EDT

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EDT: Equal Division of the Tritave (3/1, perfect twelfth). Also sometimes written as ed3.

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 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. 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 first such arrangement to be seriously studied and made into music. 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 simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [<1 1 2|, <0 2 -1|] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses MOS of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.

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 other no twos rank two temperament which 13edt "supports" is Arcturus, which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.

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.

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, 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 (and arbitrarily silly for the purposes of xenharmony, even with octaves) 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:


Individual pages for EDTs

0...99

(some pages do not exist yet)

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...199

(many pages do not exist yet)

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...299

(many pages do not exist yet)

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

A list of tritave reduced harmonics for easy comparison of JI and temperaments in tritave based systems.

Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html

EDT-EDO 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 7-limit.
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.
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¢r.
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.

Multiples of 13EDT which approximate EDO

Also, 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.

See also