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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | de = 37-EDO |
| : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-06-06 07:21:39 UTC</tt>.<br>
| | | en = 37edo |
| : The original revision id was <tt>584864853</tt>.<br>
| | | es = |
| : The revision comment was: <tt></tt><br>
| | | ja = |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | }} |
| <h4>Original Wikitext content:</h4>
| | {{Infobox ET}} |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"><span style="display: block; text-align: right;">[[xenharmonie/37edo|Deutsch]]
| | {{ED intro}} |
| </span>
| |
| 37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th [[xenharmonic/prime numbers|prime]] edo, following [[xenharmonic/31edo|31edo]] and coming before [[xenharmonic/41edo|41edo]].
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| Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[xenharmonic/porcupine|porcupine]] temperament. (It is the optimal patent val for [[Porcupine family#Porcupinefish|porcupinefish]], which is about as accurate as "13-limit porcupine" will be.) Using its alternative flat fifth, it tempers out 16875/16384, making it a [[xenharmonic/negri|negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[xenharmonic/gorgo|gorgo]]/[[xenharmonic/laconic|laconic]]).
| | == Theory == |
| | 37edo has very accurate approximations of [[harmonic]]s [[5/1|5]], [[7/1|7]], [[11/1|11]] and [[13/1|13]], making it a good choice for a [[no-threes subgroup temperaments|no-threes]] approach. Harmonic 11 is particularly accurate, being only 0.03 cents sharp. A usable approximation of [[9/1|9]] is available at 6\37 (194.6 cents) as well, and the no-3 no-15 no-21 [[23-odd-limit]] is represented [[consistent]]ly. |
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| 37 edo is also a very accurate equal tuning for Undecimation Temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a 7L+2s nonatonic MOS, which in 37-edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16 note MOS.
| | This means 37edo is useful in a number of ways. It is accurate on the 2.5.7.11.13 [[subgroup]], where it shares the same tuning as [[111edo]]. In fact, on the larger [[k*N subgroups|3*37 subgroup]], 2.27.5.7.11.13.51.57, it not only shares the same tuning as 19-limit 111edo, but tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as [[74edo]]. The native [[3/2|perfect fifth]] at 22\37 (713.5 cents) can also be used, making it a sharp-tending full [[13-limit]] system, and there is the alternative, very flat fifth at 21\37 (681.1 cents), which generates an [[2L 5s|antidiatonic]] scale. |
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| | === Odd harmonics === |
| | {{Harmonics in equal|37}} |
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| [[toc|flat]] | | === As a tuning of other temperaments === |
| ----
| | Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the [[optimal patent val]] for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as 13-limit porcupine extensions will be. Using its alternative flat fifth, it tempers out [[16875/16384]], making it a [[negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]). |
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| =Subgroups=
| | It is a good tuning of the 2.5.11.13 subgroup temperament [[barton]], especially if it is desirable to avoid approximating the perfect fifth. |
| 37edo offers close approximations to [[xenharmonic/OverToneSeries|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].
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| 12\37 = 389.2 cents
| | 37edo is also a very accurate equal tuning for [[undecimation]] temperament, which has a [[generator]] of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a [[7L 2s]] enneatonic [[mos]], which in 37edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16-note mos. |
| 30\37 = 973.0 cents
| |
| 17\37 = 551.4 cents | |
| 26\37 = 843.2 cents
| |
| [6\37edo = 194.6 cents]
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| This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger [[xenharmonic/k*N subgroups|3*37 subgroup]] 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as 74et.
| | === Subsets and supersets === |
| | 37edo is the 12th [[prime edo]], following [[31edo]] and coming before [[41edo]]. |
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| =The Two Fifths= | | [[74edo]], which doubles it, provides an alternative approximation to harmonic 3 that supports [[meantone]]. [[111edo]], which triples it, gives a very accurate approximation of harmonic 3, and manifests itself as a great higher-limit system. [[296edo]], which slices its step in eight, is a good 13-limit system. |
| The just [[xenharmonic/perfect fifth|perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo: | | |
| | === Dual fifths === |
| | The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo: |
| | |
| | The flat fifth is 21\37 = 681.1 cents (37b val) |
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| The flat fifth is 21\37 = 681.1 cents
| |
| The sharp fifth is 22\37 = 713.5 cents | | The sharp fifth is 22\37 = 713.5 cents |
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| 21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6 | | 21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6 |
| | |
| "minor third" = 10\37 = 324.3 cents | | "minor third" = 10\37 = 324.3 cents |
| | |
| "major third" = 11\37 = 356.8 cents | | "major third" = 11\37 = 356.8 cents |
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| 22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1 | | 22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1 |
| | |
| "minor third" = 8\37 = 259.5 cents | | "minor third" = 8\37 = 259.5 cents |
| | |
| "major third" = 14\37 = 454.1 cents | | "major third" = 14\37 = 454.1 cents |
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| If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[xenharmonic/The Biosphere|Biome]] temperament. | | If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The Biosphere|Oceanfront]] temperament. |
| | |
| | 37edo can only barely be considered as "dual-fifth", because the sharp fifth is 12 cents sharp of 3/2, has a regular diatonic scale, and can be interpreted as somewhat accurate regular temperaments like [[archy]] and the aforementioned oceanfront. In contrast, the flat fifth is 21 cents flat and the only low-limit interpretation is as the very inaccurate [[mavila]]. |
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| Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.
| | Since both fifths do not support [[meantone]], the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo. |
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| 37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below). | | 37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below). |
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| =Intervals= | | === Miscellaneous properties === |
| ||~ Degrees of 37edo ||~ Cents Value ||~ Approximate Ratios
| | 37edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of [[7/4]] where [[9/8]] is mapped no wider than [[8/7]] is 30\37, and the sharpest possible mapping of [[15/8]] where diamond monotone is achieveable is 34\37, where [[15/14]] is equated with [[14/13]][[~]][[13/12]] to half of [[7/6]]. Here [[5/4]] is mapped to 12\37, and [[10/9]] is mapped to 5\37. Equating both [[11/10]] and [[12/11]] with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to [[3/2]]. If the fifth was any sharper, then [[7/4]] and [[15/8]] would have to be flatter. Then 5/4 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see [[Monotonicity limits of small EDOs]]). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]]. |
| of 2.5.7.11.13.27 subgroup ||~ Ratios of 3 with | | |
| a sharp 3/2 ||~ Ratios of 3 with
| | == Intervals == |
| a flat 3/2 ||~ Ratios of 9 with
| | Inconsistent intervals are in ''italics''. |
| 194.59¢ 9/8 ||~ Ratios of 9 with
| | {| class="wikitable center-1 right-2" |
| 227.03¢ 9/8
| | |- |
| (two sharp
| | ! # |
| 3/2's) || | | ! Cents |
| || 0 || 0.00 || 1/1 || || || || ||
| | ! Approximate ratios<br>of 2.27.5.7.11.13 subgroup |
| || 1 || 32.43 || || || || || ||
| | ! Additional ratios of 3<br>with a sharp 3/2 |
| || 2 || 64.86 || 28/27, 27/26 || || || || ||
| | ! Additional ratios of 3<br>with a flat 3/2 |
| || 3 || 97.30 || || || || || ||
| | ! Additional ratios of 9<br>with 194.59 ¢ 9/8 |
| || 4 || 129.73 || 14/13 || 13/12 || 12/11 || || ||
| | |- |
| || 5 || 162.16 || 11/10 || 12/11 || 13/12 || || 10/9 || | | | 0 |
| || 6 || 194.59 || || || || 9/8, 10/9 || || | | | 0.0 |
| || 7 || 227.03 || 8/7 || || || || 9/8 || | | | 1/1 |
| || 8 || 259.46 || || 7/6 || || || || | | | |
| || 9 || 291.89 || 13/11, 32/27 || || 6/5, 7/6 || || || | | | |
| || 10 || 324.32 || || 6/5 || || || 11/9 || | | | |
| || 11 || 356.76 || 16/13, 27/22 || || || 11/9 || || | | |- |
| || 12 || 389.19 || 5/4 || || || || || | | | 1 |
| || 13 || 421.62 || 14/11 || || || 9/7 || || | | | 32.4 |
| || 14 || 454.05 || 13/10 || || || || 9/7 ||
| | | [[55/54]], [[56/55]] |
| || 15 || 486.49 || || 4/3 || || || || | | | |
| || 16 || 518.92 || 27/20 || || 4/3 || || ||
| | | |
| || 17 || 551.35 || 11/8 || || || 18/13 || ||
| | | |
| || 18 || 583.78 || 7/5 || || || || 18/13 ||
| | |- |
| || 19 || 616.22 || 10/7 || || || || 13/9 || | | | 2 |
| || 20 || 648.65 || 16/11 || || || 13/9 || || | | | 64.9 |
| || 21 || 681.08 || 40/27 || || 3/2 || || ||
| | | [[27/26]], [[28/27]] |
| || 22 || 713.51 || || 3/2 || || || ||
| | | |
| || 23 || 745.95 || 20/13 || || || || 14/9 || | | | |
| || 24 || 778.38 || 11/7 || || || 14/9 || || | | | |
| || 25 || 810.81 || 8/5 || || || || ||
| | |- |
| || 26 || 843.24 || 13/8, 44/27 || || || 18/11 || ||
| | | 3 |
| || 27 || 875.68 || || 5/3 || || || 18/11 ||
| | | 97.3 |
| || 28 || 908.11 || 22/13, 27/16 || || 5/3, 12/7 || || ||
| | | [[128/121]], [[55/52]] |
| || 29 || 940.54 || || 12/7 || || || ||
| | | [[16/15]] |
| || 30 || 972.97 || 7/4 || || || || 16/9 ||
| | | |
| || 31 || 1005.41 || || || || 16/9, 9/5 || || | | | |
| || 32 || 1037.84 || 20/11 || 11/6 || 24/13 || || 9/5 ||
| | |- |
| || 33 || 1070.27 || 13/7 || 24/13 || 11/6 || || ||
| | | 4 |
| || 34 || 1102.70 || || || || || ||
| | | 129.7 |
| || 35 || 1135.14 || 27/14, 52/27 || || || || ||
| | | [[14/13]] |
| || 36 || 1167.57 || || || || || || | | | [[13/12]], [[15/14]] |
| | | ''[[12/11]]'' |
| | | |
| | |- |
| | | 5 |
| | | 162.2 |
| | | [[11/10]] |
| | | ''[[10/9]]'', [[12/11]] |
| | | ''[[13/12]]'' |
| | | |
| | |- |
| | | 6 |
| | | 194.6 |
| | | [[28/25]] |
| | | |
| | | |
| | | [[9/8]], [[10/9]] |
| | |- |
| | | 7 |
| | | 227.0 |
| | | [[8/7]] |
| | | ''[[9/8]]'' |
| | | |
| | | |
| | |- |
| | | 8 |
| | | 259.5 |
| | | |
| | | [[7/6]], [[15/13]] |
| | | |
| | | |
| | |- |
| | | 9 |
| | | 291.9 |
| | | [[13/11]], [[32/27]] |
| | | |
| | | ''[[6/5]]'', ''[[7/6]]'' |
| | | |
| | |- |
| | | 10 |
| | | 324.3 |
| | | |
| | | [[6/5]], ''[[11/9]]'' |
| | | |
| | | |
| | |- |
| | | 11 |
| | | 356.8 |
| | | [[16/13]], [[27/22]] |
| | | |
| | | |
| | | [[11/9]] |
| | |- |
| | | 12 |
| | | 389.2 |
| | | [[5/4]] |
| | | |
| | | |
| | | |
| | |- |
| | | 13 |
| | | 421.6 |
| | | [[14/11]], [[32/25]] |
| | | |
| | | |
| | | [[9/7]] |
| | |- |
| | | 14 |
| | | 454.1 |
| | | [[13/10]] |
| | | ''[[9/7]]'' |
| | | |
| | | |
| | |- |
| | | 15 |
| | | 486.5 |
| | | |
| | | [[4/3]] |
| | | |
| | | |
| | |- |
| | | 16 |
| | | 518.9 |
| | | [[27/20]] |
| | | |
| | | ''[[4/3]]'' |
| | | |
| | |- |
| | | 17 |
| | | 551.4 |
| | | [[11/8]] |
| | | [[15/11]] |
| | | |
| | | [[18/13]] |
| | |- |
| | | 18 |
| | | 583.8 |
| | | [[7/5]] |
| | | ''[[18/13]]'' |
| | | |
| | | |
| | |- |
| | | 19 |
| | | 616.2 |
| | | [[10/7]] |
| | | ''[[13/9]]'' |
| | | |
| | | |
| | |- |
| | | 20 |
| | | 648.6 |
| | | [[16/11]] |
| | | [[22/15]] |
| | | |
| | | [[13/9]] |
| | |- |
| | | 21 |
| | | 681.1 |
| | | [[40/27]] |
| | | |
| | | ''[[3/2]]'' |
| | | |
| | |- |
| | | 22 |
| | | 713.5 |
| | | |
| | | [[3/2]] |
| | | |
| | | |
| | |- |
| | | 23 |
| | | 745.9 |
| | | [[20/13]] |
| | | ''[[14/9]]'' |
| | | |
| | | |
| | |- |
| | | 24 |
| | | 778.4 |
| | | [[11/7]], [[25/16]] |
| | | |
| | | |
| | | [[14/9]] |
| | |- |
| | | 25 |
| | | 810.8 |
| | | [[8/5]] |
| | | |
| | | |
| | | |
| | |- |
| | | 26 |
| | | 843.2 |
| | | [[13/8]], [[44/27]] |
| | | |
| | | |
| | | [[18/11]] |
| | |- |
| | | 27 |
| | | 875.7 |
| | | |
| | | [[5/3]], ''[[18/11]]'' |
| | | |
| | | |
| | |- |
| | | 28 |
| | | 908.1 |
| | | [[22/13]], [[27/16]] |
| | | |
| | | ''[[5/3]], [[12/7]]'' |
| | | |
| | |- |
| | | 29 |
| | | 940.5 |
| | | |
| | | [[12/7]], [[26/15]] |
| | | |
| | | |
| | |- |
| | | 30 |
| | | 973.0 |
| | | [[7/4]] |
| | | ''[[16/9]]'' |
| | | |
| | | |
| | |- |
| | | 31 |
| | | 1005.4 |
| | | [[25/14]] |
| | | |
| | | |
| | | [[16/9]], [[9/5]] |
| | |- |
| | | 32 |
| | | 1037.8 |
| | | [[20/11]] |
| | | ''[[9/5]]'', [[11/6]] |
| | | |
| | | |
| | |- |
| | | 33 |
| | | 1070.3 |
| | | [[13/7]] |
| | | [[24/13]], [[28/15]] |
| | | ''[[11/6]]'' |
| | | |
| | |- |
| | | 34 |
| | | 1102.7 |
| | | [[121/64]], [[104/55]] |
| | | [[15/8]] |
| | | |
| | | |
| | |- |
| | | 35 |
| | | 1135.1 |
| | | [[27/14]], [[52/27]] |
| | | |
| | | |
| | | |
| | |- |
| | | 36 |
| | | 1167.6 |
| | | |
| | | |
| | | |
| | | |
| | |- |
| | | 37 |
| | | 1200.0 |
| | | [[2/1]] |
| | | |
| | | |
| | | |
| | |} |
| | |
| | === Proposed interval names and solfèges === |
| | {| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed" |
| | |+ style="white-space: nowrap;" | Table of proposed interval names and solfèges |
| | |- |
| | ! # |
| | ! Cents |
| | ! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]] |
| | |- |
| | | 0 |
| | | 0.0 |
| | | Perfect 1sn |
| | | P1 |
| | | D |
| | |- |
| | | 1 |
| | | 32.4 |
| | | Minor 2nd |
| | | m2 |
| | | Eb |
| | |- |
| | | 2 |
| | | 64.9 |
| | | Upminor 2nd |
| | | ^m2 |
| | | ^Eb |
| | |- |
| | | 3 |
| | | 97.3 |
| | | Downmid 2nd |
| | | v~2 |
| | | ^^Eb |
| | |- |
| | | 4 |
| | | 129.7 |
| | | Mid 2nd |
| | | ~2 |
| | | Ed |
| | |- |
| | | 5 |
| | | 162.2 |
| | | Upmid 2nd |
| | | ^~2 |
| | | vvE |
| | |- |
| | | 6 |
| | | 194.6 |
| | | Downmajor 2nd |
| | | vM2 |
| | | vE |
| | |- |
| | | 7 |
| | | 227.0 |
| | | Major 2nd |
| | | M2 |
| | | E |
| | |- |
| | | 8 |
| | | 259.5 |
| | | Minor 3rd |
| | | m3 |
| | | F |
| | |- |
| | | 9 |
| | | 291.9 |
| | | Upminor 3rd |
| | | ^m3 |
| | | ^F |
| | |- |
| | | 10 |
| | | 324.3 |
| | | Downmid 3rd |
| | | v~3 |
| | | ^^F |
| | |- |
| | | 11 |
| | | 356.8 |
| | | Mid 3rd |
| | | ~3 |
| | | Ft |
| | |- |
| | | 12 |
| | | 389.2 |
| | | Upmid 3rd |
| | | ^~3 |
| | | vvF# |
| | |- |
| | | 13 |
| | | 421.6 |
| | | Downmajor 3rd |
| | | vM3 |
| | | vF# |
| | |- |
| | | 14 |
| | | 454.1 |
| | | Major 3rd |
| | | M3 |
| | | F# |
| | |- |
| | | 15 |
| | | 486.5 |
| | | Perfect 4th |
| | | P4 |
| | | G |
| | |- |
| | | 16 |
| | | 518.9 |
| | | Up 4th, dim 5th |
| | | ^4, d5 |
| | | ^G, Ab |
| | |- |
| | | 17 |
| | | 551.4 |
| | | Downmid 4th, updim 5th |
| | | v~4, ^d5 |
| | | ^^G, ^Ab |
| | |- |
| | | 18 |
| | | 583.8 |
| | | Mid 4th, downmid 5th |
| | | ~4, v~5 |
| | | Gt, ^^Ab |
| | |- |
| | | 19 |
| | | 616.2 |
| | | Mid 5th, upmid 4th |
| | | ~5, ^~4 |
| | | Ad, vvG# |
| | |- |
| | | 20 |
| | | 648.6 |
| | | Upmid 5th, downaug 5th |
| | | ^~5, vA4 |
| | | vvA, vG# |
| | |- |
| | | 21 |
| | | 681.1 |
| | | Down 5th, aug 4th |
| | | v5, A4 |
| | | vA, G# |
| | |- |
| | | 22 |
| | | 713.5 |
| | | Perfect 5th |
| | | P5 |
| | | A |
| | |- |
| | | 23 |
| | | 745.9 |
| | | Minor 6th |
| | | m6 |
| | | Bb |
| | |- |
| | | 24 |
| | | 778.4 |
| | | Upminor 6th |
| | | ^m6 |
| | | ^Bb |
| | |- |
| | | 25 |
| | | 810.8 |
| | | Downmid 6th |
| | | v~6 |
| | | ^^Bb |
| | |- |
| | | 26 |
| | | 843.2 |
| | | Mid 6th |
| | | ~6 |
| | | Bd |
| | |- |
| | | 27 |
| | | 875.7 |
| | | Upmid 6th |
| | | ^~6 |
| | | vvB |
| | |- |
| | | 28 |
| | | 908.1 |
| | | Downmajor 6th |
| | | vM6 |
| | | vB |
| | |- |
| | | 29 |
| | | 940.5 |
| | | Major 6th |
| | | M6 |
| | | B |
| | |- |
| | | 30 |
| | | 973.0 |
| | | Minor 7th |
| | | m7 |
| | | C |
| | |- |
| | | 31 |
| | | 1005.4 |
| | | Upminor 7th |
| | | ^m7 |
| | | ^C |
| | |- |
| | | 32 |
| | | 1037.8 |
| | | Downmid 7th |
| | | v~7 |
| | | ^^C |
| | |- |
| | | 33 |
| | | 1070.3 |
| | | Mid 7th |
| | | ~7 |
| | | Ct |
| | |- |
| | | 34 |
| | | 1102.7 |
| | | Upmid 7th |
| | | ^~7 |
| | | vvC# |
| | |- |
| | | 35 |
| | | 1135.1 |
| | | Downmajor 7th |
| | | vM7 |
| | | vC# |
| | |- |
| | | 36 |
| | | 1167.6 |
| | | Major 7th |
| | | M7 |
| | | C# |
| | |- |
| | | 37 |
| | | 1200.0 |
| | | Perfect 8ve |
| | | P8 |
| | | D |
| | |} |
| | |
| | == Notation == |
| | === Stein–Zimmermann–Gould notation === |
| | [[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows: |
| | {{Sharpness-sharp6-szg}} |
| | |
| | If double arrows are not desirable, arrows can be attached to quarter-tone accidentals: |
| | {{Sharpness-sharp6-qt-szg}} |
| | |
| | === Kite's ups and downs notation === |
| | 37edo can also be notated using [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. |
| | {{Sharpness-sharp6a}} |
| | |
| | Half-sharps and half-flats can be used to avoid triple arrows: |
| | {{Sharpness-sharp6b}} |
| | |
| | === Ivan Wyschnegradsky's notation === |
| | Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used: |
| | {{Sharpness-sharp6-iw}} |
| | |
| | === Sagittal notation === |
| | This notation uses the same sagittal sequence as edos [[23edo #Second-best fifth notation|23b]], [[30edo #Sagittal notation|30]], and [[44edo #Sagittal notation|44]]. |
| | |
| | ==== Evo and Revo flavors ==== |
| | <imagemap> |
| | File:37-EDO_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]] |
| | default [[File:37-EDO_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Alternative Evo flavor ==== |
| | <imagemap> |
| | File:37-EDO_Alternative_Evo_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]] |
| | default [[File:37-EDO_Alternative_Evo_Sagittal.svg]] |
| | </imagemap> |
| | |
| | ==== Evo-SZ flavor ==== |
| | <imagemap> |
| | File:37-EDO_Evo-SZ_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 300 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]] |
| | default [[File:37-EDO_Evo-SZ_Sagittal.svg]] |
| | </imagemap> |
| | |
| | == Approximation to JI == |
| | === Interval mappings === |
| | {{Q-odd-limit intervals|37}} |
| | |
| | == Regular temperament properties == |
| | {| class="wikitable center-4 center-5 center-6" |
| | |- |
| | ! rowspan="2" | [[Subgroup]] |
| | ! rowspan="2" | [[Comma list]] |
| | ! rowspan="2" | [[Mapping]] |
| | ! rowspan="2" | Optimal<br>8ve stretch (¢) |
| | ! colspan="2" | Tuning error |
| | |- |
| | ! [[TE error|Absolute]] (¢) |
| | ! [[TE simple badness|Relative]] (%) |
| | |- |
| | | 2.5 |
| | | {{Monzo| 86 -37 }} |
| | | {{Mapping| 37 86 }} |
| | | −0.619 |
| | | 0.619 |
| | | 1.91 |
| | |- |
| | | 2.5.7 |
| | | 3136/3125, 4194304/4117715 |
| | | {{Mapping| 37 86 104 }} |
| | | −0.905 |
| | | 0.647 |
| | | 2.00 |
| | |- |
| | | 2.5.7.11 |
| | | 176/175, 1375/1372, 65536/65219 |
| | | {{Mapping| 37 86 104 128 }} |
| | | −0.681 |
| | | 0.681 |
| | | 2.10 |
| | |- |
| | | 2.5.7.11.13 |
| | | 176/175, 640/637, 847/845, 1375/1372 |
| | | {{Mapping| 37 86 104 128 137 }} |
| | | −0.692 |
| | | 0.610 |
| | | 1.88 |
| | |} |
| | * 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next equal temperaments doing better in these subgroups are 109, 581, 103, 124 and 93, respectively. |
| | |
| | === Rank-2 temperaments === |
| | * [[List of 37et rank two temperaments by badness]] |
| | |
| | {| class="wikitable center-1 center-2" |
| | |- |
| | ! Generator* |
| | ! Cents* |
| | ! In patent val |
| | ! In 37b val |
| | |- |
| | | 1\37 |
| | | 32.4 |
| | | |
| | | |
| | |- |
| | | 2\37 |
| | | 64.9 |
| | | [[Sycamore]] |
| | | |
| | |- |
| | | 3\37 |
| | | 97.3 |
| | | [[Passion]] |
| | | |
| | |- |
| | | 4\37 |
| | | 129.7 |
| | | [[Twothirdtonic]] |
| | | [[Negri]] (37bd, out-of-tune) |
| | |- |
| | | 5\37 |
| | | 162.2 |
| | | [[Porcupine]] / [[porcupinefish]] |
| | | |
| | |- |
| | | 6\37 |
| | | 194.6 |
| | | [[Hemiwürschmidt]] / [[hemiwur]] |
| | | [[Hemithirds]] (37b, out-of-tune) |
| | |- |
| | | 7\37 |
| | | 227.0 |
| | | [[Semaja]] / [[gorgik]] |
| | | [[Gorgo]] (37b) |
| | |- |
| | | 8\37 |
| | | 259.5 |
| | | |
| | | [[Semaphore]] (37bd, out-of-tune) |
| | |- |
| | | 9\37 |
| | | 291.9 |
| | | [[Quasitemp]] |
| | | |
| | |- |
| | | 10\37 |
| | | 324.3 |
| | | [[Hyperkleismic]] |
| | | [[Superkleismic]] (37bc, out-of-tune) |
| | |- |
| | | 11\37 |
| | | 356.8 |
| | | [[Beatles]] |
| | | |
| | |- |
| | | 12\37 |
| | | 389.2 |
| | | [[Würschmidt]] (out-of-tune) |
| | | |
| | |- |
| | | 13\37 |
| | | 421.6 |
| | | [[Skwares]] (37dd, out-of-tune) |
| | | |
| | |- |
| | | 14\37 |
| | | 454.1 |
| | | [[Ammonite]] |
| | | |
| | |- |
| | | 15\37 |
| | | 486.5 |
| | | [[Ultrapyth]] |
| | | |
| | |- |
| | | 16\37 |
| | | 518.9 |
| | | [[Undecimation]] |
| | | [[Shallowtone]] (37b) |
| | |- |
| | | 17\37 |
| | | 551.4 |
| | | [[Freivald]], [[emka]] |
| | | |
| | |- |
| | | 18\37 |
| | | 583.8 |
| | | [[Cotritone]] |
| | | |
| | |} |
| | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct |
| | |
| | == Octave stretch or compression == |
| | 37edo's [[prime]]s 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Some compressed-octave 37edo tunings (least to most compressed) include [[161zpi]], [[ed5|86ed5]], [[ed7|104ed7]], [[ed12|133ed12]] or [[ed6|96ed6]]. |
| | |
| | == Scales == |
| | ''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]'' |
| | |
| | === [[MOS scale]]s === |
| | * [[Ammonite]][21]: 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1 |
| | * [[Beatles]][7]: 4 7 4 7 4 7 4 |
| | * Beatles[10]: 4 3 4 4 3 4 4 4 3 4 |
| | * Beatles[17]: 3 1 3 1 3 3 1 3 1 3 1 3 3 1 3 1 3 |
| | * Ultrapyth[5] (quasi-[[equipentatonic]]): 7 8 7 8 7 (''recommended mode: 8 7 7 8 7'') |
| | * Ultrapyth[7]: 7 1 7 7 7 1 7 |
| | * Ultrapyth[12]: 1 6 1 6 1 6 1 1 6 1 6 1 |
| | * Ultrapyth[17]: 1 5 1 1 1 5 1 1 5 1 1 5 1 1 1 5 1 (''great as a [[dual-fifth]] scale'') |
| | * Ultrapyth[22]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1 1 4 1 1 (''great as a [[dual-fifth]] scale'') |
| | * Passion[9]: 13 3 3 3 3 3 3 3 3 |
| | * Passion[12]: 3 3 3 3 3 3 4 3 3 3 3 3 |
| | * Passion[25]: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 (''great as a [[dual-fifth]] scale'') |
| | * Porcupine[5]: 5 17 5 5 5 |
| | * Porcupine[6]: 12 5 5 5 5 5 |
| | * Porcupine[7]: 5 5 5 7 5 5 5 |
| | * Porcupine[15]: 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 |
| | * Porcupine[22]: 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 |
| | * Twothirdtonic[7]: 13 4 4 4 4 4 4 |
| | * Twothirdtonic[8]: 9 4 4 4 4 4 4 4 |
| | * Twothirdtonic[10]: 4 4 4 4 1 4 4 4 4 4 |
| | * Twothirdtonic[19]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 |
| | |
| | === Scales by individuals === |
| | {| class="wikitable mw-collapsible mw-collapsed" |
| | |+[[Budjarn Lambeth]]'s scales |
| | |'''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''' |
| | |
| | |
| | * Opalised ammonite{{idio}} (modmos of Ammonite[8]): 5 4 6 5 2 5 4 6 |
| | * [[User:BudjarnLambeth/Antechinus|Antechinus]]{{idio}} (''nonoctave period'') |
| | * [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]]{{idio}} (''octave-reduced ver.: 5 3 13 9 7'') |
| | * Approximated [[pelog]] lima: 4 5 12 4 12 |
| | |
| | |
| | * Flattened ionian pentatonic: 12 3 6 12 4 |
| | * Flattened major: 6 6 3 6 6 6 4 |
| | * Flattened major pentatonic: 6 6 9 6 10 |
| | * Sharpened natural minor: 7 3 6 6 3 6 6 |
| | * Sharpened harmonic minor: 7 3 6 6 3 9 3 |
| | * Sharpened pentatonic minor: 10 6 6 9 6 |
| | * Superharmonic minor pentatatonic I: 7 3 12 13 2 |
| | * Superharmonic minor pentatatonic II: 10 6 6 13 2 |
| | * Flattened hexatonic minor: 6 3 6 6 9 7 |
| | * Flattened phrygian dominant: 2 9 4 6 3 6 7 |
| | * Sharpened blues aeolian hexatonic: 10 6 3 3 3 12 |
| | * Flattened blues aeolian pentatonic: 9 6 6 3 13 |
| | * Sharpened blues aeolian pentatonic: 10 12 3 6 6 |
| | * Sharpened blues dorian hexatonic: 10 6 6 6 3 6 |
| | * Extrasharp blues dorian hexatonic: 10 6 6 6 4 5 |
| | * Roughened augmented: 10 2 10 2 11 2 |
| | * Flattened cosmic: 15 6 3 6 7 (''approximated from [[32afdo]]'') |
| | * Sharpened Hirajoshi: 7 3 12 3 12 |
| | * Sharpened Akebono I: 7 3 12 6 9 |
| | * Roughened Javanese pentachordal: 2 8 9 2 16 |
| | * Sharpened underpass: 10 12 7 2 6 (''approximated from [[10afdo]]'') |
| | |
| | |
| | * ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]'' |
| | * ''The scales listed in: [[Oceanfront scales]]'' (not all Budjarn's) |
| | |} |
| | |
| | === Equally spaced scales === |
| | * [[37ed4]] (''every 2 steps''): 2 2 2... |
| | * [[Square root of 13 over 10]] (''every 7 steps''): 7 7 7... |
| | * ''Every 8 steps (see below)'' |
| | |
| | === Every 8 steps of 37edo === |
| | {| class="wikitable center-1 right-2" |
| | |+ |
| | !Degrees |
| | !Cents |
| | !Approximate Ratios<br>of 6.7.11.20.27 subgroup |
| | !Additional Ratios |
| | |- |
| | |0 |
| | |0.000 |
| | |[[1/1]] |
| | | |
| | |- |
| | |1 |
| | |259.46 |
| | |[[7/6]] |
| | | |
| | |- |
| | |2 |
| | |518.92 |
| | |[[27/20]] |
| | | |
| | |- |
| | |3 |
| | |778.38 |
| | |[[11/7]] |
| | | |
| | |- |
| | |4 |
| | |1037.84 |
| | |[[20/11]], [[11/6]] |
| | | |
| | |- |
| | |5 |
| | |1297.30 |
| | | |
| | |[[19/9]] |
| | |- |
| | |6 |
| | |1556.76 |
| | |[[27/11]] |
| | | |
| | |- |
| | |7 |
| | |1816.22 |
| | |[[20/7]] |
| | | |
| | |- |
| | |8 |
| | |2075.68 |
| | |[[10/3]] |
| | | |
| | |- |
| | |9 |
| | |2335.14 |
| | |[[27/7]] |
| | | |
| | |- |
| | |10 |
| | |2594.59 |
| | |[[9/2]] |
| | | |
| | |- |
| | |11 |
| | |2854.05 |
| | | |
| | |[[26/5]] |
| | |- |
| | |12 |
| | |3113.51 |
| | |[[6/1]] |
| | | |
| | |- |
| | |13 |
| | |3372.97 |
| | |[[7/1]] |
| | | |
| | |- |
| | |14 |
| | |3632.43 |
| | | |
| | | |
| | |- |
| | |15 |
| | |3891.89 |
| | | |
| | |[[19/2]] |
| | |- |
| | |16 |
| | |4151.35 |
| | |[[11/1]] |
| | | |
| | |- |
| | |17 |
| | |4410.81 |
| | | |
| | | |
| | |- |
| | |18 |
| | |4670.27 |
| | | |
| | | |
| | |- |
| | |19 |
| | |4929.73 |
| | | |
| | | |
| | |- |
| | |20 |
| | |5189.19 |
| | |[[20/1]] |
| | | |
| | |- |
| | |21 |
| | |5448.65 |
| | | |
| | | |
| | |- |
| | |22 |
| | |5708.11 |
| | |[[27/1]] |
| | | |
| | |} |
| | |
| | == Instruments == |
| | |
| | ; Lumatone |
| | * [[Lumatone mapping for 37edo]] |
| | |
| | ; Fretted instruments |
| | * [[Skip fretting system 37 2 7]] |
| | |
| | == Music == |
| | === Modern renderings === |
| | ; {{W|Alessandro Marcello}} and {{w|Johann Sebastian Bach}} |
| | * [https://www.youtube.com/watch?v=HTAobydvC20 ''Oboe Concerto in D minor'', BWV 974] (1715) – arranged for oboe & organ by [[Claudi Meneghin]] (2022) |
| | |
| | ; {{W|Pietro Domenico Paradies}} |
| | * [https://www.youtube.com/watch?v=AJ2sa-fRqbE "Toccata" from ''Harpsichord Sonata in A major''] – arranged for organ by Claudi Meneghin (2023) |
| | |
| | === 21st century === |
| | ; [[Beheld]] |
| | * [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023) |
| | |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/TEzitpGJvt0 ''37edo''] (2023) |
| | * [https://www.youtube.com/shorts/e7dLJTsS3PQ ''37edo''] (2025) |
| | * [https://www.youtube.com/shorts/m9hmiH8zong ''37edo jam''] (2025) |
| | * [https://www.youtube.com/shorts/mVRbcB2hoBU ''37edo prelude''] (2026) |
| | * [https://www.youtube.com/shorts/Jt6_r6r3lGY ''37edo improv''] (2026) |
| | |
| | ; [[Francium]] |
| | * [https://www.youtube.com/watch?v=jpPjVouoq3E ''5 days in''] (2023) |
| | * [https://www.youtube.com/watch?v=ngxSiuVadls ''A Dark Era Arises''] (2023) – in Porcupine[15], 37edo tuning |
| | * [https://www.youtube.com/watch?v=U93XFJJ1aXw ''Two Faced People''] (2025) – in Twothirdtonic[10], 37edo tuning |
| | |
| | ; [[groundfault]] |
| | * From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube] |
| | ** "The Life Unreachable" |
| | ** "Not This Time" |
| | |
| | ; [[Andrew Heathwaite]] |
| | * From [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012) |
| | ** "Shorn Brown" |
| | ** "Jellybear" |
| | |
| | ; [[Aaron Krister Johnson]] |
| | * [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 ''Toccata Bianca 37EDO'']{{dead link}} |
| | |
| | ; [[JUMBLE]] |
| | * [https://www.youtube.com/watch?v=taT1DClJ2KM ''Tyrian and Gold''] (2024) |
| | |
| | ; [[Fitzgerald Lee]] |
| | * [https://www.youtube.com/watch?v=Nr0cUJcL4SU ''Bittersweet End''] (2025) |
|
| |
|
| =Scales= | | ; [[Mandrake]] |
| | * [https://www.youtube.com/watch?v=iL_4nRZBJDc ''What if?''] (2023) |
|
| |
|
| [[xenharmonic/MOS Scales of 37edo|MOS Scales of 37edo]] | | ; [[Claudi Meneghin]] |
| | * [https://www.youtube.com/watch?v=7dU8eyGbt9I ''Deck The Halls''] (2022) |
| | * [https://www.youtube.com/watch?v=hpjZZXFM_Fk ''Little Fugue on Happy Birthday''] (2022) – in Passion, 37edo tuning |
| | * [https://www.youtube.com/watch?v=SgHY3snZ5bs ''Fugue on an Original Theme''] (2022) |
|
| |
|
| [[xenharmonic/roulette6|roulette6]] | | ; [[Micronaive]] |
| [[xenharmonic/roulette7|roulette7]] | | * [https://www.youtube.com/watch?v=TMVRYLvg_cA No.27.50] (2022) |
| [[xenharmonic/roulette13|roulette13]]
| |
| [[xenharmonic/roulette19|roulette19]]
| |
|
| |
|
| [[xenharmonic/Chromatic pairs#Shoe|Shoe]] | | ; [[Herman Miller]] |
| | * [https://soundcloud.com/morphosyntax-1/luck-of-the-draw ''Luck of the Draw''] (2023) |
|
| |
|
| [[xenharmonic/37ED4|37ED4]] | | ; [[Joseph Monzo]] |
| | * [https://youtube.com/watch?v=QERRKsbbWUQ ''The Kog Sisters''] (2014) |
| | * [https://www.youtube.com/watch?v=BfP8Ig94kE0 ''Afrikan Song''] (2016) |
|
| |
|
| [[xenharmonic/square root of 13 over 10|The Square Root of 13/10]] | | ; [[Mundoworld]] |
| | * ''Reckless Discredit'' (2021) – [https://www.youtube.com/watch?v=ovgsjSoHOkg YouTube] · [https://mundoworld.bandcamp.com/track/reckless-discredit Bandcamp] |
|
| |
|
| =Linear temperaments= | | ; [[Ray Perlner]] |
| [[List of 37et rank two temperaments by badness]] | | * [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] (2020) – in Porcupine, 37edo tuning |
| | * [https://www.youtube.com/watch?v=j8C9ECvfyQM ''Fugue for Brass in 37EDO sssLsss "Dingoian"''] (2022) – in Porcupine[7], 37edo tuning |
| | * [https://www.youtube.com/watch?v=_xfvNKUu8gY ''Fugue for Klezmer Band in 37EDO Porcupine<nowiki>[</nowiki>7<nowiki>]</nowiki> sssssLs "Lemurian"''] (2023) – in Porcupine[7], 37edo tuning |
|
| |
|
| ||~ Generator ||~ "Sharp 3/2" temperaments ||~ "Flat 3/2" temperaments (37b val) ||
| | ; [[Phanomium]] |
| || 1\37 || || ||
| | * [https://www.youtube.com/watch?v=2otxZqUrvHc ''Elevated Floors''] (2025) |
| || 2\37 || [[xenharmonic/Sycamore family|Sycamore]] || ||
| | * [https://www.youtube.com/watch?v=BbexOU-9700 ''cat jam 37''] (2025) |
| || 3\37 || [[xenharmonic/Passion|Passion]] || ||
| |
| || 4\37 || [[xenharmonic/Twothirdtonic|Twothirdtonic]] || [[xenharmonic/Negri|Negri]] ||
| |
| || 5\37 || [[xenharmonic/Porcupine|Porcupine]]/[[xenharmonic/The Biosphere#Oceanfront-Oceanfront%20Children-Porcupinefish|porcupinefish]] || ||
| |
| || 6\37 |||| [[xenharmonic/Chromatic pairs#Roulette|Roulette]] ||
| |
| || 7\37 || [[xenharmonic/Semaja|Semaja]] || [[xenharmonic/Gorgo|Gorgo]]/[[xenharmonic/Laconic|Laconic]] ||
| |
| || 8\37 || || [[semiphore|Semiphore]] ||
| |
| || 9\37 || || ||
| |
| || 10\37 || || ||
| |
| || 11\37 || [[xenharmonic/Beatles|Beatles]] || ||
| |
| || 12\37 || [[xenharmonic/Würschmidt|Würschmidt]] (out-of-tune) || ||
| |
| || 13\37 || || ||
| |
| || 14\37 || [[xenharmonic/Ammonite|Ammonite]] || ||
| |
| || 15\37 || [[The Biosphere#Oceanfront-Oceanfront%20Children-Ultrapyth|Ultrapyth]], **not** [[xenharmonic/superpyth|superpyth]] || ||
| |
| || 16\37 || || **Not** [[xenharmonic/mavila|mavila]] (this is "undecimation") ||
| |
| || 17\37 || [[xenharmonic/Emka|Emka]] || ||
| |
| || 18\37 || || ||
| |
|
| |
|
| | ; [[Togenom]] |
| | * "Canals of Mars" from ''Xenharmonics, Vol. 5'' (2024) – [https://open.spotify.com/track/7v2dpCjiRKUfVVBZw8aWSf Spotify] |[https://togenom.bandcamp.com/track/canals-of-mars Bandcamp] | [https://www.youtube.com/watch?v=qPcEl_bifC0 YouTube] |
|
| |
|
| ==Music in 37edo==
| | ; [[Uncreative Name]] |
| [[http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3|Toccata Bianca 37edo]] by [[http://www.akjmusic.com/|Aaron Krister Johnson]] | | * [https://www.youtube.com/watch?v=rE9L56yZ1Kw ''Winter''] (2025) |
| [[@http://andrewheathwaite.bandcamp.com/track/shorn-brown|Shorn Brown]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3|play]] and [[@http://andrewheathwaite.bandcamp.com/track/jellybear|Jellybear]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3|play]] by [[xenharmonic/Andrew Heathwaite|Andrew Heathwaite]]
| |
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3|The Kog Sisters]] by [[Joe Monzo]]
| |
| ==Links==
| |
| [[http://tonalsoft.com/enc/number/37-edo/37edo.aspx|37edo at Tonalsoft]]</pre></div>
| |
| <h4>Original HTML content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>37edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/37edo">Deutsch</a><br />
| |
| </span><br />
| |
| 37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th <a class="wiki_link" href="http://xenharmonic.wikispaces.com/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31edo</a> and coming before <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41edo</a>.<br />
| |
| <br />
| |
| Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/porcupine">porcupine</a> temperament. (It is the optimal patent val for <a class="wiki_link" href="/Porcupine%20family#Porcupinefish">porcupinefish</a>, which is about as accurate as &quot;13-limit porcupine&quot; will be.) Using its alternative flat fifth, it tempers out 16875/16384, making it a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/negri">negri</a> tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/gorgo">gorgo</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/laconic">laconic</a>).<br />
| |
| <br />
| |
| 37 edo is also a very accurate equal tuning for Undecimation Temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a 7L+2s nonatonic MOS, which in 37-edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16 note MOS.<br />
| |
| <br />
| |
| <br />
| |
| <!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Subgroups">Subgroups</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#The Two Fifths">The Two Fifths</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
| |
| <!-- ws:end:WikiTextTocRule:22 --><hr />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->Subgroups</h1>
| |
| 37edo offers close approximations to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/OverToneSeries">harmonics</a> 5, 7, 11, and 13 [and a usable approximation of 9 as well].<br />
| |
| <br />
| |
| 12\37 = 389.2 cents<br />
| |
| 30\37 = 973.0 cents<br />
| |
| 17\37 = 551.4 cents<br />
| |
| 26\37 = 843.2 cents<br />
| |
| [6\37edo = 194.6 cents]<br />
| |
| <br />
| |
| This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">3*37 subgroup</a> 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as 74et.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="The Two Fifths"></a><!-- ws:end:WikiTextHeadingRule:2 -->The Two Fifths</h1>
| |
| The just <a class="wiki_link" href="http://xenharmonic.wikispaces.com/perfect%20fifth">perfect fifth</a> of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:<br />
| |
| <br />
| |
| The flat fifth is 21\37 = 681.1 cents<br />
| |
| The sharp fifth is 22\37 = 713.5 cents<br />
| |
| <br />
| |
| 21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6<br />
| |
| &quot;minor third&quot; = 10\37 = 324.3 cents<br />
| |
| &quot;major third&quot; = 11\37 = 356.8 cents<br />
| |
| <br />
| |
| 22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1<br />
| |
| &quot;minor third&quot; = 8\37 = 259.5 cents<br />
| |
| &quot;major third&quot; = 14\37 = 454.1 cents<br />
| |
| <br />
| |
| If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Biosphere">Biome</a> temperament.<br />
| |
| <br />
| |
| Interestingly, the &quot;major thirds&quot; of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.<br />
| |
| <br />
| |
| 37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below).<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | ; [[Stephen Weigel]] |
| <tr>
| | * [https://www.youtube.com/watch?v=71yBnSVBsJk ''Leap Day Cloo''] (2025) |
| <th>Degrees of 37edo<br />
| |
| </th>
| |
| <th>Cents Value<br />
| |
| </th>
| |
| <th>Approximate Ratios<br />
| |
| of 2.5.7.11.13.27 subgroup<br />
| |
| </th>
| |
| <th>Ratios of 3 with<br />
| |
| a sharp 3/2<br />
| |
| </th>
| |
| <th>Ratios of 3 with<br />
| |
| a flat 3/2<br />
| |
| </th>
| |
| <th>Ratios of 9 with<br />
| |
| 194.59¢ 9/8<br />
| |
| </th>
| |
| <th>Ratios of 9 with<br />
| |
| 227.03¢ 9/8<br />
| |
| (two sharp<br /> | |
| 3/2's)<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0.00<br />
| |
| </td>
| |
| <td>1/1<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>32.43<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>64.86<br />
| |
| </td>
| |
| <td>28/27, 27/26<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>97.30<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>129.73<br />
| |
| </td>
| |
| <td>14/13<br />
| |
| </td>
| |
| <td>13/12<br />
| |
| </td>
| |
| <td>12/11<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>162.16<br />
| |
| </td>
| |
| <td>11/10<br />
| |
| </td>
| |
| <td>12/11<br />
| |
| </td>
| |
| <td>13/12<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>10/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>194.59<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>9/8, 10/9<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>227.03<br />
| |
| </td>
| |
| <td>8/7<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>9/8<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>259.46<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>7/6<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>291.89<br />
| |
| </td>
| |
| <td>13/11, 32/27<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>6/5, 7/6<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>324.32<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>6/5<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>11/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>356.76<br />
| |
| </td>
| |
| <td>16/13, 27/22<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>11/9<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>389.19<br />
| |
| </td>
| |
| <td>5/4<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>421.62<br />
| |
| </td>
| |
| <td>14/11<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>9/7<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>454.05<br />
| |
| </td>
| |
| <td>13/10<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>9/7<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>486.49<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>4/3<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>518.92<br />
| |
| </td>
| |
| <td>27/20<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>4/3<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>551.35<br />
| |
| </td>
| |
| <td>11/8<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>18/13<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>583.78<br />
| |
| </td>
| |
| <td>7/5<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>18/13<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>616.22<br />
| |
| </td>
| |
| <td>10/7<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>13/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>648.65<br />
| |
| </td>
| |
| <td>16/11<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>13/9<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>681.08<br />
| |
| </td>
| |
| <td>40/27<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>3/2<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>713.51<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>3/2<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>745.95<br />
| |
| </td>
| |
| <td>20/13<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>14/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>778.38<br />
| |
| </td>
| |
| <td>11/7<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>14/9<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>810.81<br />
| |
| </td>
| |
| <td>8/5<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>843.24<br />
| |
| </td>
| |
| <td>13/8, 44/27<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>18/11<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>875.68<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>5/3<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>18/11<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>908.11<br />
| |
| </td>
| |
| <td>22/13, 27/16<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>5/3, 12/7<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>940.54<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>12/7<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>972.97<br />
| |
| </td>
| |
| <td>7/4<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>16/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
| |
| <td>1005.41<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>16/9, 9/5<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32<br />
| |
| </td>
| |
| <td>1037.84<br />
| |
| </td>
| |
| <td>20/11<br />
| |
| </td>
| |
| <td>11/6<br />
| |
| </td>
| |
| <td>24/13<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>9/5<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>33<br />
| |
| </td>
| |
| <td>1070.27<br />
| |
| </td>
| |
| <td>13/7<br />
| |
| </td>
| |
| <td>24/13<br />
| |
| </td>
| |
| <td>11/6<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>34<br />
| |
| </td>
| |
| <td>1102.70<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>35<br />
| |
| </td>
| |
| <td>1135.14<br />
| |
| </td>
| |
| <td>27/14, 52/27<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>36<br />
| |
| </td>
| |
| <td>1167.57<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | ; [[Xeno*n*]] |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->Scales</h1>
| | * [https://www.youtube.com/watch?v=_m5u4VviMXw ''Galantean Drift''] (2025) |
| <br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS%20Scales%20of%2037edo">MOS Scales of 37edo</a><br />
| |
| <br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/roulette6">roulette6</a><br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/roulette7">roulette7</a><br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/roulette13">roulette13</a><br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/roulette19">roulette19</a><br />
| |
| <br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chromatic%20pairs#Shoe">Shoe</a><br />
| |
| <br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/37ED4">37ED4</a><br />
| |
| <br />
| |
| <a class="wiki_link" href="http://xenharmonic.wikispaces.com/square%20root%20of%2013%20over%2010">The Square Root of 13/10</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Linear temperaments</h1>
| |
| <a class="wiki_link" href="/List%20of%2037et%20rank%20two%20temperaments%20by%20badness">List of 37et rank two temperaments by badness</a><br />
| |
| <br />
| |
|
| |
|
| | == See also == |
| | * [[User:Unque/37edo Composition Theory|Unque's approach]] |
|
| |
|
| <table class="wiki_table">
| | == External links == |
| <tr>
| | * [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37-edo / 37-et / 37-tone equal-temperament] on [[Tonalsoft Encyclopedia]] |
| <th>Generator<br />
| |
| </th>
| |
| <th>&quot;Sharp 3/2&quot; temperaments<br />
| |
| </th>
| |
| <th>&quot;Flat 3/2&quot; temperaments (37b val)<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>1\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Sycamore%20family">Sycamore</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Passion">Passion</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Twothirdtonic">Twothirdtonic</a><br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Negri">Negri</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Porcupine">Porcupine</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Biosphere#Oceanfront-Oceanfront%20Children-Porcupinefish">porcupinefish</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6\37<br />
| |
| </td>
| |
| <td colspan="2"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chromatic%20pairs#Roulette">Roulette</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Semaja">Semaja</a><br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Gorgo">Gorgo</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Laconic">Laconic</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/semiphore">Semiphore</a><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Beatles">Beatles</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/W%C3%BCrschmidt">Würschmidt</a> (out-of-tune)<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ammonite">Ammonite</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/The%20Biosphere#Oceanfront-Oceanfront%20Children-Ultrapyth">Ultrapyth</a>, <strong>not</strong> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/superpyth">superpyth</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><strong>Not</strong> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/mavila">mavila</a> (this is &quot;undecimation&quot;)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17\37<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Emka">Emka</a><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18\37<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | [[Category:Listen]] |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Linear temperaments-Music in 37edo"></a><!-- ws:end:WikiTextHeadingRule:10 -->Music in 37edo</h2>
| |
| <a class="wiki_link_ext" href="http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3" rel="nofollow">Toccata Bianca 37edo</a> by <a class="wiki_link_ext" href="http://www.akjmusic.com/" rel="nofollow">Aaron Krister Johnson</a><br />
| |
| <a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shorn-brown" rel="nofollow" target="_blank">Shorn Brown</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3" rel="nofollow">play</a> and <a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/jellybear" rel="nofollow" target="_blank">Jellybear</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Andrew%20Heathwaite">Andrew Heathwaite</a><br />
| |
| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3" rel="nofollow">The Kog Sisters</a> by <a class="wiki_link" href="/Joe%20Monzo">Joe Monzo</a><br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Linear temperaments-Links"></a><!-- ws:end:WikiTextHeadingRule:12 -->Links</h2>
| |
| <a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/37-edo/37edo.aspx" rel="nofollow">37edo at Tonalsoft</a></body></html></pre></div>
| |