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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-11-09 22:45:49 UTC</tt>.<br>
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| : The original revision id was <tt>273855666</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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="color: #007a23; font-family: 'Times New Roman',Times,serif; font-size: 113%;">39 tone equal temperament</span>=
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| **39-EDO, 39-ED2** or **39-tET** divides the Octave (Duple 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view seems to have attracted the attention of the Armodue school, an Italian group that use the scheme of [[xenharmonic/7L 2s|Superdiatonic]] LLLsLLLLs like a basical scale for notation and theory, implemented in [[xenharmonic/16edo|16-ED2]], and, allied systems ([[xenharmonic/25edo|25-ED2]] [1/3-tone]; [[xenharmonic/41edo|41-ED2]] [1/5-tone]; [[xenharmonic/55edo|55]] and [[xenharmonic/57edo|57]] ED2s [1/7-tones]; [[xenharmonic/71edo|71]] and [[xenharmonic/73edo|73]] ED2s [1/9-tones]; [[xenharmonic/87edo|87]] and [[xenharmonic/89edo|89]] ED2s [1/11-tones] & [[xenharmonic/101edo|101]] and [[xenharmonic/103edo|103]] ED2s [1/13-tones]). **Hornbostel Temperaments** is included too on the list: [[xenharmonic/23edo|23-ED2]] [1/3-tone]; 39-ED2 [1/5-tone]; [[xenharmonic/62edo|62-ED2]] [1/8-tone]; [[xenharmonic/85edo|85-ED2]] [1/11-tone] and larger: [[xenharmonic/131edo|131-ED2]] [1/17-tone]; [[xenharmonic/177edo|177-ED2]] [1/23-tone]; [[xenharmonic/200edo|200-ED2]] [1/26-tone] & [[xenharmonic/223edo|223-ED2]] [1/29-tone]. Note that 101, 131, 177 & 200 ED2s are tempered systems that Alexei Ogolevets was proposing in his List of Temperaments.
| | == Theory == |
| However, its 23\39 fifth, 5.737 Cents sharp, is in much better tune than the Mavila fifth which like all Mavila fifths is very, very flat, in this case, 25 Cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is <39 62 91 110 135|.
| | 39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}. |
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| ==__**39-EDO Intervals**__==
| | A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|Charles Clagget}} (Ireland, 1740?–1820), a little extract [http://ml.oxfordjournals.org/content/76/2/291.extract.jpg here]. |
| || **NOMENCLATURE** || | |
| || * **t** = Semisharp (1/5-tone up)
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| * **b** = Flat (3/5-tone down)
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| * **#** = Sharp (3/5-tone up)
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| * **v** = Semiflat (1/5-tone down) ||
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| || **Degrees** || **Armodue note** || **Cents size** || **[[xenharmonic/Nearest just interval|Nearest Just I]]nterval** || **Cents value** || **Error** ||
| | As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the [[diatonic semitone]] is [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates. |
| || 0 || **1** || 0 || **1/1** || 0 || **None** ||
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| || 1 || 1t || 30.7692 || 57/56 || 30.6421 || +0.1271 ||
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| || 2 || 2b || 61.5385 || 29/28 || 60.7513 || +0.7872 ||
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| || 3 || 1# || 92.3077 || 39/37 || 91.1386 || +1.1691 ||
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| || 4 || 2v || 123.0769 || 44/41 || 122.2555 || +0.8214 ||
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| || 5 || 2 || 153.8462 || 35/32 || 155.1396 || -1.2934 ||
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| || 6 || 2t || 184.6154 || 10/9 || 182.4037 || +2.2117 ||
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| || **7·** || **3b** || **215.3846** || **17/15** || **216.6867** || **-1.3021** ||
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| || 8 || 2# || 246.1538 || 15/13 || 247.7411 || -1.5873 ||
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| || 9 || 3v || 276.9231 || 27/23 || 277.5907 || -0.6676 ||
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| || 10 || 3 || 307.6923 || 43/36 || 307.6077 || +0.0846 ||
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| || 11 || 3t || 338.4615 || 17/14 || 336.1295 || +2.332 ||
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| || **12·** || **4b** || **369.2308** || **26/21** || **369.7468** || **-0.516** ||
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| || 13 || 3# || 400 || 34/27 || 399.0904 || +0.9096 ||
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| || 14 || 4v || 430.7692 || 41/32 || 429.0624 || +1.7068 ||
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| || 15 || 4 || 461.5385 || 30/23 || 459.9944 || +1.5441 ||
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| || 16 || 4t (5v) || 492.3077 || 85/64 || 491.2691 || +1.0386 ||
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| || **17·** || **5** || **523.0769** || **23/17** || **523.3189** || **-0.242** ||
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| || 18 || 5t || 553.8462 || 11/8 || 551.3179 || +2.5283 ||
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| || 19 || 6b || 584.6154 || 7/5 || 582.5122 || +2.1032 ||
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| || 20 || 5# || 615.3846 || 10/7 || 617.4878 || -2.1032 ||
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| || 21 || 6v || 646.1538 || 16/11 || 648.6821 || -2.5283 ||
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| || **22·** || **6** || **676.9231** || **34/23** || **676.6811** || **+0.242** ||
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| || 23 || 6t || 707.6923 || 128/85 || 708.7309 || -1.0386 ||
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| || 24 || 7b || 738.4615 || 23/15 || 740.0056 || -1.5441 ||
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| || 25 || 6# || 769.2308 || 64/41 || 770.9376 || -1.7068 ||
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| || 26 || 7v || 800 || 27/17 || 800.9096 || -0.9096 ||
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| || **27·** || **7** || **830.7692** || **21/13** || **830.2532** || **+0.516** ||
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| || 28 || 7t || 861.5385 || 28/17 || 863.8705 || -2.332 ||
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| || 29 || 8b || 892.3077 || 72/43 || 892.3923 || -0.0846 ||
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| || 30 || 7# || 923.0769 || 46/27 || 922.4093 || +0.6676 ||
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| || 31 || 8v || 953.8462 || 26/15 || 952.2589 || +1.5873 ||
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| || **32·** || **8** || **984.6154** || **30/17** || **983.3133** || **+1.3021** ||
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| || 33 || 8t || 1015.3846 || 9/5 || 1017.5963 || -2.2117 ||
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| || 34 || 9b || 1046.1538 || 64/35 || 1044.8604 || +1.2934 ||
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| || 35 || 8# || 1076.9231 || 41/22 || 1077.7445 || -0.8214 ||
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| || 36 || 9v || 1107.6923 || 74/39 || 1108.8614 || -1.1691 ||
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| || 37 || 9 || 1138.4615 || 56/29 || 1139.2487 || -0.7872 ||
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| || 38 || 9t (1v) || 1169.2308 || 112/57 || 1169.3579 || -0.1271 ||
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| || **39··(or 0)** || **1** || **1200** || **2/1** || **1200** || **None** ||
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| ==__Instruments (prototypes):__==
| | Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat. |
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| [[image:xenharmonic/TECLADO_39-EDO.PNG width="811" height="472" caption="Armodue-Hornbostel 1/5-tone keyboard prototype"]] | | 39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic). |
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| [[image:xenharmonic/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png width="1002" height="307" caption="39-ED2 fretboard visualization"]]
| | === Odd harmonics === |
| [[image:Diapasón_para_Bajo_Precision_ó_Jazz_Tricésanonafónico.png width="1010" height="93" caption="Tricesanonaphonic fretboard for P or J Bass"]]
| | {{Harmonics in equal|39}} |
| [[image:Diapasón_para_Gitarra_StratoTricésanonafónico.png width="998" height="117" caption="Tricesanonaphonic fretboard for standard electric guitars"]]
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| ==**__39 tone equal [[xenharmonic/modes|modes]]__:**==
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| 15 15 9 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 1s|2L 1s]]
| | === Octave stretch === |
| 14 14 11 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 1s|2L 1s]]
| | 39edo's approximations of harmonics 3, 5, 7, and 11 can all be improved by slightly [[octave shrinking|compressing the octave]], using tunings such as [[62edt]] or [[101ed6]]. [[equal tuning|18ed11/8]], a heavily compressed version of 39edo where the harmonics 13 and 17 are brought to tune at the cost of a worse 11, is also a possible choice. |
| 13 13 13 = [[xenharmonic/3edo|3edo]]
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| 11 11 11 6 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 1s|3L 1s]]
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| 10 10 10 9 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 1s|3L 1s]]
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| 11 3 11 11 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 2s|3L 2s (father)]]
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| 11 3 11 3 11 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="color: #660000; cursor: pointer;">[[xenharmonic/3L 2s|3L 2s (father)]]</span>
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| 9 6 9 9 6 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 2s|3L 2s (father)]]
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| 9 6 9 6 9 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="color: #660000; cursor: pointer;">[[xenharmonic/3L 2s|3L 2s (father)]]</span>
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| 9 9 9 9 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/4L 1s|4L 1s (bug)]]
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| 9 3 9 9 9 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="color: #660000; cursor: pointer;">[[xenharmonic/4L 1s|4L 1s (bug)]]</span>
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| 8 8 8 8 7 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/4L 1s|4L 1s (bug)]]
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| 10 3 10 3 10 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 3s|3L 3s (augmented)]]
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| 9 4 9 4 9 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 3s|3L 3s (augmented)]]
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| 8 5 8 5 8 5 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 3s|3L 3s (augmented)]]
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| 7 7 7 7 7 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/5L 1s|5L 1s (Grumpy hexatonic)]] | |
| 7 4 7 7 7 7 - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type <span style="cursor: pointer;">[[xenharmonic/5L 1s|5L 1s (Grumpy hexatonic)]]</span>
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| 3 9 3 9 3 9 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 4s|3L 4s (mosh)]]
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| 5 5 7 5 5 5 7 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 5s|2L 5s (mavila)]]
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| 5 5 5 7 5 5 7 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 5s|2L 5s (mavila)]]
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| 5 7 5 5 7 5 5 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/2L 5s|2L 5s (mavila)]]
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| 6 3 6 6 3 6 6 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/5L 3s|5L 3s (unfair father)]]
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| 5 5 5 5 5 5 5 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 1s|7L 1s (Grumpy octatonic)]]
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| 5 4 5 5 5 5 5 5 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 1s|7L 1s (Grumpy octatonic)]]
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| **5 5 5 2 5 5 5 5 2** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 2s|7L 2s (mavila superdiatonic)]]
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| 5 5 2 5 5 5 2 5 5 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 2s|7L 2s (mavila superdiatonic)]]
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| 5 5 3 5 5 3 5 5 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/6L 3s|6L 3s (unfair augmented)]]
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| 5 4 4 5 4 4 5 4 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 6s|3L 6s (fair augmented)]]
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| 4 4 4 4 4 4 4 4 4 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/9L 1s|9L 1s (Grumpy decatonic)]]
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| 4 4 3 4 4 4 4 4 4 4 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/9L 1s|9L 1s (Grumpy decatonic)]]
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| **3 3 5 3 3 3 5 3 3 3 5** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 8s|3L 8s (undecimal anti-triatonic)]]
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| 3 3 3 3 3 3 3 3 3 3 3 3 3 = [[xenharmonic/13edo|13edo]]
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| **3 3 3 2 3 3 3 3 2 3 3 3 3 2** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/11L 3s|11L 3s (tetradecimal triatonic)]]
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| 3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/9L 6s|9L 6s]]
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| 3 2 3 2 3 2 2 3 2 3 2 3 2 3 2 2 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/7L 9s|7L 9s]]
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| **2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 2 3** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/5L 12s|5L 12s]]
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| 2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/3L 15s|3L 15s]]
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| **3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3** - <span style="cursor: pointer;">[[xenharmonic/MOSScales|MOS]]</span> of type [[xenharmonic/10L 9s|10L 9s]]
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| 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/19L 1s|19L 1s]]
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| 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/17L 5s|17L 5s]]
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| **2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 1** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/16L 7s|16L 7s]]
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| 2 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 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/13L 13s|13L 13s]]
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| **2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1** - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/10L 19s|10L 19s]]
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| 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/8L 23s|8L 23s]]
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| 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 - [[xenharmonic/MOSScales|MOS]] of type [[xenharmonic/6L 27s|6L 27s]]</pre></div>
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| <h4>Original HTML content:</h4>
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| <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>39edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x39 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #007a23; font-family: 'Times New Roman',Times,serif; font-size: 113%;">39 tone equal temperament</span></h1>
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| <strong>39-EDO, 39-ED2</strong> or <strong>39-tET</strong> divides the Octave (Duple 2/1) in 39 equal parts of 30.76923 Cents each one. If we take 22\39 as a fifth, can be used in Mavila Temperament, and from that point of view seems to have attracted the attention of the Armodue school, an Italian group that use the scheme of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%202s">Superdiatonic</a> LLLsLLLLs like a basical scale for notation and theory, implemented in <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16edo">16-ED2</a>, and, allied systems (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/25edo">25-ED2</a> [1/3-tone]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41-ED2</a> [1/5-tone]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/55edo">55</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/57edo">57</a> ED2s [1/7-tones]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/71edo">71</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/73edo">73</a> ED2s [1/9-tones]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/87edo">87</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/89edo">89</a> ED2s [1/11-tones] &amp; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/101edo">101</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/103edo">103</a> ED2s [1/13-tones]). <strong>Hornbostel Temperaments</strong> is included too on the list: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/23edo">23-ED2</a> [1/3-tone]; 39-ED2 [1/5-tone]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/62edo">62-ED2</a> [1/8-tone]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/85edo">85-ED2</a> [1/11-tone] and larger: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/131edo">131-ED2</a> [1/17-tone]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/177edo">177-ED2</a> [1/23-tone]; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/200edo">200-ED2</a> [1/26-tone] &amp; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/223edo">223-ED2</a> [1/29-tone]. Note that 101, 131, 177 &amp; 200 ED2s are tempered systems that Alexei Ogolevets was proposing in his List of Temperaments.<br />
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| However, its 23\39 fifth, 5.737 Cents sharp, is in much better tune than the Mavila fifth which like all Mavila fifths is very, very flat, in this case, 25 Cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, 128/125, and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds 64/63 and 126/125 to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting augene temperament, and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out 99/98 and 121/120 also. This better choice for 39et is &lt;39 62 91 110 135|.<br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x39 tone equal temperament-39-EDO Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>39-EDO Intervals</strong></u></h2>
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| |
|
| <table class="wiki_table">
| | There are also some nearby [[zeta peak index]] (ZPI) tunings which can be used for this same purpose: 171zpi, 172zpi and 173zpi. The main zeta peak index page details all three tunings. |
| <tr>
| |
| <td><strong>NOMENCLATURE</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><ul><li><strong>t</strong> = Semisharp (1/5-tone up)</li><li><strong>b</strong> = Flat (3/5-tone down)</li><li><strong>#</strong> = Sharp (3/5-tone up)</li><li><strong>v</strong> = Semiflat (1/5-tone down)</li></ul></td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | === Subsets and supersets === |
| | Since 39 factors into {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics. |
|
| |
|
| | == Intervals == |
| | {| class="wikitable center-all right-2 left-3 right-9 right-10" |
| | |- |
| | ! Steps |
| | ! Cents |
| | ! Approximate ratios* |
| | ! colspan="3" | [[Ups and downs notation]] |
| | ! colspan="3" | [[Nearest just interval]] <br>(Ratio, cents, error) |
| | |- |
| | | 0 |
| | | 0.0 |
| | | [[1/1]] |
| | | P1 |
| | | perfect unison |
| | | D |
| | | 1/1 |
| | | 0.00 |
| | | None |
| | |- |
| | | 1 |
| | | 30.8 |
| | | ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]] |
| | | ^1, <br>vm2 |
| | | up unison, <br>downminor 2nd |
| | | ^D, <br>vEb |
| | | 57/56 |
| | | 30.64 |
| | | +0.1271 |
| | |- |
| | | 2 |
| | | 61.5 |
| | | [[28/27]], [[33/32]], ''[[49/48]]'' |
| | | m2 |
| | | minor 2nd |
| | | Eb |
| | | 29/28 |
| | | 60.75 |
| | | +0.7872 |
| | |- |
| | | 3 |
| | | 92.3 |
| | | ''[[16/15]]'', [[21/20]], ''[[25/24]]'' |
| | | ^m2 |
| | | upminor 2nd |
| | | ^Eb |
| | | 39/37 |
| | | 91.14 |
| | | +1.1691 |
| | |- |
| | | 4 |
| | | 123.1 |
| | | [[15/14]] |
| | | ^^m2 |
| | | dupminor 2nd |
| | | ^^Eb |
| | | 44/41 |
| | | 122.26 |
| | | +0.8214 |
| | |- |
| | | 5 |
| | | 153.8 |
| | | [[11/10]], [[12/11]] |
| | | vvM2 |
| | | dudmajor 2nd |
| | | vvE |
| | | 35/32 |
| | | 155.14 |
| | | -1.2934 |
| | |- |
| | | 6 |
| | | 184.6 |
| | | [[10/9]] |
| | | vM2 |
| | | downmajor 2nd |
| | | vE |
| | | 10/9 |
| | | 182.40 |
| | | +2.2117 |
| | |- |
| | | 7 |
| | | 215.4 |
| | | [[9/8]], ''[[8/7]]'' |
| | | M2 |
| | | major 2nd |
| | | E |
| | | 17/15 |
| | | 216.69 |
| | | -1.3021 |
| | |- |
| | | 8 |
| | | 246.2 |
| | | [[81/70]] |
| | | ^M2, <br>vm3 |
| | | upmajor 2nd, <br>downminor 3rd |
| | | ^E, <br>vF |
| | | 15/13 |
| | | 247.74 |
| | | -1.5873 |
| | |- |
| | | 9 |
| | | 276.9 |
| | | [[7/6]] |
| | | m3 |
| | | minor 3rd |
| | | F |
| | | 27/23 |
| | | 277.59 |
| | | -0.6676 |
| | |- |
| | | 10 |
| | | 307.7 |
| | | [[6/5]] |
| | | ^m3 |
| | | upminor 3rd |
| | | ^F |
| | | 43/36 |
| | | 307.61 |
| | | +0.0846 |
| | |- |
| | | 11 |
| | | 338.5 |
| | | [[11/9]] |
| | | ^^m3 |
| | | dupminor 3rd |
| | | ^^F |
| | | 17/14 |
| | | 336.13 |
| | | +2.3320 |
| | |- |
| | | 12 |
| | | 369.2 |
| | | [[27/22]] |
| | | vvM3 |
| | | dudmajor 3rd |
| | | vvF# |
| | | 26/21 |
| | | 369.75 |
| | | -0.5160 |
| | |- |
| | | 13 |
| | | 400.0 |
| | | [[5/4]] |
| | | vM3 |
| | | downmajor 3rd |
| | | vF# |
| | | 34/27 |
| | | 399.09 |
| | | +0.9096 |
| | |- |
| | | 14 |
| | | 430.8 |
| | | [[9/7]], [[14/11]] |
| | | M3 |
| | | major 3rd |
| | | F# |
| | | 41/32 |
| | | 429.06 |
| | | +1.7068 |
| | |- |
| | | 15 |
| | | 461.5 |
| | | [[35/27]] |
| | | v4 |
| | | down 4th |
| | | vG |
| | | 30/23 |
| | | 459.99 |
| | | +1.5441 |
| | |- |
| | | 16 |
| | | 492.3 |
| | | [[4/3]] |
| | | P4 |
| | | perfect 4th |
| | | G |
| | | 85/64 |
| | | 491.27 |
| | | +1.0386 |
| | |- |
| | | 17 |
| | | 523.1 |
| | | [[27/20]] |
| | | ^4 |
| | | up 4th |
| | | ^G |
| | | 23/17 |
| | | 523.32 |
| | | -0.2420 |
| | |- |
| | | 18 |
| | | 553.8 |
| | | [[11/8]] |
| | | ^^4 |
| | | dup 4th |
| | | ^^G |
| | | 11/8 |
| | | 551.32 |
| | | +2.5283 |
| | |- |
| | | 19 |
| | | 584.6 |
| | | [[7/5]] |
| | | vvA4, <br>^d5 |
| | | dudaug 4th, <br>updim 5th |
| | | vvG#, <br>^Ab |
| | | 7/5 |
| | | 582.51 |
| | | +2.1032 |
| | |- |
| | | 20 |
| | | 615.4 |
| | | [[10/7]] |
| | | vA4, <br>^^d5 |
| | | downaug 4th, <br>dupdim 5th |
| | | vG#, <br>^^Ab |
| | | 10/7 |
| | | 617.49 |
| | | -2.1032 |
| | |- |
| | | 21 |
| | | 646.2 |
| | | [[16/11]] |
| | | vv5 |
| | | dud 5th |
| | | vvA |
| | | 16/11 |
| | | 648.68 |
| | | -2.5283 |
| | |- |
| | | 22 |
| | | 676.9 |
| | | [[40/27]] |
| | | v5 |
| | | down 5th |
| | | vA |
| | | 34/23 |
| | | 676.68 |
| | | +0.2420 |
| | |- |
| | | 23 |
| | | 707.7 |
| | | [[3/2]] |
| | | P5 |
| | | perfect 5th |
| | | A |
| | | 128/85 |
| | | 708.73 |
| | | -1.0386 |
| | |- |
| | | 24 |
| | | 738.5 |
| | | [[54/35]] |
| | | ^5 |
| | | up 5th |
| | | A^ |
| | | 23/15 |
| | | 740.01 |
| | | -1.5441 |
| | |- |
| | | 25 |
| | | 769.2 |
| | | [[11/7]], [[14/9]] |
| | | m6 |
| | | minor 6th |
| | | Bb |
| | | 64/41 |
| | | 770.94 |
| | | -1.7068 |
| | |- |
| | | 26 |
| | | 800.0 |
| | | [[8/5]] |
| | | ^m6 |
| | | upminor 6th |
| | | ^Bb |
| | | 27/17 |
| | | 800.91 |
| | | -0.9096 |
| | |- |
| | | 27 |
| | | 830.8 |
| | | [[44/27]] |
| | | ^^m6 |
| | | dupminor 6th |
| | | ^^Bb |
| | | 21/13 |
| | | 830.25 |
| | | +0.5160 |
| | |- |
| | | 28 |
| | | 861.5 |
| | | [[18/11]] |
| | | vvM6 |
| | | dudmajor 6th |
| | | vvB |
| | | 28/17 |
| | | 863.87 |
| | | -2.3320 |
| | |- |
| | | 29 |
| | | 892.3 |
| | | [[5/3]] |
| | | vM6 |
| | | downmajor 6th |
| | | vB |
| | | 72/43 |
| | | 892.39 |
| | | -0.0846 |
| | |- |
| | | 30 |
| | | 923.1 |
| | | [[12/7]] |
| | | M6 |
| | | major 6th |
| | | B |
| | | 46/27 |
| | | 922.41 |
| | | +0.6676 |
| | |- |
| | | 31 |
| | | 953.8 |
| | | [[140/81]] |
| | | ^M6, <br>vm7 |
| | | upmajor 6th, <br>downminor 7th |
| | | ^B, <br>vC |
| | | 26/15 |
| | | 952.26 |
| | | +1.5873 |
| | |- |
| | | 32 |
| | | 984.6 |
| | | ''[[7/4]]'', [[16/9]] |
| | | m7 |
| | | minor 7th |
| | | C |
| | | 30/17 |
| | | 983.31 |
| | | +1.3021 |
| | |- |
| | | 33 |
| | | 1015.4 |
| | | [[9/5]] |
| | | ^m7 |
| | | upminor 7th |
| | | ^C |
| | | 9/5 |
| | | 1017.60 |
| | | -2.2117 |
| | |- |
| | | 34 |
| | | 1046.2 |
| | | [[11/6]], [[20/11]] |
| | | ^^m7 |
| | | dupminor 7th |
| | | ^^C |
| | | 64/35 |
| | | 1044.86 |
| | | +1.2934 |
| | |- |
| | | 35 |
| | | 1076.9 |
| | | [[28/15]] |
| | | vvM7 |
| | | dudmajor 7th |
| | | vvC# |
| | | 41/22 |
| | | 1077.74 |
| | | -0.8214 |
| | |- |
| | | 36 |
| | | 1107.7 |
| | | ''[[15/8]]'', [[40/21]], ''[[48/25]]'' |
| | | vM7 |
| | | downmajor 7th |
| | | vC# |
| | | 74/39 |
| | | 1108.86 |
| | | -1.1691 |
| | |- |
| | | 37 |
| | | 1138.5 |
| | | [[27/14]], ''[[96/49]]'', [[64/33]] |
| | | M7 |
| | | major 7th |
| | | C# |
| | | 56/29 |
| | | 1139.25 |
| | | -0.7872 |
| | |- |
| | | 38 |
| | | 1169.2 |
| | | ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]] |
| | | ^M7, <br>v8 |
| | | upmajor 7th, <br>down 8ve |
| | | ^C#, <br>vD |
| | | 112/57 |
| | | 1169.36 |
| | | -0.1271 |
| | |- |
| | | 39 |
| | | 1200.0 |
| | | [[2/1]] |
| | | P8 |
| | | perfect 8ve |
| | | D |
| | | 2/1 |
| | | 1200.00 |
| | | None |
| | |} |
| | <nowiki/>* 11-limit in the 39d val, inconsistent intervals in ''italic'' |
|
| |
|
| <table class="wiki_table">
| | Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]]. |
| <tr>
| |
| <td><strong>Degrees</strong><br />
| |
| </td>
| |
| <td><strong>Armodue note</strong><br />
| |
| </td>
| |
| <td><strong>Cents size</strong><br />
| |
| </td>
| |
| <td><strong><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Nearest%20just%20interval">Nearest Just I</a>nterval</strong><br />
| |
| </td>
| |
| <td><strong>Cents value</strong><br />
| |
| </td>
| |
| <td><strong>Error</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td><strong>1</strong><br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td><strong>1/1</strong><br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td><strong>None</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>1t<br />
| |
| </td>
| |
| <td>30.7692<br />
| |
| </td>
| |
| <td>57/56<br />
| |
| </td>
| |
| <td>30.6421<br />
| |
| </td>
| |
| <td>+0.1271<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>2b<br />
| |
| </td>
| |
| <td>61.5385<br />
| |
| </td>
| |
| <td>29/28<br />
| |
| </td>
| |
| <td>60.7513<br />
| |
| </td>
| |
| <td>+0.7872<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>1#<br />
| |
| </td>
| |
| <td>92.3077<br />
| |
| </td>
| |
| <td>39/37<br />
| |
| </td>
| |
| <td>91.1386<br />
| |
| </td>
| |
| <td>+1.1691<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>2v<br />
| |
| </td>
| |
| <td>123.0769<br />
| |
| </td>
| |
| <td>44/41<br />
| |
| </td>
| |
| <td>122.2555<br />
| |
| </td>
| |
| <td>+0.8214<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>2<br />
| |
| </td>
| |
| <td>153.8462<br />
| |
| </td>
| |
| <td>35/32<br />
| |
| </td>
| |
| <td>155.1396<br />
| |
| </td>
| |
| <td>-1.2934<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>2t<br />
| |
| </td>
| |
| <td>184.6154<br />
| |
| </td>
| |
| <td>10/9<br />
| |
| </td>
| |
| <td>182.4037<br />
| |
| </td>
| |
| <td>+2.2117<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>7·</strong><br />
| |
| </td>
| |
| <td><strong>3b</strong><br />
| |
| </td>
| |
| <td><strong>215.3846</strong><br />
| |
| </td>
| |
| <td><strong>17/15</strong><br />
| |
| </td>
| |
| <td><strong>216.6867</strong><br />
| |
| </td>
| |
| <td><strong>-1.3021</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>2#<br />
| |
| </td>
| |
| <td>246.1538<br />
| |
| </td>
| |
| <td>15/13<br />
| |
| </td>
| |
| <td>247.7411<br />
| |
| </td>
| |
| <td>-1.5873<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>3v<br />
| |
| </td>
| |
| <td>276.9231<br />
| |
| </td>
| |
| <td>27/23<br />
| |
| </td>
| |
| <td>277.5907<br />
| |
| </td>
| |
| <td>-0.6676<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>3<br />
| |
| </td>
| |
| <td>307.6923<br />
| |
| </td>
| |
| <td>43/36<br />
| |
| </td>
| |
| <td>307.6077<br />
| |
| </td>
| |
| <td>+0.0846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>3t<br />
| |
| </td>
| |
| <td>338.4615<br />
| |
| </td>
| |
| <td>17/14<br />
| |
| </td>
| |
| <td>336.1295<br />
| |
| </td>
| |
| <td>+2.332<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>12·</strong><br />
| |
| </td>
| |
| <td><strong>4b</strong><br />
| |
| </td>
| |
| <td><strong>369.2308</strong><br />
| |
| </td>
| |
| <td><strong>26/21</strong><br />
| |
| </td>
| |
| <td><strong>369.7468</strong><br />
| |
| </td>
| |
| <td><strong>-0.516</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>3#<br />
| |
| </td>
| |
| <td>400<br />
| |
| </td>
| |
| <td>34/27<br />
| |
| </td>
| |
| <td>399.0904<br />
| |
| </td>
| |
| <td>+0.9096<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>4v<br />
| |
| </td>
| |
| <td>430.7692<br />
| |
| </td>
| |
| <td>41/32<br />
| |
| </td>
| |
| <td>429.0624<br />
| |
| </td>
| |
| <td>+1.7068<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>4<br />
| |
| </td>
| |
| <td>461.5385<br />
| |
| </td>
| |
| <td>30/23<br />
| |
| </td>
| |
| <td>459.9944<br />
| |
| </td>
| |
| <td>+1.5441<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>4t (5v)<br />
| |
| </td>
| |
| <td>492.3077<br />
| |
| </td>
| |
| <td>85/64<br />
| |
| </td>
| |
| <td>491.2691<br />
| |
| </td>
| |
| <td>+1.0386<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>17·</strong><br />
| |
| </td>
| |
| <td><strong>5</strong><br />
| |
| </td>
| |
| <td><strong>523.0769</strong><br />
| |
| </td>
| |
| <td><strong>23/17</strong><br />
| |
| </td>
| |
| <td><strong>523.3189</strong><br />
| |
| </td>
| |
| <td><strong>-0.242</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>5t<br />
| |
| </td>
| |
| <td>553.8462<br />
| |
| </td>
| |
| <td>11/8<br />
| |
| </td>
| |
| <td>551.3179<br />
| |
| </td>
| |
| <td>+2.5283<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>6b<br />
| |
| </td>
| |
| <td>584.6154<br />
| |
| </td>
| |
| <td>7/5<br />
| |
| </td>
| |
| <td>582.5122<br />
| |
| </td>
| |
| <td>+2.1032<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>5#<br />
| |
| </td>
| |
| <td>615.3846<br />
| |
| </td>
| |
| <td>10/7<br />
| |
| </td>
| |
| <td>617.4878<br />
| |
| </td>
| |
| <td>-2.1032<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>6v<br />
| |
| </td>
| |
| <td>646.1538<br />
| |
| </td>
| |
| <td>16/11<br />
| |
| </td>
| |
| <td>648.6821<br />
| |
| </td>
| |
| <td>-2.5283<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>22·</strong><br />
| |
| </td>
| |
| <td><strong>6</strong><br />
| |
| </td>
| |
| <td><strong>676.9231</strong><br />
| |
| </td>
| |
| <td><strong>34/23</strong><br />
| |
| </td>
| |
| <td><strong>676.6811</strong><br />
| |
| </td>
| |
| <td><strong>+0.242</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>6t<br />
| |
| </td>
| |
| <td>707.6923<br />
| |
| </td>
| |
| <td>128/85<br />
| |
| </td>
| |
| <td>708.7309<br />
| |
| </td>
| |
| <td>-1.0386<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>7b<br />
| |
| </td>
| |
| <td>738.4615<br />
| |
| </td>
| |
| <td>23/15<br />
| |
| </td>
| |
| <td>740.0056<br />
| |
| </td>
| |
| <td>-1.5441<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>6#<br />
| |
| </td>
| |
| <td>769.2308<br />
| |
| </td>
| |
| <td>64/41<br />
| |
| </td>
| |
| <td>770.9376<br />
| |
| </td>
| |
| <td>-1.7068<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>7v<br />
| |
| </td>
| |
| <td>800<br />
| |
| </td>
| |
| <td>27/17<br />
| |
| </td>
| |
| <td>800.9096<br />
| |
| </td>
| |
| <td>-0.9096<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>27·</strong><br />
| |
| </td>
| |
| <td><strong>7</strong><br />
| |
| </td>
| |
| <td><strong>830.7692</strong><br />
| |
| </td>
| |
| <td><strong>21/13</strong><br />
| |
| </td>
| |
| <td><strong>830.2532</strong><br />
| |
| </td>
| |
| <td><strong>+0.516</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>7t<br />
| |
| </td>
| |
| <td>861.5385<br />
| |
| </td>
| |
| <td>28/17<br />
| |
| </td>
| |
| <td>863.8705<br />
| |
| </td>
| |
| <td>-2.332<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>8b<br />
| |
| </td>
| |
| <td>892.3077<br />
| |
| </td>
| |
| <td>72/43<br />
| |
| </td>
| |
| <td>892.3923<br />
| |
| </td>
| |
| <td>-0.0846<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>7#<br />
| |
| </td>
| |
| <td>923.0769<br />
| |
| </td>
| |
| <td>46/27<br />
| |
| </td>
| |
| <td>922.4093<br />
| |
| </td>
| |
| <td>+0.6676<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
| |
| <td>8v<br />
| |
| </td>
| |
| <td>953.8462<br />
| |
| </td>
| |
| <td>26/15<br />
| |
| </td>
| |
| <td>952.2589<br />
| |
| </td>
| |
| <td>+1.5873<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>32·</strong><br />
| |
| </td>
| |
| <td><strong>8</strong><br />
| |
| </td>
| |
| <td><strong>984.6154</strong><br />
| |
| </td>
| |
| <td><strong>30/17</strong><br />
| |
| </td>
| |
| <td><strong>983.3133</strong><br />
| |
| </td>
| |
| <td><strong>+1.3021</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>33<br />
| |
| </td>
| |
| <td>8t<br />
| |
| </td>
| |
| <td>1015.3846<br />
| |
| </td>
| |
| <td>9/5<br />
| |
| </td>
| |
| <td>1017.5963<br />
| |
| </td>
| |
| <td>-2.2117<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>34<br />
| |
| </td>
| |
| <td>9b<br />
| |
| </td>
| |
| <td>1046.1538<br />
| |
| </td>
| |
| <td>64/35<br />
| |
| </td>
| |
| <td>1044.8604<br />
| |
| </td>
| |
| <td>+1.2934<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>35<br />
| |
| </td>
| |
| <td>8#<br />
| |
| </td>
| |
| <td>1076.9231<br />
| |
| </td>
| |
| <td>41/22<br />
| |
| </td>
| |
| <td>1077.7445<br />
| |
| </td>
| |
| <td>-0.8214<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>36<br />
| |
| </td>
| |
| <td>9v<br />
| |
| </td>
| |
| <td>1107.6923<br />
| |
| </td>
| |
| <td>74/39<br />
| |
| </td>
| |
| <td>1108.8614<br />
| |
| </td>
| |
| <td>-1.1691<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>37<br />
| |
| </td>
| |
| <td>9<br />
| |
| </td>
| |
| <td>1138.4615<br />
| |
| </td>
| |
| <td>56/29<br />
| |
| </td>
| |
| <td>1139.2487<br />
| |
| </td>
| |
| <td>-0.7872<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>38<br />
| |
| </td>
| |
| <td>9t (1v)<br />
| |
| </td>
| |
| <td>1169.2308<br />
| |
| </td>
| |
| <td>112/57<br />
| |
| </td>
| |
| <td>1169.3579<br />
| |
| </td>
| |
| <td>-0.1271<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><strong>39··(or 0)</strong><br />
| |
| </td>
| |
| <td><strong>1</strong><br />
| |
| </td>
| |
| <td><strong>1200</strong><br />
| |
| </td>
| |
| <td><strong>2/1</strong><br />
| |
| </td>
| |
| <td><strong>1200</strong><br />
| |
| </td>
| |
| <td><strong>None</strong><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | == Notation == |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x39 tone equal temperament-Instruments (prototypes):"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Instruments (prototypes):</u></h2>
| | === Ups and downs notation === |
| <br />
| | 39edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down). |
| <!-- ws:start:WikiTextLocalImageRule:604:&lt;img src=&quot;http://xenharmonic.wikispaces.com/file/view/TECLADO_39-EDO.PNG/258413072/811x472/TECLADO_39-EDO.PNG&quot; alt=&quot;Armodue-Hornbostel 1/5-tone keyboard prototype&quot; title=&quot;Armodue-Hornbostel 1/5-tone keyboard prototype&quot; style=&quot;height: 472px; width: 811px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="http://xenharmonic.wikispaces.com/file/view/TECLADO_39-EDO.PNG/258413072/811x472/TECLADO_39-EDO.PNG" alt="TECLADO_39-EDO.PNG" title="TECLADO_39-EDO.PNG" style="height: 472px; width: 811px;" /></td></tr><tr><td class="imageCaption">Armodue-Hornbostel 1/5-tone keyboard prototype</td></tr></table><!-- ws:end:WikiTextLocalImageRule:604 --><br />
| | {{Sharpness-sharp5a}} |
| <br />
| | Another notation uses [[Alternative symbols for ups and downs notation #Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: |
| <!-- ws:start:WikiTextLocalImageRule:605:&lt;img src=&quot;http://xenharmonic.wikispaces.com/file/view/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png/258445130/1002x307/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png&quot; alt=&quot;39-ED2 fretboard visualization&quot; title=&quot;39-ED2 fretboard visualization&quot; style=&quot;height: 307px; width: 1002px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="http://xenharmonic.wikispaces.com/file/view/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png/258445130/1002x307/Custom_700mm_5-str_Tricesanonaphonic_Guitar.png" alt="Custom_700mm_5-str_Tricesanonaphonic_Guitar.png" title="Custom_700mm_5-str_Tricesanonaphonic_Guitar.png" style="height: 307px; width: 1002px;" /></td></tr><tr><td class="imageCaption">39-ED2 fretboard visualization</td></tr></table><!-- ws:end:WikiTextLocalImageRule:605 --><br />
| | {{Sharpness-sharp5}} |
| <!-- ws:start:WikiTextLocalImageRule:606:&lt;img src=&quot;/file/view/Diapas%C3%B3n_para_Bajo_Precision_%C3%B3_Jazz_Tric%C3%A9sanonaf%C3%B3nico.png/273853908/1010x93/Diapas%C3%B3n_para_Bajo_Precision_%C3%B3_Jazz_Tric%C3%A9sanonaf%C3%B3nico.png&quot; alt=&quot;Tricesanonaphonic fretboard for P or J Bass&quot; title=&quot;Tricesanonaphonic fretboard for P or J Bass&quot; style=&quot;height: 93px; width: 1010px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Diapas%C3%B3n_para_Bajo_Precision_%C3%B3_Jazz_Tric%C3%A9sanonaf%C3%B3nico.png/273853908/1010x93/Diapas%C3%B3n_para_Bajo_Precision_%C3%B3_Jazz_Tric%C3%A9sanonaf%C3%B3nico.png" alt="Diapasón_para_Bajo_Precision_ó_Jazz_Tricésanonafónico.png" title="Diapasón_para_Bajo_Precision_ó_Jazz_Tricésanonafónico.png" style="height: 93px; width: 1010px;" /></td></tr><tr><td class="imageCaption">Tricesanonaphonic fretboard for P or J Bass</td></tr></table><!-- ws:end:WikiTextLocalImageRule:606 --><br />
| | === Sagittal notation === |
| <!-- ws:start:WikiTextLocalImageRule:607:&lt;img src=&quot;/file/view/Diapas%C3%B3n_para_Gitarra_StratoTric%C3%A9sanonaf%C3%B3nico.png/273854620/998x117/Diapas%C3%B3n_para_Gitarra_StratoTric%C3%A9sanonaf%C3%B3nico.png&quot; alt=&quot;Tricesanonaphonic fretboard for standard electric guitars&quot; title=&quot;Tricesanonaphonic fretboard for standard electric guitars&quot; style=&quot;height: 117px; width: 998px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Diapas%C3%B3n_para_Gitarra_StratoTric%C3%A9sanonaf%C3%B3nico.png/273854620/998x117/Diapas%C3%B3n_para_Gitarra_StratoTric%C3%A9sanonaf%C3%B3nico.png" alt="Diapasón_para_Gitarra_StratoTricésanonafónico.png" title="Diapasón_para_Gitarra_StratoTricésanonafónico.png" style="height: 117px; width: 998px;" /></td></tr><tr><td class="imageCaption">Tricesanonaphonic fretboard for standard electric guitars</td></tr></table><!-- ws:end:WikiTextLocalImageRule:607 --><br />
| | This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]]. |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x39 tone equal temperament-39 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:6 --><strong><u>39 tone equal <a class="wiki_link" href="http://xenharmonic.wikispaces.com/modes">modes</a></u>:</strong></h2> | | |
| <br />
| | ==== Evo flavor ==== |
| 15 15 9 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%201s">2L 1s</a><br />
| | <imagemap> |
| 14 14 11 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%201s">2L 1s</a><br />
| | File:39-EDO_Evo_Sagittal.svg |
| 13 13 13 = <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3edo">3edo</a><br />
| | desc none |
| 11 11 11 6 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%201s">3L 1s</a><br />
| | rect 80 0 300 50 [[Sagittal_notation]] |
| 10 10 10 9 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%201s">3L 1s</a><br />
| | rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| 11 3 11 11 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%202s">3L 2s (father)</a><br />
| | rect 20 80 120 106 [[81/80]] |
| 11 3 11 3 11 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="color: #660000; cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%202s">3L 2s (father)</a></span><br />
| | rect 120 80 240 106 [[33/32]] |
| 9 6 9 9 6 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%202s">3L 2s (father)</a><br />
| | default [[File:39-EDO_Evo_Sagittal.svg]] |
| 9 6 9 6 9 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="color: #660000; cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%202s">3L 2s (father)</a></span><br /> | | </imagemap> |
| 9 9 9 9 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/4L%201s">4L 1s (bug)</a><br /> | | |
| 9 3 9 9 9 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="color: #660000; cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/4L%201s">4L 1s (bug)</a></span><br />
| | ==== Revo flavor ==== |
| 8 8 8 8 7 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/4L%201s">4L 1s (bug)</a><br />
| | <imagemap> |
| 10 3 10 3 10 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%203s">3L 3s (augmented)</a><br />
| | File:39-EDO_Revo_Sagittal.svg |
| 9 4 9 4 9 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%203s">3L 3s (augmented)</a><br />
| | desc none |
| 8 5 8 5 8 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%203s">3L 3s (augmented)</a><br />
| | rect 80 0 300 50 [[Sagittal_notation]] |
| 7 7 7 7 7 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%201s">5L 1s (Grumpy hexatonic)</a><br />
| | rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| 7 4 7 7 7 7 - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%201s">5L 1s (Grumpy hexatonic)</a></span><br />
| | rect 20 80 120 106 [[81/80]] |
| 3 9 3 9 3 9 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%204s">3L 4s (mosh)</a><br />
| | rect 120 80 240 106 [[33/32]] |
| 5 5 7 5 5 5 7 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%205s">2L 5s (mavila)</a><br />
| | default [[File:39-EDO_Revo_Sagittal.svg]] |
| 5 5 5 7 5 5 7 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%205s">2L 5s (mavila)</a><br />
| | </imagemap> |
| 5 7 5 5 7 5 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2L%205s">2L 5s (mavila)</a><br />
| | |
| 6 3 6 6 3 6 6 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%203s">5L 3s (unfair father)</a><br />
| | === Armodue notation === |
| 5 5 5 5 5 5 5 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%201s">7L 1s (Grumpy octatonic)</a><br />
| | ; Armodue nomenclature 5;2 relation |
| 5 4 5 5 5 5 5 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%201s">7L 1s (Grumpy octatonic)</a><br />
| | * '''‡''' = Semisharp (1/5-tone up) |
| <strong>5 5 5 2 5 5 5 5 2</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%202s">7L 2s (mavila superdiatonic)</a><br />
| | * '''b''' = Flat (3/5-tone down) |
| 5 5 2 5 5 5 2 5 5 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%202s">7L 2s (mavila superdiatonic)</a><br />
| | * '''#''' = Sharp (3/5-tone up) |
| 5 5 3 5 5 3 5 5 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/6L%203s">6L 3s (unfair augmented)</a><br />
| | * '''v''' = Semiflat (1/5-tone down) |
| 5 4 4 5 4 4 5 4 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%206s">3L 6s (fair augmented)</a><br />
| | |
| 4 4 4 4 4 4 4 4 4 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9L%201s">9L 1s (Grumpy decatonic)</a><br />
| | {| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed" |
| 4 4 3 4 4 4 4 4 4 4 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9L%201s">9L 1s (Grumpy decatonic)</a><br />
| | |- |
| <strong>3 3 5 3 3 3 5 3 3 3 5</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%208s">3L 8s (undecimal anti-triatonic)</a><br />
| | ! colspan="2" | # |
| 3 3 3 3 3 3 3 3 3 3 3 3 3 = <a class="wiki_link" href="http://xenharmonic.wikispaces.com/13edo">13edo</a><br />
| | ! Cents |
| <strong>3 3 3 2 3 3 3 3 2 3 3 3 3 2</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/11L%203s">11L 3s (tetradecimal triatonic)</a><br />
| | ! Armodue notation |
| 3 2 3 3 2 3 2 3 3 2 3 2 3 3 2 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9L%206s">9L 6s</a><br />
| | ! Associated ratios |
| 3 2 3 2 3 2 2 3 2 3 2 3 2 3 2 2 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7L%209s">7L 9s</a><br />
| | |- |
| <strong>2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 2 3</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5L%2012s">5L 12s</a><br />
| | | 0 |
| 2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3L%2015s">3L 15s</a><br />
| | | |
| <strong>3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3</strong> - <span style="cursor: pointer;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/10L%209s">10L 9s</a><br />
| | | 0.0 |
| 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19L%201s">19L 1s</a><br />
| | | 1 |
| 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17L%205s">17L 5s</a><br />
| | | [[1/1]] |
| <strong>2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 2 1</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16L%207s">16L 7s</a><br />
| | |- |
| 2 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 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/13L%2013s">13L 13s</a><br />
| | | 1 |
| <strong>2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1</strong> - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/10L%2019s">10L 19s</a><br />
| | | |
| 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8L%2023s">8L 23s</a><br />
| | | 30.8 |
| 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 - <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a> of type <a class="wiki_link" href="http://xenharmonic.wikispaces.com/6L%2027s">6L 27s</a></body></html></pre></div>
| | | 1‡ (9#) |
| | | |
| | |- |
| | | 2 |
| | | |
| | | 61.5 |
| | | 2b |
| | | |
| | |- |
| | | 3 |
| | | |
| | | 92.3 |
| | | 1# |
| | | |
| | |- |
| | | 4 |
| | | |
| | | 123.1 |
| | | 2v |
| | | |
| | |- |
| | | 5 |
| | | |
| | | 153.8 |
| | | 2 |
| | | 11/10~12/11 |
| | |- |
| | | 6 |
| | | |
| | | 184.6 |
| | | 2‡ |
| | | |
| | |- |
| | | 7 |
| | | · |
| | | 215.4 |
| | | 3b |
| | | 8/7 |
| | |- |
| | | 8 |
| | | |
| | | 246.2 |
| | | 2# |
| | | |
| | |- |
| | | 9 |
| | | |
| | | 276.9 |
| | | 3v |
| | | |
| | |- |
| | | 10 |
| | | |
| | | 307.7 |
| | | 3 |
| | | 6/5~7/6 |
| | |- |
| | | 11 |
| | | |
| | | 338.5 |
| | | 3‡ |
| | | |
| | |- |
| | | 12 |
| | | · |
| | | 369.2 |
| | | 4b |
| | | 5/4 |
| | |- |
| | | 13 |
| | | |
| | | 400.0 |
| | | 3# |
| | | |
| | |- |
| | | 14 |
| | | |
| | | 430.8 |
| | | 4v (5b) |
| | | |
| | |- |
| | | 15 |
| | | |
| | | 461.5 |
| | | 4 |
| | | |
| | |- |
| | | 16 |
| | | |
| | | 492.3 |
| | | 4‡ (5v) |
| | | |
| | |- |
| | | 17 |
| | | · |
| | | 523.1 |
| | | 5 |
| | | 4/3~11/8 |
| | |- |
| | | 18 |
| | | |
| | | 553.8 |
| | | 5‡ (4#) |
| | | |
| | |- |
| | | 19 |
| | | |
| | | 584.6 |
| | | 6b |
| | | 10/7 |
| | |- |
| | | 20 |
| | | |
| | | 615.4 |
| | | 5# |
| | | 7/5 |
| | |- |
| | | 21 |
| | | |
| | | 646.2 |
| | | 6v |
| | | |
| | |- |
| | | 22 |
| | | · |
| | | 676.9 |
| | | 6 |
| | | 3/2~16/11 |
| | |- |
| | | 23 |
| | | |
| | | 707.7 |
| | | 6‡ |
| | | |
| | |- |
| | | 24 |
| | | |
| | | 738.5 |
| | | 7b |
| | | |
| | |- |
| | | 25 |
| | | |
| | | 769.2 |
| | | 6# |
| | | |
| | |- |
| | | 26 |
| | | |
| | | 800.0 |
| | | 7v |
| | | |
| | |- |
| | | 27 |
| | | · |
| | | 830.8 |
| | | 7 |
| | | 8/5 |
| | |- |
| | | 28 |
| | | |
| | | 861.5 |
| | | 7‡ |
| | | |
| | |- |
| | | 29 |
| | | |
| | | 892.3 |
| | | 8b |
| | | 5/3~12/7 |
| | |- |
| | | 30 |
| | | |
| | | 923.1 |
| | | 7# |
| | | |
| | |- |
| | | 31 |
| | | |
| | | 953.8 |
| | | 8v |
| | | |
| | |- |
| | | 32 |
| | | · |
| | | 984.6 |
| | | 8 |
| | | 7/4 |
| | |- |
| | | 33 |
| | | |
| | | 1015.4 |
| | | 8‡ |
| | | |
| | |- |
| | | 34 |
| | | |
| | | 1046.2 |
| | | 9b |
| | | 11/6~20/11 |
| | |- |
| | | 35 |
| | | |
| | | 1076.9 |
| | | 8# |
| | | |
| | |- |
| | | 36 |
| | | |
| | | 1107.7 |
| | | 9v (1b) |
| | | |
| | |- |
| | | 37 |
| | | |
| | | 1138.5 |
| | | 9 |
| | | |
| | |- |
| | | 38 |
| | | |
| | | 1169.2 |
| | | 9‡ (1v) |
| | | |
| | |- |
| | | 39 |
| | | ·· |
| | | 1200.0 |
| | | 1 |
| | | 2/1 |
| | |} |
| | |
| | == 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]] (%) |
| | |- |
| | | 2.3 |
| | | {{Monzo| 62 -39 }} |
| | | {{Mapping| 39 62 }} |
| | | −1.81 |
| | | 1.81 |
| | | 5.88 |
| | |- |
| | | 2.3.5 |
| | | 128/125, 1594323/1562500 |
| | | {{Mapping| 39 62 91 }} |
| | | −3.17 |
| | | 2.42 |
| | | 7.89 |
| | |- |
| | | 2.3.5.7 |
| | | 64/63, 126/125, 2430/2401 |
| | | {{Mapping| 39 62 91 110 }} (39d) |
| | | −3.78 |
| | | 2.35 |
| | | 7.65 |
| | |- |
| | | 2.3.5.7.11 |
| | | 64/63, 99/98, 121/120, 126/125 |
| | | {{Mapping| 39 62 91 110 135 }} (39d) |
| | | −3.17 |
| | | 2.43 |
| | | 7.91 |
| | |} |
| | |
| | === Rank-2 temperaments === |
| | {| class="wikitable center-all left-4 left-5" |
| | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator |
| | |- |
| | ! Periods <br />per 8ve |
| | ! Generator* |
| | ! Cents* |
| | ! Temperament |
| | ! Mos scales |
| | |- |
| | | 1 |
| | | 1\39 |
| | | 30.8 |
| | | |
| | | |
| | |- |
| | | 1 |
| | | 2\39 |
| | | 61.5 |
| | | [[Unicorn]] (39d) |
| | | [[1L 18s]], [[19L 1s]] |
| | |- |
| | | 1 |
| | | 4\39 |
| | | 123.1 |
| | | [[Negri]] (39c) |
| | | [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]] |
| | |- |
| | | 1 |
| | | 5\39 |
| | | 153.8 |
| | | |
| | | [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]] |
| | |- |
| | | 1 |
| | | 7\39 |
| | | 215.4 |
| | | [[Machine]] (39d) |
| | | [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]] |
| | |- |
| | | 1 |
| | | 8\39 |
| | | 246.2 |
| | | [[Immunity]] (39) / [[immunized]] (39d) |
| | | [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]] |
| | |- |
| | | 1 |
| | | 10\39 |
| | | 307.7 |
| | | [[Familia]] (39df) |
| | | [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]] |
| | |- |
| | | 1 |
| | | 11\39 |
| | | 338.5 |
| | | [[Amity]] (39) / [[accord]] (39d) |
| | | [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]] |
| | |- |
| | | 1 |
| | | 14\39 |
| | | 430.8 |
| | | [[Hamity]] (39df) |
| | | [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]] |
| | |- |
| | | 1 |
| | | 16\39 |
| | | 492.3 |
| | | [[Quasisuper]] (39d) |
| | | [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]] |
| | |- |
| | | 1 |
| | | 17\39 |
| | | 523.1 |
| | | [[Mavila]] (39bc) |
| | | [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]] |
| | |- |
| | | 1 |
| | | 19\39 |
| | | 584.6 |
| | | [[Pluto]] (39d) |
| | | [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]] |
| | |- |
| | | 3 |
| | | 1\39 |
| | | 30.8 |
| | | |
| | | |
| | |- |
| | | 3 |
| | | 2\39 |
| | | 61.5 |
| | | |
| | | [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]] |
| | |- |
| | | 3 |
| | | 6\39 |
| | | 184.6 |
| | | [[Terrain]] / [[mirkat]] (39df) |
| | | [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]] |
| | |- |
| | | 3 |
| | | 8\39 <br>(5\39) |
| | | 246.2 <br>(153.8) |
| | | [[Triforce]] (39) |
| | | [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]] |
| | |- |
| | | 3 |
| | | 16\39 <br>(3\39) |
| | | 492.3 <br>(92.3) |
| | | [[Augene]] (39d) |
| | | [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]] |
| | |- |
| | | 3 |
| | | 17\39 <br>(4\39) |
| | | 523.1 <br>(123.0) |
| | | [[Deflated]] (39bd) |
| | | [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]] |
| | |- |
| | | 13 |
| | | 16\39 <br>(1\39) |
| | | 492.3 <br>(30.8) |
| | | [[Tridecatonic]] |
| | | [[13L 13s]] |
| | |} |
| | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct |
| | |
| | == 39edo and world music == |
| | 39edo is a good candidate for a "universal tuning" in that it offers reasonable approximations of many different world music [[approaches to musical tuning|traditions]]; it is one of the simplest edos that can make this claim. Because of this, composers wishing to combine multiple world music traditions (for example, [[gamelan]] with [[maqam]] singing) within one unified framework might find 39edo an interesting possibility. |
| | |
| | === Western === |
| | 39edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic mos: 7 7 2 7 7 7 2). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing. |
| | |
| | Another option is to use a [[modmos]], such as 7 6 3 7 6 7 3; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either 7 3 3 3 7 3 3 7 3 (a [[modmos]] of type [[3L 6s]]) or 4 3 6 3 4 3 6 4 3 3. There are other modmos scales that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time. |
| | |
| | The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|semitones]], which makes them more effective for melody than their counterparts in 12edo or meantone systems. |
| | |
| | Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that [[xenharmonic]] to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the 7 6 3 7 6 7 3 modmos), for example. |
| | |
| | === Indian === |
| | A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth). |
| | |
| | === Arabic, Turkish, Iranian === |
| | While [[Arabic, Turkish, Persian music|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because: |
| | |
| | * It has two types of "neutral" seconds (154 and 185 cents) |
| | * It has two minor seconds (92 and 123 cents), which when added together give a whole tone (215 cents) |
| | |
| | whereas neither 17edo nor 24edo satisfy these properties. |
| | |
| | 39edo will likely be more suited to some middle-eastern scales than others. Specifically, Turkish music (in which the Rast makam has a "major-like" wide neutral third and a wide "neutral" second approaching 10/9), will likely be especially well suited to 39edo. |
| | |
| | === Blues / Jazz / African-American === |
| | The [[harmonic seventh]] ("[[barbershop]] seventh") [[tetrad]] is reasonably well approximated in 39edo, and some temperaments (augene in particular) give scales that are liberally supplied with them. John Coltrane might have loved augene (→ [[Wikipedia: Coltrane changes]]). |
| | |
| | [[Tritone]] substitution, which is a major part of jazz and blues harmony, is more complicated in 39edo because there are two types of tritones. Therefore, the tritone substitution of one seventh chord will need to be a different type of seventh chord. However, this also opens new possibilities; if the substituted chord is of a more consonant type than the original, then the tritone substitution may function as a ''resolution'' rather than a suspension. |
| | |
| | Blue notes, rather than being considered inflections, can be notated as accidentals instead; for example, a "blue major third" can be identified as either of the two neutral thirds. There are two possible [[mapping]]s for [[7/4]] which are about equal in closeness. The sharp mapping is the normal one because it works better with the [[5/4]] and [[3/2]], but using the flat one instead (as an accidental) allows for another type of blue note. |
| | |
| | === Other === |
| | 39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator. Pelog can also be approximated as 4 5 13 4 13. |
| | |
| | It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] mos (which is 9 7 7 9 7). [[Slendro]] can be approximated using that scale or using something like the [[quasi-equal]] 8 8 8 8 7 or 8 8 7 8 8. |
| | |
| | One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6. |
| | |
| | Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated. |
| | |
| | == Instruments == |
| | |
| | === Lumatone mapping === |
| | See [[Lumatone mapping for 39edo]] |
| | |
| | === Skip fretting === |
| | '''Skip fretting system 39 2 5''' is a [[skip-fretting]] system for [[39edo]]. All examples on this page are for 7-string [[Guitar|guitar.]] |
| | |
| | ; Prime harmonics |
| | 1/1: string 2 open |
| | |
| | 2/1: string 5 fret 12 and string 7 fret 7 |
| | |
| | 3/2: string 3 fret 9 and string 5 fret 4 |
| | |
| | 5/4: string 1 fret 9 and string 3 fret 4 |
| | |
| | 7/4: string 5 fret 8 and string 7 fret 3 |
| | |
| | 11/8: string 2 fret 9 and string 4 fret 4 |
| | |
| | === Prototypes === |
| | [[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]] |
| | |
| | ''An illustrative image of a 39edo keyboard'' |
| | |
| | [[File:Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|alt=Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|826x203px|Custom_700mm_5-str_Tricesanonaphonic_Guitar.png]] |
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| | ''39edo fretboard visualization'' |
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| | == Music == |
| | ; [[Bryan Deister]] |
| | * [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023) |
| | * [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025) |
| | * [https://www.youtube.com/shorts/4y11CWLIHNA ''Sinner's Finale - Genshin Impact (microtonal cover in 39edo)''] (2025) |
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| | ; [[Randy Wells]] |
| | * [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021) |
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| | [[Category:Listen]] |