16edo
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<span style="display: block; text-align: right;">[[xenharmonie/16edo|Deutsch]] - [[16平均律|日本語]] </span> [[toc]] **16-EDO** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step. =Images= [[image:16edo wheel 01.png width="325" height="325"]] =Intervals= 16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always higher pitched than the latter. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. The second approach is to preserve the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, major is narrower than minor, etc. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly". || Degree || Cents ||= Approximate Ratios* || Interval Names Just ||= Interval Names Simplified ||||= Interval names, major wider than minor ||||= Interval names, major narrower than minor || || 0 || 0 ||= 1/1 || Unison ||= unison ||= unison ||= D ||= unison ||= D || || 1 || 75 ||= 28/27, 27/26 || Subminor 2nd ||= min 2nd ||= aug 1, dim 2nd ||= D#, Eb ||= dim 1, aug 2nd ||= Db, E# || || 2 || 150 ||= 12/11, 35/32 || Neutral 2nd ||= maj 2nd ||= min 2nd ||= E ||= maj 2nd ||= E || || 3 || 225 ||= 8/7 || Supermajor 2nd, Septimal Whole-Tone ||= perf 2nd ||= maj 2nd ||= E# ||= min 2nd ||= Eb || || 4 || 300 ||= 19/16, 32/27 || Minor 3rd ||= min 3rd ||= min 3rd ||= Fb ||= maj 3rd ||= F# || || 5 || 375 ||= 5/4, 26/21 || Major 3rd ||= maj 3rd ||= maj 3rd ||= F ||= min 3rd ||= F || || 6 || 450 ||= 13/10, 35/27 || Sub-4th, Supermajor 3rd ||= min 4th ||= aug 3rd, dim 4th ||= F#, Gb ||= dim 3rd, aug 4th ||= Fb, G# || || 7 || 525 ||= 27/20, 52/35, 256/189 || Wide 4th ||= maj 4th ||= perf 4th ||= G ||= perf 4th ||= G || || 8 || 600 ||= 7/5, 10/7 || Tritone ||= aug 4th, dim 5th ||= aug 4th, dim 5th ||= G#, Ab ||= dim 4th, aug 5th ||= Gb, A# || || 9 || 675 ||= 40/27, 35/26, 189/128 || Narrow 5th ||= min 5th ||= perf 5th ||= A ||= perf 5th ||= A || || 10 || 750 ||= 20/13, 54/35 || Super-5th, Subminor 6th ||= maj 5th ||= aug 5th, dim 6th ||= A#, Bb ||= dim 5th, aug 6th ||= Ab, B# || || 11 || 825 ||= 8/5, 21/13 || Minor 6th ||= min 6th ||= min 6th ||= B ||= maj 6th ||= B || || 12 || 900 ||= 27/16, 32/19 || Major 6th ||= maj 6th ||= maj 6th ||= B# ||= min 6th ||= Bb || || 13 || 975 ||= 7/4 || Subminor 7th, Septimal Minor 7th ||= perf 7th ||= min 7th ||= Cb ||= maj 7th ||= C# || || 14 || 1050 ||= 11/6, 64/35 || Neutral 7th ||= min 7th ||= maj 7th ||= C ||= min 7th ||= C || || 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th ||= maj 7th ||= aug 7th, dim 8ve ||= C# ||= dim 7th, aug 8ve ||= D# || || 16 || 1200 ||= 2/1 || octave ||= octave ||= 8ve ||= D ||= 8ve ||= D || *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. ==Selected just intervals by error== The following table shows how [[Just-24|some prominent just intervals]] are represented in 16-EDO (ordered by absolute error). || **Interval, complement** || **Error (abs., in [[cent|cents]])** || ||= [[12_11|12/11]], [[11_6|11/6]] ||= 0.637 || ||= [[13_10|13/10]], [[20_13|20/13]] ||= 4.214 || ||= [[8_7|8/7]], [[7_4|7/4]] ||= 6.174 || ||= [[13_11|13/11]], [[22_13|22/13]] ||= 10.790 || ||= [[5_4|5/4]], [[8_5|8/5]] ||= 11.314 || ||= [[13_12|13/12]], [[24_13|24/13]] ||= 11.427 || ||= [[15_11|15/11]], [[22_15|22/15]] ||= 11.951 || ||= [[9_7|9/7]], [[14_9|14/9]] ||= 14.916 || ||= [[11_10|11/10]], [[20_11|20/11]] ||= 15.004 || ||= [[16_13|16/13]], [[13_8|13/8]] ||= 15.528 || ||= [[6_5|6/5]], [[5_3|5/3]] ||= 15.641 || ||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 || ||= [[9_8|9/8]], [[16_9|16/9]] ||= 21.090 || ||= [[14_13|14/13]], [[13_7|13/7]] ||= 21.702 || ||= [[15_13|15/13]], [[26_15|26/15]] ||= 22.741 || ||= [[11_8|11/8]], [[16_11|16/11]] ||= 26.318 || ||= [[4_3|4/3]], [[3_2|3/2]] ||= 26.955 || ||= [[11_9|11/9]], [[18_11|18/11]] ||= 27.592 || ||= [[15_14|15/14]], [[28_15|28/15]] ||= 30.557 || ||= [[10_9|10/9]], [[9_5|9/5]] ||= 32.404 || ||= [[14_11|14/11]], [[11_7|11/7]] ||= 32.492 || ||= [[7_6|7/6]], [[12_7|12/7]] ||= 33.129 || ||= [[18_13|18/13]], [[13_9|13/9]] ||= 36.618 || ||= [[16_15|16/15]], [[15_8|15/8]] ||= 36.731 || It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant. [[media type="custom" key="24810096"]] [[file:16ed2-001.svg]] =Hexadecaphonic Octave Theory= The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third. 16-EDO is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "**Magic family of scales**". [[Easley Blackwood]] writes of 16-EDO: "16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh." Another interesting approach can include two interwoven [[8edo|8-EDO]] scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's **Mavila** as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th ([[19_16|19/16]], approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below). [[image:http://www.ronsword.com/161928%20copy.jpg width="316" height="179"]] The interval between the 28th & 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19. =Hexadecaphonic Notation= 16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO. [[image:http://ronsword.com/DSgoldsmith_piece.jpg width="1008" height="342"]] ---- In 16-EDO diatonic scales are dissonant and "shimmery" because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes. Moment of Symmetry Scales like Mavila [7] (or "Inverse/Anti-Diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale. The 6-line 16-EDO "Mavila-[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16-EDO keyboard. [[23edo|23-EDO]] also works with the Mavila-[9] 6-line staff, notated as 1/3 tones of 16-EDO. If the 9-note "Nonatonic" MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, "[[Decave]]". [[Paul Erlich]] writes, "Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80." ==16 Tone Piano Layout Based on the Anti-diatonic Scale== [[image:16-EDO-PIano-Diagram.png width="748" height="293"]] This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower. =Rank two temperaments= [[List of 16et rank two temperaments by badness]] ||~ Periods per octave ||~ Generator ||~ Temperaments || || 1 || 1\16 ||= [[Valentine]], [[Slurpee|slurpee]] || || 1 || 3\16 ||= [[Gorgo]] || || 1 || 5\16 ||= Messed-up [[Magic|magic]]/muggles || || 1 || 7\16 ||= [[Mavila]]/armodue || || 2 || 1\16 ||= [[Bipelog]] || || 2 || 3\16 ||= [[Lemba]], [[Astrology|astrology]] || || 4 || 1\16 ||= [[Diminished]]/demolished || || 8 || 1\16 ||= || **Mavila** || [5]: || 5 2 5 2 2 || || || [7]: || 3 2 2 3 2 2 2 || [[media type="file" key="MavilaAntidiatonic16edo.mp3" width="240" height="20"]] || || [9]: || 1 2 2 2 1 2 2 2 2 || [[media type="file" key="MavilaSuperdiatonic16edo.mp3" width="240" height="20"]] || **Diminished** [8]: 1 3 1 3 1 3 1 3 [12]: 1 1 2 1 1 2 1 1 2 1 1 2 **Magic** [7]: 1 4 1 4 1 4 1 [10]: 1 3 1 1 3 1 1 1 3 1 [13]: 1 1 2 1 1 1 2 1 1 1 2 1 1 **Cynder/Gorgo** [5]: 3 3 4 3 3 [6]: 3 3 1 3 3 3 [11]: 1 2 1 2 1 2 1 2 1 2 1 **Lemba** [6]: 3 2 3 3 2 3 [10]: 2 1 2 1 2 2 1 2 1 2 [[Igliashon Jones]] writes: "The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third and the "fourth" is the same as the distance between the "fourth" and the "fifth" (i.e. near a 12/11)...This mean(s) that 135/128 (the difference between 16/15 and 9/8) is tempered out...." ==Harmonizing Mavila/Armodue in 16 EDO== Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option. However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads 0-975-975 and 0-1050-1050 are also obvious possible chords. Their characteristic metallic sound has earned them the name "Metallic triads". ===MOS that support metallic harmony=== Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-1050. Mavila 9 introduces two more soft triads while the Wilson scale introduces two more hard triads. See [[Metallic Harmony]]. =Commas= 16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] < 16 25 37 45 55 59 |.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || ||= 135/128 ||< | -7 3 1 > ||> 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma || ||= 648/625 ||< | 3 4 -4 > ||> 62.57 ||= Major Diesis ||= Diminished Comma ||= || ||= 3125/3072 ||< | -10 -1 5 > ||> 29.61 ||= Small Diesis ||= Magic Comma ||= || ||= ||< | 23 6 -14 > ||> 3.34 ||= Vishnuzma ||= Semisuper ||= || ||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||= ||= || ||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= || ||= 50/49 ||< | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||= || ||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||= ||= || ||= 3125/3087 ||< | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||= ||= || ||= 126/125 ||< | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||= || ||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||= ||= || ||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||= ||= || ||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || ||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= ||= || ||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||= ||= || ||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= || ||= 3025/3024 ||< | -4 -3 2 -1 2 > ||> 0.57 ||= Lehmerisma ||= ||= || =**Armodue Theory (4-line staff)**= [[http://www.armodue.com/ricerche.htm|Armodue]]: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions. Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki. =Books/Literature= Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011 Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning) Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning) =Compositions= [[@http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&songID=12370649&showPlayer=true|Prenestyna Highway]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]] [[@http://lastsacrament.bandcamp.com/album/enantiodromia|Enantiodromia (album)]] by Last Sacrament [[@http://www.cdbaby.com/cd/aeternamusic|Tribute to Armodue]] by Aeterna [[http://www.io.com/%7Ehmiller/midi/16tet.mid|Etude in 16-tone equal tuning]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3|play]] ([[@http://soonlabel.com/xenharmonic/archives/2604|organ version]]) by [[Herman Miller|Herman Miller ]] [[http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3|16-tone steel string acoustic diddle]] by Ron Sword [[http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3|Armodue78]] by [[@Jean-Pierre Poulin]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Fiale/ffffiale+palestrinamortafantasiaquasiunasonata.mp3|Palestrina Morta, fantasia quasi una sonata]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3|Comets Over Flatland 5]] by [[Randy Winchester]] [[http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3|Malathion]] by [[http://chrisvaisvil.com/?p=2358#comments|Chris Vaisvil ]] [[http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3|Being of Vesta]] by [[http://chrisvaisvil.com/?p=3061|Chris Vaisvil]] [[http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3|Thin Ice]] by [[Chris Vaisvil]] ; [[http://chrisvaisvil.com/thin-ice-for-alto-female-choir-harp-and-percussion-in-adaptive-ji-16-edo-and-8-edo/|information on the composition]] [[file:Mavila Jazz Rhodes 1.mp3|Mavila Jazz Groove]] by [[https://soundcloud.com/pianofreak96|William Lynch]] [[file:mavila4.mp3|Cold, Dark Night for a Dance ]]by [[https://soundcloud.com/pianofreak96|William Lynch]] [[https://soundcloud.com/fff-fiale/in-sospensione-neutra|In Sospensione Neutra by Fabrizio Fulvio Fiale]]
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<html><head><title>16edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo">Deutsch</a> - <a class="wiki_link" href="/16%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
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<!-- ws:start:WikiTextTocRule:29:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Images">Images</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Intervals-Selected just intervals by error">Selected just intervals by error</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Notation">Hexadecaphonic Notation</a></div>
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale">16 Tone Piano Layout Based on the Anti-diatonic Scale</a></div>
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div>
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO">Harmonizing Mavila/Armodue in 16 EDO</a></div>
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 3em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony">MOS that support metallic harmony</a></div>
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a></div>
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#Books/Literature">Books/Literature</a></div>
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div>
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --></div>
<!-- ws:end:WikiTextTocRule:43 --><strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br />
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<!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc0"><a name="Images"></a><!-- ws:end:WikiTextHeadingRule:3 -->Images</h1>
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<!-- ws:start:WikiTextHeadingRule:5:<h1> --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals</h1>
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16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always higher pitched than the latter. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.<br />
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The second approach is to preserve the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, major is narrower than minor, etc. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".<br />
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<table class="wiki_table">
<tr>
<td>Degree<br />
</td>
<td>Cents<br />
</td>
<td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
<td>Interval <br />
Names<br />
Just<br />
</td>
<td style="text-align: center;">Interval<br />
Names<br />
Simplified<br />
</td>
<td colspan="2" style="text-align: center;">Interval names,<br />
major wider <br />
than minor<br />
</td>
<td colspan="2" style="text-align: center;">Interval names, <br />
major narrower<br />
than minor<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
<td>Unison<br />
</td>
<td style="text-align: center;">unison<br />
</td>
<td style="text-align: center;">unison<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">unison<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>75<br />
</td>
<td style="text-align: center;">28/27, 27/26<br />
</td>
<td>Subminor 2nd<br />
</td>
<td style="text-align: center;">min 2nd<br />
</td>
<td style="text-align: center;">aug 1, dim 2nd<br />
</td>
<td style="text-align: center;">D#, Eb<br />
</td>
<td style="text-align: center;">dim 1, aug 2nd<br />
</td>
<td style="text-align: center;">Db, E#<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>150<br />
</td>
<td style="text-align: center;">12/11, 35/32<br />
</td>
<td>Neutral 2nd<br />
</td>
<td style="text-align: center;">maj 2nd<br />
</td>
<td style="text-align: center;">min 2nd<br />
</td>
<td style="text-align: center;">E<br />
</td>
<td style="text-align: center;">maj 2nd<br />
</td>
<td style="text-align: center;">E<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>225<br />
</td>
<td style="text-align: center;">8/7<br />
</td>
<td>Supermajor 2nd,<br />
Septimal Whole-Tone<br />
</td>
<td style="text-align: center;">perf 2nd<br />
</td>
<td style="text-align: center;">maj 2nd<br />
</td>
<td style="text-align: center;">E#<br />
</td>
<td style="text-align: center;">min 2nd<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>300<br />
</td>
<td style="text-align: center;">19/16, 32/27<br />
</td>
<td>Minor 3rd<br />
</td>
<td style="text-align: center;">min 3rd<br />
</td>
<td style="text-align: center;">min 3rd<br />
</td>
<td style="text-align: center;">Fb<br />
</td>
<td style="text-align: center;">maj 3rd<br />
</td>
<td style="text-align: center;">F#<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>375<br />
</td>
<td style="text-align: center;">5/4, 26/21<br />
</td>
<td>Major 3rd<br />
</td>
<td style="text-align: center;">maj 3rd<br />
</td>
<td style="text-align: center;">maj 3rd<br />
</td>
<td style="text-align: center;">F<br />
</td>
<td style="text-align: center;">min 3rd<br />
</td>
<td style="text-align: center;">F<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>450<br />
</td>
<td style="text-align: center;">13/10, 35/27<br />
</td>
<td>Sub-4th,<br />
Supermajor 3rd<br />
</td>
<td style="text-align: center;">min 4th<br />
</td>
<td style="text-align: center;">aug 3rd, dim 4th<br />
</td>
<td style="text-align: center;">F#, Gb<br />
</td>
<td style="text-align: center;">dim 3rd, aug 4th<br />
</td>
<td style="text-align: center;">Fb, G#<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>525<br />
</td>
<td style="text-align: center;">27/20, 52/35, 256/189<br />
</td>
<td>Wide 4th<br />
</td>
<td style="text-align: center;">maj 4th<br />
</td>
<td style="text-align: center;">perf 4th<br />
</td>
<td style="text-align: center;">G<br />
</td>
<td style="text-align: center;">perf 4th<br />
</td>
<td style="text-align: center;">G<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>600<br />
</td>
<td style="text-align: center;">7/5, 10/7<br />
</td>
<td>Tritone<br />
</td>
<td style="text-align: center;">aug 4th,<br />
dim 5th<br />
</td>
<td style="text-align: center;">aug 4th, dim 5th<br />
</td>
<td style="text-align: center;">G#, Ab<br />
</td>
<td style="text-align: center;">dim 4th, aug 5th<br />
</td>
<td style="text-align: center;">Gb, A#<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>675<br />
</td>
<td style="text-align: center;">40/27, 35/26, 189/128<br />
</td>
<td>Narrow 5th<br />
</td>
<td style="text-align: center;">min 5th<br />
</td>
<td style="text-align: center;">perf 5th<br />
</td>
<td style="text-align: center;">A<br />
</td>
<td style="text-align: center;">perf 5th<br />
</td>
<td style="text-align: center;">A<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>750<br />
</td>
<td style="text-align: center;">20/13, 54/35<br />
</td>
<td>Super-5th,<br />
Subminor 6th<br />
</td>
<td style="text-align: center;">maj 5th<br />
</td>
<td style="text-align: center;">aug 5th, dim 6th<br />
</td>
<td style="text-align: center;">A#, Bb<br />
</td>
<td style="text-align: center;">dim 5th, aug 6th<br />
</td>
<td style="text-align: center;">Ab, B#<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>825<br />
</td>
<td style="text-align: center;">8/5, 21/13<br />
</td>
<td>Minor 6th<br />
</td>
<td style="text-align: center;">min 6th<br />
</td>
<td style="text-align: center;">min 6th<br />
</td>
<td style="text-align: center;">B<br />
</td>
<td style="text-align: center;">maj 6th<br />
</td>
<td style="text-align: center;">B<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>900<br />
</td>
<td style="text-align: center;">27/16, 32/19<br />
</td>
<td>Major 6th<br />
</td>
<td style="text-align: center;">maj 6th<br />
</td>
<td style="text-align: center;">maj 6th<br />
</td>
<td style="text-align: center;">B#<br />
</td>
<td style="text-align: center;">min 6th<br />
</td>
<td style="text-align: center;">Bb<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>975<br />
</td>
<td style="text-align: center;">7/4<br />
</td>
<td>Subminor 7th,<br />
Septimal Minor 7th<br />
</td>
<td style="text-align: center;">perf 7th<br />
</td>
<td style="text-align: center;">min 7th<br />
</td>
<td style="text-align: center;">Cb<br />
</td>
<td style="text-align: center;">maj 7th<br />
</td>
<td style="text-align: center;">C#<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>1050<br />
</td>
<td style="text-align: center;">11/6, 64/35<br />
</td>
<td>Neutral 7th<br />
</td>
<td style="text-align: center;">min 7th<br />
</td>
<td style="text-align: center;">maj 7th<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">min 7th<br />
</td>
<td style="text-align: center;">C<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1125<br />
</td>
<td style="text-align: center;">27/14, 52/27<br />
</td>
<td>Supermajor 7th<br />
</td>
<td style="text-align: center;">maj 7th<br />
</td>
<td style="text-align: center;">aug 7th, dim 8ve<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">dim 7th, aug 8ve<br />
</td>
<td style="text-align: center;">D#<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>1200<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
<td>octave<br />
</td>
<td style="text-align: center;">octave<br />
</td>
<td style="text-align: center;">8ve<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">8ve<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
</table>
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:<h2> --><h2 id="toc2"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:7 -->Selected just intervals by error</h2>
The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 16-EDO (ordered by absolute error).<br />
<table class="wiki_table">
<tr>
<td><strong>Interval, complement</strong><br />
</td>
<td><strong>Error (abs., in <a class="wiki_link" href="/cent">cents</a>)</strong><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a>, <a class="wiki_link" href="/11_6">11/6</a><br />
</td>
<td style="text-align: center;">0.637<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_10">13/10</a>, <a class="wiki_link" href="/20_13">20/13</a><br />
</td>
<td style="text-align: center;">4.214<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
<td style="text-align: center;">6.174<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_11">13/11</a>, <a class="wiki_link" href="/22_13">22/13</a><br />
</td>
<td style="text-align: center;">10.790<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br />
</td>
<td style="text-align: center;">11.314<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_12">13/12</a>, <a class="wiki_link" href="/24_13">24/13</a><br />
</td>
<td style="text-align: center;">11.427<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/22_15">22/15</a><br />
</td>
<td style="text-align: center;">11.951<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br />
</td>
<td style="text-align: center;">14.916<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_10">11/10</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
<td style="text-align: center;">15.004<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_13">16/13</a>, <a class="wiki_link" href="/13_8">13/8</a><br />
</td>
<td style="text-align: center;">15.528<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br />
</td>
<td style="text-align: center;">15.641<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br />
</td>
<td style="text-align: center;">17.488<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br />
</td>
<td style="text-align: center;">21.090<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/13_7">13/7</a><br />
</td>
<td style="text-align: center;">21.702<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_13">15/13</a>, <a class="wiki_link" href="/26_15">26/15</a><br />
</td>
<td style="text-align: center;">22.741<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br />
</td>
<td style="text-align: center;">26.318<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br />
</td>
<td style="text-align: center;">26.955<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br />
</td>
<td style="text-align: center;">27.592<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/28_15">28/15</a><br />
</td>
<td style="text-align: center;">30.557<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br />
</td>
<td style="text-align: center;">32.404<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/11_7">11/7</a><br />
</td>
<td style="text-align: center;">32.492<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br />
</td>
<td style="text-align: center;">33.129<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/18_13">18/13</a>, <a class="wiki_link" href="/13_9">13/9</a><br />
</td>
<td style="text-align: center;">36.618<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_8">15/8</a><br />
</td>
<td style="text-align: center;">36.731<br />
</td>
</tr>
</table>
It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant.<br />
<!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/24810096?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;24810096&quot;" title="Custom Media"/> --><object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/16ed2-001.svg">alt : Your browser has no SVG support.</object><!-- ws:end:WikiTextMediaRule:0 --><br />
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<!-- ws:start:WikiTextHeadingRule:9:<h1> --><h1 id="toc3"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:9 -->Hexadecaphonic Octave Theory</h1>
<br />
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.<br />
<br />
16-EDO is also a tuning for the <a class="wiki_link" href="/Jubilismic%20clan">no-threes 7-limit temperament tempering out 50/49</a>. This has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "<strong>Magic family of scales</strong>".<br />
<br />
<a class="wiki_link" href="/Easley%20Blackwood">Easley Blackwood</a> writes of 16-EDO:<br />
"16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh."<br />
<br />
Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br />
<!-- ws:start:WikiTextRemoteImageRule:911:<img src="http://www.ronsword.com/161928%20copy.jpg" alt="" title="" style="height: 179px; width: 316px;" /> --><img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" /><!-- ws:end:WikiTextRemoteImageRule:911 --><br />
<br />
The interval between the 28th & 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:11:<h1> --><h1 id="toc4"><a name="Hexadecaphonic Notation"></a><!-- ws:end:WikiTextHeadingRule:11 -->Hexadecaphonic Notation</h1>
<br />
16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br />
<!-- ws:start:WikiTextRemoteImageRule:912:<img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="" title="" style="height: 342px; width: 1008px;" /> --><img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="external image DSgoldsmith_piece.jpg" title="external image DSgoldsmith_piece.jpg" style="height: 342px; width: 1008px;" /><!-- ws:end:WikiTextRemoteImageRule:912 --><br />
<hr />
In 16-EDO diatonic scales are dissonant and "shimmery" because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br />
Moment of Symmetry Scales like Mavila [7] (or "Inverse/Anti-Diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale. The 6-line 16-EDO "Mavila-[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16-EDO keyboard. <a class="wiki_link" href="/23edo">23-EDO</a> also works with the Mavila-[9] 6-line staff, notated as 1/3 tones of 16-EDO. If the 9-note "Nonatonic" MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, "<a class="wiki_link" href="/Decave">Decave</a>".<br />
<br />
<a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> writes,<br />
"Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a<br />
unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80."<br />
<br />
<!-- ws:start:WikiTextHeadingRule:13:<h2> --><h2 id="toc5"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"></a><!-- ws:end:WikiTextHeadingRule:13 -->16 Tone Piano Layout Based on the Anti-diatonic Scale</h2>
<!-- ws:start:WikiTextLocalImageRule:910:<img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="" title="" style="height: 293px; width: 748px;" /> --><img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /><!-- ws:end:WikiTextLocalImageRule:910 --><br />
This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:15:<h1> --><h1 id="toc6"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:15 -->Rank two temperaments</h1>
<a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br />
<br />
<table class="wiki_table">
<tr>
<th>Periods<br />
per octave<br />
</th>
<th>Generator<br />
</th>
<th>Temperaments<br />
</th>
</tr>
<tr>
<td>1<br />
</td>
<td>1\16<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Valentine">Valentine</a>, <a class="wiki_link" href="/Slurpee">slurpee</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>3\16<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Gorgo">Gorgo</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>5\16<br />
</td>
<td style="text-align: center;">Messed-up <a class="wiki_link" href="/Magic">magic</a>/muggles<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>7\16<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Mavila">Mavila</a>/armodue<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>1\16<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Bipelog">Bipelog</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>3\16<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Lemba">Lemba</a>, <a class="wiki_link" href="/Astrology">astrology</a><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>1\16<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Diminished">Diminished</a>/demolished<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>1\16<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<strong>Mavila</strong><br />
<table class="wiki_table">
<tr>
<td>[5]:<br />
</td>
<td>5 2 5 2 2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>[7]:<br />
</td>
<td>3 2 2 3 2 2 2<br />
</td>
<td><!-- ws:start:WikiTextMediaRule:1:<img src="http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaAntidiatonic16edo.mp3?h=20&w=240" class="WikiMedia WikiMediaFile" id="wikitext@@media@@type=&quot;file&quot; key=&quot;MavilaAntidiatonic16edo.mp3&quot; width=&quot;240&quot; height=&quot;20&quot;" title="Local Media File"height="20" width="240"/> --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FMavilaAntidiatonic16edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:1 --><br />
</td>
</tr>
<tr>
<td>[9]:<br />
</td>
<td>1 2 2 2 1 2 2 2 2<br />
</td>
<td><!-- ws:start:WikiTextMediaRule:2:<img src="http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaSuperdiatonic16edo.mp3?h=20&w=240" class="WikiMedia WikiMediaFile" id="wikitext@@media@@type=&quot;file&quot; key=&quot;MavilaSuperdiatonic16edo.mp3&quot; width=&quot;240&quot; height=&quot;20&quot;" title="Local Media File"height="20" width="240"/> --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FMavilaSuperdiatonic16edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:2 --><br />
</td>
</tr>
</table>
<strong>Diminished</strong><br />
[8]: 1 3 1 3 1 3 1 3<br />
[12]: 1 1 2 1 1 2 1 1 2 1 1 2<br />
<strong>Magic</strong><br />
[7]: 1 4 1 4 1 4 1<br />
[10]: 1 3 1 1 3 1 1 1 3 1<br />
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1<br />
<strong>Cynder/Gorgo</strong><br />
[5]: 3 3 4 3 3<br />
[6]: 3 3 1 3 3 3<br />
[11]: 1 2 1 2 1 2 1 2 1 2 1<br />
<strong>Lemba</strong><br />
[6]: 3 2 3 3 2 3<br />
[10]: 2 1 2 1 2 2 1 2 1 2<br />
<br />
<a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a> writes: "The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third<br />
and the "fourth" is the same as the distance between the "fourth" and the "fifth" (i.e. near a 12/11)...This mean(s) that<br />
135/128 (the difference between 16/15 and 9/8) is tempered out...."<br />
<br />
<!-- ws:start:WikiTextHeadingRule:17:<h2> --><h2 id="toc7"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"></a><!-- ws:end:WikiTextHeadingRule:17 -->Harmonizing Mavila/Armodue in 16 EDO</h2>
<br />
Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.<br />
However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use<br />
it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric<br />
sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads<br />
0-975-975 and 0-1050-1050 are also obvious possible chords.<br />
Their characteristic metallic sound has earned them the name "Metallic triads".<br />
<br />
<!-- ws:start:WikiTextHeadingRule:19:<h3> --><h3 id="toc8"><a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"></a><!-- ws:end:WikiTextHeadingRule:19 -->MOS that support metallic harmony</h3>
Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords<br />
are wide symmetrical triads 0-1050-1050. Mavila 9 introduces two more soft triads while the Wilson scale introduces two more hard triads.<br />
<br />
See <a class="wiki_link" href="/Metallic%20Harmony">Metallic Harmony</a>.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:21:<h1> --><h1 id="toc9"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:21 -->Commas</h1>
16 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes <a class="wiki_link" href="/val">val</a> < 16 25 37 45 55 59 |.)<br />
<table class="wiki_table">
<tr>
<th>Comma<br />
</th>
<th>Monzo<br />
</th>
<th>Value (Cents)<br />
</th>
<th>Name 1<br />
</th>
<th>Name 2<br />
</th>
<th>Name 3<br />
</th>
</tr>
<tr>
<td style="text-align: center;">135/128<br />
</td>
<td style="text-align: left;">| -7 3 1 ><br />
</td>
<td style="text-align: right;">92.18<br />
</td>
<td style="text-align: center;">Major Chroma<br />
</td>
<td style="text-align: center;">Major Limma<br />
</td>
<td style="text-align: center;">Pelogic Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">648/625<br />
</td>
<td style="text-align: left;">| 3 4 -4 ><br />
</td>
<td style="text-align: right;">62.57<br />
</td>
<td style="text-align: center;">Major Diesis<br />
</td>
<td style="text-align: center;">Diminished Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3125/3072<br />
</td>
<td style="text-align: left;">| -10 -1 5 ><br />
</td>
<td style="text-align: right;">29.61<br />
</td>
<td style="text-align: center;">Small Diesis<br />
</td>
<td style="text-align: center;">Magic Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: left;">| 23 6 -14 ><br />
</td>
<td style="text-align: right;">3.34<br />
</td>
<td style="text-align: center;">Vishnuzma<br />
</td>
<td style="text-align: center;">Semisuper<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">36/35<br />
</td>
<td style="text-align: left;">| 2 2 -1 -1 ><br />
</td>
<td style="text-align: right;">48.77<br />
</td>
<td style="text-align: center;">Septimal Quarter Tone<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">525/512<br />
</td>
<td style="text-align: left;">| -9 1 2 1 ><br />
</td>
<td style="text-align: right;">43.41<br />
</td>
<td style="text-align: center;">Avicennma<br />
</td>
<td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">50/49<br />
</td>
<td style="text-align: left;">| 1 0 2 -2 ><br />
</td>
<td style="text-align: right;">34.98<br />
</td>
<td style="text-align: center;">Tritonic Diesis<br />
</td>
<td style="text-align: center;">Jubilisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">64827/64000<br />
</td>
<td style="text-align: left;">| -9 3 -3 4 ><br />
</td>
<td style="text-align: right;">22.23<br />
</td>
<td style="text-align: center;">Squalentine<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3125/3087<br />
</td>
<td style="text-align: left;">| 0 -2 5 -3 ><br />
</td>
<td style="text-align: right;">21.18<br />
</td>
<td style="text-align: center;">Gariboh<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">126/125<br />
</td>
<td style="text-align: left;">| 1 2 -3 1 ><br />
</td>
<td style="text-align: right;">13.79<br />
</td>
<td style="text-align: center;">Septimal Semicomma<br />
</td>
<td style="text-align: center;">Starling Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1029/1024<br />
</td>
<td style="text-align: left;">| -10 1 0 3 ><br />
</td>
<td style="text-align: right;">8.43<br />
</td>
<td style="text-align: center;">Gamelisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">6144/6125<br />
</td>
<td style="text-align: left;">| 11 1 -3 -2 ><br />
</td>
<td style="text-align: right;">5.36<br />
</td>
<td style="text-align: center;">Porwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">121/120<br />
</td>
<td style="text-align: left;">| -3 -1 -1 0 2 ><br />
</td>
<td style="text-align: right;">14.37<br />
</td>
<td style="text-align: center;">Biyatisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">176/175<br />
</td>
<td style="text-align: left;">| 4 0 -2 -1 1 ><br />
</td>
<td style="text-align: right;">9.86<br />
</td>
<td style="text-align: center;">Valinorsma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">385/384<br />
</td>
<td style="text-align: left;">| -7 -1 1 1 1 ><br />
</td>
<td style="text-align: right;">4.50<br />
</td>
<td style="text-align: center;">Keenanisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">441/440<br />
</td>
<td style="text-align: left;">| -3 2 -1 2 -1 ><br />
</td>
<td style="text-align: right;">3.93<br />
</td>
<td style="text-align: center;">Werckisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3025/3024<br />
</td>
<td style="text-align: left;">| -4 -3 2 -1 2 ><br />
</td>
<td style="text-align: right;">0.57<br />
</td>
<td style="text-align: center;">Lehmerisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:23:<h1> --><h1 id="toc10"><a name="Armodue Theory (4-line staff)"></a><!-- ws:end:WikiTextHeadingRule:23 --><strong>Armodue Theory (4-line staff)</strong></h1>
<a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.<br />
Translations of parts of the Armodue pages can be found <a class="wiki_link" href="/Armodue">here</a> on this wiki.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:25:<h1> --><h1 id="toc11"><a name="Books/Literature"></a><!-- ws:end:WikiTextHeadingRule:25 -->Books/Literature</h1>
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011<br />
Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br />
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:27:<h1> --><h1 id="toc12"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:27 -->Compositions</h1>
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&songID=12370649&showPlayer=true" rel="nofollow" target="_blank">Prenestyna Highway</a> by <a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank">Fabrizio Fulvio Fausto Fiale</a><br />
<a class="wiki_link_ext" href="http://lastsacrament.bandcamp.com/album/enantiodromia" rel="nofollow" target="_blank">Enantiodromia (album)</a> by Last Sacrament<br />
<a class="wiki_link_ext" href="http://www.cdbaby.com/cd/aeternamusic" rel="nofollow" target="_blank">Tribute to Armodue</a> by Aeterna<br />
<a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow">Etude in 16-tone equal tuning</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3" rel="nofollow">play</a> (<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2604" rel="nofollow" target="_blank">organ version</a>) by <a class="wiki_link" href="/Herman%20Miller">Herman Miller </a><br />
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3" rel="nofollow">16-tone steel string acoustic diddle</a> by Ron Sword<br />
<a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3" rel="nofollow">Armodue78</a> by <a class="wiki_link" href="/Jean-Pierre%20Poulin" target="_blank">Jean-Pierre Poulin</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Fiale/ffffiale+palestrinamortafantasiaquasiunasonata.mp3" rel="nofollow">Palestrina Morta, fantasia quasi una sonata</a> by <a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank">Fabrizio Fulvio Fausto Fiale</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3" rel="nofollow">Comets Over Flatland 5</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3" rel="nofollow">Malathion</a> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2358#comments" rel="nofollow">Chris Vaisvil </a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3" rel="nofollow">Being of Vesta</a> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=3061" rel="nofollow">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3" rel="nofollow">Thin Ice</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a> ; <a class="wiki_link_ext" href="http://chrisvaisvil.com/thin-ice-for-alto-female-choir-harp-and-percussion-in-adaptive-ji-16-edo-and-8-edo/" rel="nofollow">information on the composition</a><br />
<a href="/file/view/Mavila%20Jazz%20Rhodes%201.mp3/533097734/Mavila%20Jazz%20Rhodes%201.mp3" onclick="ws.common.trackFileLink('/file/view/Mavila%20Jazz%20Rhodes%201.mp3/533097734/Mavila%20Jazz%20Rhodes%201.mp3');">Mavila Jazz Groove</a> by <a class="wiki_link_ext" href="https://soundcloud.com/pianofreak96" rel="nofollow">William Lynch</a><br />
<a href="/file/view/mavila4.mp3/533097942/mavila4.mp3" onclick="ws.common.trackFileLink('/file/view/mavila4.mp3/533097942/mavila4.mp3');">Cold, Dark Night for a Dance </a>by <a class="wiki_link_ext" href="https://soundcloud.com/pianofreak96" rel="nofollow">William Lynch</a><br />
<a class="wiki_link_ext" href="https://soundcloud.com/fff-fiale/in-sospensione-neutra" rel="nofollow">In Sospensione Neutra by Fabrizio Fulvio Fiale</a></body></html>