2\135 of an octave, but technically does not entirely miss 2/1 due to having a step of ~17.8¢. | |
108edt | 68edo | Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 7-limit. |
109edt | 69edo | 109edt is 69edo with ~4 cent stretched octaves (rough correspondence). Patent vals match through the 5-limit. |
110edt | 110edt falls exactly halfway between 69 and 70 edos, so it resembles the scale with generator 2\139 of an octave, but technically does not entirely miss 2/1 due to having a step of ~17.3¢. | |
111edt | 70edo | 111edt is 70edo with ~0.57 cent compressed octaves. Patent vals match through the 67-limit. |
112edt | Neither 112edt nor 113edt is equivalent to 71edo. | |
113edt | ||
114edt | 72edo | Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 19-limit. |
115edt | 115edt falls exactly halfway between 72 and 73 edos, so it strongly resembles the scale with generator 2\145 of an octave, but technically does not entirely miss 2/1 due to having a step of ~16.6¢. | |
116edt | 73edo | 116edt is 73edo with ~3.1 cent compressed octaves. Patent vals match through the 11-limit, though products of of any two of 5, 7 and 11 rather than 5, 7 and 11 themselves can be said to match with a straight face. |
117edt | 74edo | 117edt is 74edo with ~2.95 cent stretched octaves, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
118edt | 118edt falls exactly halfway between 74 and 75 edos, so it strongly resembles the scale with generator 2\149 of an octave, but technically does not entirely miss 2/1 due to having a step of ~16.1¢. | |
119edt | 75edo | 119edt is 75edo with ~1.3 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11. |
120edt | 76edo | Same ~4.6 cent octave stretch as 30edt~19edo (rough correspondence). Patent vals match through the 7-limit, though the 7s cannot be said to match with a straight face. |
121edt | 121edt falls exactly halfway between 76 and 77 edos, so it resembles the scale with generator 2\153 of an octave, but technically does not entirely miss 2/1 due to having a step of ~15.7¢. | |
122edt | 77edo | 122edt is 77edo with ~0.41 cent stretched octaves. Patent vals match through the 37-limit. |
123edt | Same ~6.1 cent octave stretch as 41edt~26edo, but actually more strongly resembles the scale with generator 2\155 of an octave. | |
124edt | 78edo | Same ~3.6 cent octave compression as 62edt~39edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
125edt | 79edo | 125edt is 79edo with ~2 cent stretched octaves. Patent vals match through the 13-limit, with the exception of 7. |
126edt | 126edt falls exactly halfway between 79 and 80 edos, so it strongly resembles the scale with generator 2\159 of an octave, but technically does not entirely miss 2/1 due to having a step of ~15.1¢. | |
127edt | 80edo | 127edt is 80edo with ~1.9 cent compressed octaves. Patent vals match through the 11-limit. |
128edt | 81edo | 128edt is 81edo with ~3.6 cent stretched octaves. Patent vals match through the 7-limit, though the 5s cannot be said to match with a straight face. |
129edt | Same ~5.7 cent octave compression as 43edt~27edo, but actually more strongly resembles the scale with generator 2\163 of an octave. | |
130edt | 82edo | Same ~0.31 cent octave compression as 65edt~41edo. Patent vals match through the 43-limit, with the exception of 13. |
131edt | 131edt falls exactly halfway between 82 and 83 edos, so it resembles the scale with generator 2\165 of an octave, but technically does not entirely miss 2/1 due to having a step of ~14.5¢. | |
132edt | 83edo | 132edt is 83edo with ~4.1 cent compressed octaves (rough correspondence). Patent vals match through the 5-limit. |
133edt | 84edo | Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 7-limit. |
134edt | 134edt falls exactly halfway between 84 and 85 edos, so it strongly resembles the scale with generator 2\169 of an octave, but technically does not entirely miss 2/1 due to having a step of ~14.2¢. | |
135edt | 85edo | Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 7-limit, with the exception of 5. |
136edt | 86edo | Same ~2.7 cent octave stretch as 68edt~43edo. |
137edt | 137edt falls exactly halfway between 86 and 87 edos, so it strongly resembles the scale with generator 2\173 of an octave, but technically does not entirely miss 2/1 due to having a step of ~13.9¢. | |
138edt | 87edo | Same ~0.94 cent octave compression as 46edt~29edo. |
139edt | 139edt is 88edo with a ~4.1 cent stretched octave, but also 175ed4 with a ~5.55 cent compressed 4/1. | |
140edt | Same ~4.5 cent octave compression as 35edt~22edo, but actually equally strongly resembles the scale with generator 2\177 of an octave. | |
141edt | 89edo | 141edt is 89edo with ~0.52 cent stretched octaves. |
142edt | 142edt falls exactly halfway between 89 and 90 edos, so it strongly resembles the scale with generator 2\179 of an octave, but technically does not entirely miss 2/1 due to having a step of ~13.4¢. | |
143edt | 90edo | 143edt is 90edo with ~3 cent compressed octaves. |
144edt | 91edo | 144edt is 91edo with ~1.9 cent stretched octaves. |
145edt | 145edt falls exactly halfway between 91 and 92 edos, so it strongly resembles the scale with generator 2\183 of an octave, but technically does not entirely miss 2/1 due to having a step of ~13.1¢. | |
146edt | 92edo | Same ~1.5 cent octave compression as 73edt~46edo. |
147edt | 93edo | Same ~3.3 cent octave compression as 49edt~31edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
148edt | 148edt falls exactly halfway between 93 and 94 edos, so it strongly resembles the scale with generator 2\187 of an octave, but technically does not entirely miss 2/1 due to having a step of ~12.85¢. | |
149edt | 94edo | 149edt is 94edo with ~0.11 cent compressed octaves. |
150edt | Same ~4.6 cent octave stretch as 30edt~19edo, but actually equally strongly resembles the scale with generator 2\189 of an octave. | |
151edt | 151edt is 95edo with a ~3.4 cent compressed octave, but also 191ed4 with a ~5.7 cent stretched 4/1. | |
152edt | 96edo | Same ~1.2 cent octave stretch as 19edt~12edo. |
153edt | 153edt falls exactly halfway between 96 and 97 edos, so it strongly resembles the scale with generator 2\193 of an octave, but technically does not entirely miss 2/1 due to having a step of ~12.4¢. | |
154edt | 97edo | 154edt is 97edo with ~2 cent compressed octaves. |
155edt | 98edo | 155edt is 98edo with ~2.5 cent stretched octaves. |
156edt | Same ~5.2 cent octave stretch as 78edt~49edo, but actually equally strongly resembles the scale with generator 2\197 of an octave. | |
157edt | 99edo | 157edt is 99edo with ~0.68 cent compressed octaves. |
158edt | Same ~3.6 cent octave stretch as 79edt~50edo, but actually equally strongly resembles the scale with generator 2\199 of an octave. | |
159edt | 159edt is 100edo with a ~3.8 cent compressed octave, but also 201ed4 with a ~4.4 cent stretched 4/1. | |
160edt | 101edo | 160edt is 101edo with ~0.61 cent stretched octaves. |
161edt | 161edt falls exactly halfway between 101 and 102 edos, so it strongly resembles the scale with generator 2\203 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.8¢. | |
162edt | 102edo | Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 5-limit. |
163edt | 103edo | 163edt is 103edo with ~1.85 cent stretched octaves. |
164edt | Same ~6.1 cent octave stretch as 41edt~26edo, but actually more strongly resembles the scale with generator 2\207 of an octave. | |
165edt | 104edo | 165edt is 104edo with ~1.2 cent compressed octaves. |
166edt | 105edo | 166edt is 105edo with ~3 cent stretched octaves. |
167edt | 167edt falls exactly halfway between 105 and 106 edos, so it strongly resembles the scale with generator 2\211 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.4¢. | |
168edt | 106edo | Same ~0.04 cent octave stretch as 84edt~53edo. |
169edt | 169edt falls exactly halfway between 106 and 107 edos, so it strongly resembles the scale with generator 2\213 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.25¢. | |
170edt | 107edo | 170edt is 107edo with ~2.9 cent compressed octaves. |
171edt | 108edo | Same ~1.2 cent octave stretch as 19edt~12edo. |
172edt | 172edt falls exactly halfway between 108 and 109 edos, so it strongly resembles the scale with generator 2\217 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.1¢. | |
173edt | 109edo | 173edt is 109edo with ~1.7 cent compressed octaves. |
174edt | 110edo | 174edt is 110edo with ~2.4 cent stretched octaves. |
175edt | Same ~4.5 cent octave compression as 35edt~22edo, but actually more strongly resembles the scale with generator 2\221 of an octave. | |
176edt | 111edo | 176edt is 111edo with ~0.47 cent compressed octaves. |
177edt | 177edt is 112edo with a ~3.5 cent compressed octave, but also 223ed4 with a ~3.75 cent compressed 4/1. | |
178edt | Same ~3.3 cent octave stretch as 89edt~56edo, but actually more strongly resembles the scale with generator 2\225 of an octave. | |
179edt | 113edo | 179edt is 113edo with ~0.68 cent stretched octaves. |
180edt | Same ~4.6 cent octave stretch as 30edt~19edo, but actually more strongly resembles the scale with generator 2\227 of an octave. | |
181edt | 114edo | 181edt is 114edo with ~2.1 cent compressed octaves. |
182edt | 115edo | 182edt is 115edo with ~1.8 cent stretched octaves. |
183edt | 183edt falls exactly halfway between 115 and 116 edos, so it strongly resembles the scale with generator 2\231 of an octave, but technically does not entirely miss 2/1 due to having a step of ~10.4¢. | |
184edt | 116edo | Same ~0.94 cent octave compression as 46edt~29edo. |
185edt | 117edo | 185edt is 117edo with a ~2.9 cent stretched octave. |
186edt | Same ~3.6 cent octave compression as 62edt~39edo, but actually more strongly resembles the scale with generator 2\235 of an octave. | |
187edt | 118edo | 187edt is 118edo with ~0.16 cent stretched octaves. |
188edt | 188edt is 119edo with ~3.9 cent stretched octaves, but also 237ed4 with an ~2.3 cent compressed 4/1. | |
189edt | 119edo | Same ~2.5 cent octave compression as 27edt~17edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit. |
190edt | 120edo | Same ~1.2 cent octave stretch as 19edt~12edo. |
191edt | 191edt falls exactly halfway between 120 and 121 edos, so it strongly resembles the scale with generator 2\241 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.96¢. | |
192edt | 121edo | 192edt is 121edo with ~1.4 cent compressed octaves. |
193edt | 122edo | 193edt is 122edo with ~2.3 cent stretched octaves. |
194edt | Same ~3.9 cent octave compression as 97edt~61edo, but actually more strongly resembles the scale with generator 2\245 of an octave. | |
195edt | 123edo | Same ~0.31 cent octave compression as 65edt~41edo. |
196edt | Same ~3.3 cent octave compression as 49edt~31edo, but actually more strongly resembles the scale with generator 2\247 of an octave. | |
197edt | 197edt is 124edo with a ~2.9 cent compressed octave, but also 249ed4 with a ~4 cent stretched 4/1. | |
198edt | 125edo | 198edt is 125edo with ~0.73 cent stretched octaves. |
199edt | 199edt falls exactly halfway between 125 and 126 edos, so it strongly resembles the scale with generator 2\251 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.56¢. | |
200edt | 126edo | Same ~1.8 cent octave compression as 100edt~63edo. |
201edt | 127edo | 201edt is 127edo with ~1.7 cent compressed octave. |
202edt | 202edt falls exactly halfway between 127 and 128 edos, so it strongly resembles the scale with generator 2\255 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.42¢. | |
203edt | 128edo | 203edt is 128edo with ~0.74 cent compressed octaves. |
204edt | Same ~2.7 cent octave stretch as 68edt~43edo, but actually more strongly resembles the scale with generator 2\257 of an octave. | |
205edt | 205edt is 129edo with a ~3.2 cent compressed octave, but also 259ed4 with a ~3 cent stretched 4/1. | |
206edt | 130edo | Same ~2.4 cent octave stretch as 103edt~65edo. |
207edt | 207edt falls exactly halfway between 130 and 131 edos, so it strongly resembles the scale with generator 2\261 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.19¢. | |
208edt | 131edo | 208edt is 131edo with ~2.1 cent compressed octaves. |
209edt | 132edo | Same ~1.2 cent octave stretch as 19edt~12edo. |
210edt | 210edt falls exactly halfway between 132 and 133 edos, so it strongly resembles the scale with generator 2\265 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.06¢. | |
211edt | 133edo | 211edt is 133edo with ~1.1 cent compressed octaves. |
212edt | 134edo | Same ~2.2 cent octave stretch as 106edt~67edo, but patent vals surprisingly actually match through the 7-limit, though the 7s nevertheless cannot be said to match with a straight face. |
213edt | 213edt falls exactly halfway between 134 and 135 edos, so it strongly resembles the scale with generator 2\269 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.92¢ | |
214edt | 135edo | 214edt is 135edo with ~0.17 cent compressed octaves. |
215edt | 215edt falls exactly halfway between 135 and 136 edos, so it strongly resembles the scale with generator 2\271 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.85¢. | |
216edt | 136edo | Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 5-limit. |
217edt | 137edo | 217edt is 137edo with ~0.77 cent stretched octaves. |
218edt | 218edt falls exactly halfway between 137 and 138 edos, so it strongly resembles the scale with generator 2\275 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.725¢. | |
219edt | 138edo | Same ~1.5 cent octave compression as 73edt~46edo. |
220edt | 139edo | 220edt is 139edo with ~1.6 cent stretched octaves |
221edt | 221edt falls exactly halfway between 139 and 140 edos, so it strongly resembles the scale with generator 2\279 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.61¢. | |
222edt | 140edo | Same ~0.57 cent octave compression as 111edt~70edo. |
223edt | Neither 223edt nor 224edt is equivalent to 141edo. | |
224edt | ||
225edt | 142edo | 225edt is 142edo with ~0.345 cent stretched octaves. |
226edt | 226edt falls exactly halfway between 142 and 143 edos, so it strongly resembles the scale with generator 2\285 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.42¢. | |
227edt | 143edo | 227edt is 143edo with ~1.85 cent compressed octaves. |
228edt | 144edo | Same ~1.2 cent octave stretch as 19edt~12edo. |
229edt | 229edt falls exactly halfway between 144 and 145 edos, so it strongly resembles the scale with generator 2\279 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.305¢. | |
230edt | 145edo | Same ~0.94 cent octave compression as 46edt~29edo. |
231edt | Neither 231edt nor 232edt is equivalent to 146edo. | |
232edt | ||
233edt | 147edo | 233edt is 147edo with ~.05 cent compressed octaves. |
234edt | Same ~2.95 cent octave stretch as 117edt~74edo, but actually more strongly resembles the scale with generator 2\295 of an octave. | |
235edt | 235edt is 148edo with a ~2.2 cent compressed octave, but also 297ed4 with a ~3.75 cent stretched 4/1. | |
236edt | 149edo | 236edt is 149edo with a ~0.81 cent stretched octave |
237edt | Same ~3.8 cent octave stretch as 79edt~50edo, but actually more strongly resembles the scale with generator 2/299 of an octave. | |
238edt | 150edo | Same ~1.9 cent octave compression as 119edt~75edo, but actually do not start matching patent vals until 11. |
239edt | 151edo | 239edt is 151edo with ~1.65 cent stretched octaves. |
240edt | 240edt falls exactly halfway between 151 and 152 edos, so it strongly resembles the scale with generator 2\303 of an octave, but technically does not entirely miss 2/1 due to having a step of ~7.925¢. | |
241edt | 152edo | 241edt is 152edo with ~0.43 cent compressed octaves. |
242edt | Neither 242edt nor 243edt is equivalent to 153edo. | |
243edt | ||
244edt | 154edo | Same ~0.41 cent octave stretch as 122edt~77edo, but actually do not start matching patent vals until 7. |
245edt | 245edt falls exactly halfway between 154 and 155 edos, so it strongly resembles the scale with generator 2\309 of an octave, but technically does not entirely miss 2/1 due to having a step of ~7.76¢. | |
246edt | 155edo | 246edt is 155edo with ~1.6 cent compressed octaves. |
247edt | 156edo | Same ~1.2 cent octave stretch as 19edt~12edo. |
248edt | 248edt falls exactly halfway between 156 and 157 edos, so it strongly resembles the scale with generator 2\313 of an octave, but technically does not entirely miss 2/1 due to having a step of ~7.67¢. | |
249edt | 157edo | 249edt is 157edo with ~.775 cent compressed octaves. |
250edt | Same ~2 cent octave stretch as 125edt~79edo, but actually more strongly resembles the scale with generator 2\315 of an octave. | |
251edt | 251edt is 158edo with a ~2.8 cent compressed octave, but also 317ed4 with a ~2.1 cent stretched 4/1. | |
252edt | 159edo | Same ~0.04 cent octave stretch as 84edt~53edo. |