21edo
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=21 equal divisions of the octave= Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other ET <26. ==21-EDO as a temperament:== In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals. In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it. The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance. || **Degree** || **Cents** **Value** || **7-EDO** **Notation** || **5L3s** **Notation** ||= **D.-R. Interval** **Types** ||= **Approximate** **Ratios *1** ||= <span style="display: block; text-align: center;">**Approximate**</span><span style="display: block; text-align: center;">**Ratios *2**</span> ||= <span style="display: block; text-align: center;">**Approximate**</span>**Ratios *3** || || 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 || || 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 || || 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 || || 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 || || 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 || || 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 || || 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 || || 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 || || 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 || || 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 || || 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 || || 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 || || 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 || || 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 || || 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 || || 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 || || 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 || || 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 || || 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 || || 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 || || 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 || || 21 || 1200 || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 || *1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament *2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament *3: based on treating 21-EDO as 13-limit laconic temperament **21-tone scales:** [[augment6]] [[augment9]] [[augment12]] ==Triadic Harmony in 21-EDO:== One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series: ||= **Steps** ||= **Cents** ||= **Ratio** || ||= 0-5-10 ||= 0-286-571 ||= 23:27:32 || ||= 0-4-11 ||= 0-229-629 ||= 7:8:10 || ||= 0-6-11 ||= 0-343-629 ||= 9:11:13 || ||= 0-5-13 ||= 0-286-743 ||= 11:13:17 || ||= 0-8-13 ||= 0-457-743 ||= 13:17:20 || ||= 0-5-15 ||= 0-286-857 ||= 11:13:18 || ==Moment-of-Symmetry Scales in 21-EDO:== Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period. For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example. For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales. ==Tetrachordal Scales in 21-EDO== While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways: ||= Step Pattern ||= Cents ||= Name* || ||= 3, 3, 3 ||= 0-171-343-514 ||= Equal diatonic || ||= 4, 3, 2 ||= 0-229-400-514 ||= Hard diatonic || ||= 4, 4, 1 ||= 0-229-457-514 ||= Soft diatonic || ||= 5, 3, 1 ||= 0-286-457-514 ||= Soft chromatic || ||= 5, 2, 2 ||= 0-286-400-514 ||= Hard chromatic || ||= 6, 2, 1 ||= 0-343-457-514 ||= Hard enharmonic || ||= 7, 1, 1 ||= 0-400-457-514 ||= Soft enharmonic || *these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better! The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah. ==Rank two temperaments== [[List of 21edo rank two temperaments by badness]] ||~ Periods per octave ||~ Generator ||~ Temperaments || || 1 || 1\21 || [[Escapade family#Escapade|Escapade]] || || 1 || 2\21 || [[Gamelismic clan#Miracle|Miracle]] || || 1 || 4\21 || [[Slendric]]/[[Gamelismic clan#Gorgo|Gorgo]]/[[Gamelismic clan#Gidorah|Gidorah]] || || 1 || 5\21 || [[Mint temperaments#Subklei|Subklei]] || || 1 || 8\21 || [[Chromatic pairs#Tridec|Tridec]] || || 1 || 10\21 || [[Marvel temperaments#Triton|Triton]] || || 3 || 1\21 || || || 3 || 2\21 || [[Augmented family|Augmented]]/[[August]] || || 3 || 3\21 || || || 7 || 1\21 || [[Apotome family|Whitewood]] || ==13-limit Commas== 21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 || ||= 2187/2048 ||< | -11 7 > ||> 113.69 ||= Apotome ||= || ||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma || ||= 9931568/9752117 ||< | -25 7 6 > ||> 31.57 ||= Ampersand's Comma || || ||= 9193891/9143623 ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma || || ||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||= || ||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone || || ||= 9859966/9733137 ||< | -10 7 8 -7 > ||> 22.41 ||= Blackjackisma ||= || ||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||= || ||= 225/224 ||< | -5 2 2 -1 > ||> 7.71 ||= Septimal Kleisma ||= Marvel Comma || ||= 16875/16807 ||< | 0 3 4 -5 > ||> 6.99 ||= Mirkwai ||= || ||= 2401/2400 ||< | -5 -1 -2 4 > ||> 0.72 ||= Breedsma ||= || ||= 394839/394762 ||< | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma || ||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= || ||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= || ||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||= || =**Books / Literature:**= Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009. [[image:http://www.ronsword.com/images/ron1.jpg width="254" height="188"]][[image:http://www.swordguitars.com/21tetsm.JPG width="363" height="191"]] **//21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)//** =**Compositions/Listening:**= [[@http://www.ronsword.com/sounds/21_improv.mp3|Short Clip of 21-edo Acoustic]] by [[Ron Sword]] [[@http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3|Open tuning Drone Improvisation in 21-edo]] by Ron Sword [[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715|Anomalous Readings]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3|play]] by [[Andrew Heathwaite]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3|Comets Over Flatland 15]] by [[Randy Winchester]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3|Comets Over Flatland 18]] by [[Randy Winchester]] [[@http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco|L'esatonale ubriaco (the drunk hexatonal)]], ALIENAMENTE by [[xenharmonic/Fabrizio Fiale|Fabrizio Fulvio Fausto Fiale]]
Original HTML content:
<html><head><title>21edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x21 equal divisions of the octave"></a><!-- ws:end:WikiTextHeadingRule:0 -->21 equal divisions of the octave</h1>
<br />
Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other ET <26.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x21 equal divisions of the octave-21-EDO as a temperament:"></a><!-- ws:end:WikiTextHeadingRule:2 -->21-EDO as a temperament:</h2>
In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.<br />
<br />
In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.<br />
<br />
The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit <a class="wiki_link" href="/Laconic%20Family">laconic</a> temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.<br />
<br />
<table class="wiki_table">
<tr>
<td><strong>Degree</strong><br />
</td>
<td><strong>Cents</strong><br />
<strong>Value</strong><br />
</td>
<td><strong>7-EDO</strong><br />
<strong>Notation</strong><br />
</td>
<td><strong>5L3s</strong><br />
<strong>Notation</strong><br />
</td>
<td style="text-align: center;"><strong>D.-R. Interval</strong><br />
<strong>Types</strong><br />
</td>
<td style="text-align: center;"><strong>Approximate</strong><br />
<strong>Ratios *1</strong><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;"><strong>Approximate</strong></span><span style="display: block; text-align: center;"><strong>Ratios *2</strong></span><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;"><strong>Approximate</strong></span><strong>Ratios *3</strong><br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>C<br />
</td>
<td>C<br />
</td>
<td style="text-align: center;">Unison<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>57.143<br />
</td>
<td>C^/Dvv<br />
</td>
<td>C#<br />
</td>
<td style="text-align: center;">Subminor 2nd<br />
</td>
<td style="text-align: center;">28/27, 30/29<br />
</td>
<td style="text-align: center;">35/34, 36/35<br />
</td>
<td style="text-align: center;">64/63<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>114.286<br />
</td>
<td>C^^/Dv<br />
</td>
<td>Db<br />
</td>
<td style="text-align: center;">Minor 2nd<br />
</td>
<td style="text-align: center;">16/15, 15/14, 29/27<br />
</td>
<td style="text-align: center;">18/17<br />
</td>
<td style="text-align: center;">16/15, 25/24<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>171.429<br />
</td>
<td>D<br />
</td>
<td>D<br />
</td>
<td style="text-align: center;">Submajor 2nd<br />
</td>
<td style="text-align: center;">10/9, 32/29<br />
</td>
<td style="text-align: center;">10/9,11/10<br />
</td>
<td style="text-align: center;">9/8<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>228.571<br />
</td>
<td>D^/Evv<br />
</td>
<td>D#<br />
</td>
<td style="text-align: center;">Supermajor 2nd<br />
</td>
<td style="text-align: center;">8/7<br />
</td>
<td style="text-align: center;">8/7<br />
</td>
<td style="text-align: center;">8/7, 10/9, 11/10<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>285.714<br />
</td>
<td>D^^/Ev<br />
</td>
<td>Eb<br />
</td>
<td style="text-align: center;">Subminor 3rd<br />
</td>
<td style="text-align: center;">27/23, 32/27<br />
</td>
<td style="text-align: center;">13/11, 20/17<br />
</td>
<td style="text-align: center;">6/5, 7/6<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>342.857<br />
</td>
<td>E<br />
</td>
<td>E<br />
</td>
<td style="text-align: center;">Neutral 3rd<br />
</td>
<td style="text-align: center;">28/23<br />
</td>
<td style="text-align: center;">11/9<br />
</td>
<td style="text-align: center;">16/13<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>400<br />
</td>
<td>E^/Fvv<br />
</td>
<td>E#/Fb<br />
</td>
<td style="text-align: center;">Major 3rd<br />
</td>
<td style="text-align: center;">29/23<br />
</td>
<td style="text-align: center;">44/35<br />
</td>
<td style="text-align: center;">5/4, 9/7, 11/9, 14/11<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>457.143<br />
</td>
<td>E^^/Fv<br />
</td>
<td>F<br />
</td>
<td style="text-align: center;">Third-Fourth<br />
</td>
<td style="text-align: center;">30/23<br />
</td>
<td style="text-align: center;">13/10, 17/13, 22/17<br />
</td>
<td style="text-align: center;">13/10<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>514.286<br />
</td>
<td>F<br />
</td>
<td>F#<br />
</td>
<td style="text-align: center;">Acute 4th<br />
</td>
<td style="text-align: center;">161/120, 256/189<br />
</td>
<td style="text-align: center;">35/26<br />
</td>
<td style="text-align: center;">4/3, 18/13<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>571.429<br />
</td>
<td>F^/Gvv<br />
</td>
<td>Gb<br />
</td>
<td style="text-align: center;">Narrow Tritone<br />
</td>
<td style="text-align: center;">32/23<br />
</td>
<td style="text-align: center;">18/13<br />
</td>
<td style="text-align: center;">7/5, 11/8<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>628.571<br />
</td>
<td>F^^/Gv<br />
</td>
<td>G<br />
</td>
<td style="text-align: center;">Wide Tritone<br />
</td>
<td style="text-align: center;">23/16<br />
</td>
<td style="text-align: center;">13/9<br />
</td>
<td style="text-align: center;">10/7, 16/11<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>685.714<br />
</td>
<td>G<br />
</td>
<td>G#<br />
</td>
<td style="text-align: center;">Grave 5th<br />
</td>
<td style="text-align: center;">189/128, 240/161<br />
</td>
<td style="text-align: center;">52/35<br />
</td>
<td style="text-align: center;">3/2, 13/9<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>742.857<br />
</td>
<td>G^/Avv<br />
</td>
<td>Hb<br />
</td>
<td style="text-align: center;">Fifth-Sixth<br />
</td>
<td style="text-align: center;">23/15<br />
</td>
<td style="text-align: center;">17/11, 20/13, 26/17<br />
</td>
<td style="text-align: center;">20/13<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>800<br />
</td>
<td>G^^/Av<br />
</td>
<td>H<br />
</td>
<td style="text-align: center;">Minor 6th<br />
</td>
<td style="text-align: center;">46/29<br />
</td>
<td style="text-align: center;">35/22<br />
</td>
<td style="text-align: center;">8/5, 11/7, 14/9, 18/11<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>857.143<br />
</td>
<td>A<br />
</td>
<td>H#/Ab<br />
</td>
<td style="text-align: center;">Neutral 6th<br />
</td>
<td style="text-align: center;">23/14<br />
</td>
<td style="text-align: center;">18/11<br />
</td>
<td style="text-align: center;">13/8<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>914.286<br />
</td>
<td>A^/Bvv<br />
</td>
<td>A<br />
</td>
<td style="text-align: center;">Supermajor 6th<br />
</td>
<td style="text-align: center;">27/16, 46/27<br />
</td>
<td style="text-align: center;">17/10, 22/13<br />
</td>
<td style="text-align: center;">5/3, 12/7<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>971.429<br />
</td>
<td>A^^/Bv<br />
</td>
<td>A#<br />
</td>
<td style="text-align: center;">Subminor 7th<br />
</td>
<td style="text-align: center;">7/4<br />
</td>
<td style="text-align: center;">7/4<br />
</td>
<td style="text-align: center;">7/4, 9/5, 20/11<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>1028.571<br />
</td>
<td>B<br />
</td>
<td>Bb<br />
</td>
<td style="text-align: center;">Supraminor 7th<br />
</td>
<td style="text-align: center;">29/16, 9/5<br />
</td>
<td style="text-align: center;">9/5, 20/11<br />
</td>
<td style="text-align: center;">16/9<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>1085.714<br />
</td>
<td>B^/Cvv<br />
</td>
<td>B<br />
</td>
<td style="text-align: center;">Major 7th<br />
</td>
<td style="text-align: center;">15/8<br />
</td>
<td style="text-align: center;">17/9<br />
</td>
<td style="text-align: center;">15/8, 48/25<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>1142.857<br />
</td>
<td>B^^/Cv<br />
</td>
<td>B#/Cb<br />
</td>
<td style="text-align: center;">Supermajor 7th<br />
</td>
<td style="text-align: center;">27/14, 29/15<br />
</td>
<td style="text-align: center;">35/18, 68/35<br />
</td>
<td style="text-align: center;">63/32<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>1200<br />
</td>
<td>C<br />
</td>
<td>C<br />
</td>
<td style="text-align: center;">Octave<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
</tr>
</table>
<br />
*1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament<br />
*2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament<br />
*3: based on treating 21-EDO as 13-limit laconic temperament<br />
<br />
<strong>21-tone scales:</strong><br />
<a class="wiki_link" href="/augment6">augment6</a><br />
<a class="wiki_link" href="/augment9">augment9</a><br />
<a class="wiki_link" href="/augment12">augment12</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x21 equal divisions of the octave-Triadic Harmony in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:4 -->Triadic Harmony in 21-EDO:</h2>
<br />
One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:<br />
<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;"><strong>Steps</strong><br />
</td>
<td style="text-align: center;"><strong>Cents</strong><br />
</td>
<td style="text-align: center;"><strong>Ratio</strong><br />
</td>
</tr>
<tr>
<td style="text-align: center;">0-5-10<br />
</td>
<td style="text-align: center;">0-286-571<br />
</td>
<td style="text-align: center;">23:27:32<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0-4-11<br />
</td>
<td style="text-align: center;">0-229-629<br />
</td>
<td style="text-align: center;">7:8:10<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0-6-11<br />
</td>
<td style="text-align: center;">0-343-629<br />
</td>
<td style="text-align: center;">9:11:13<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0-5-13<br />
</td>
<td style="text-align: center;">0-286-743<br />
</td>
<td style="text-align: center;">11:13:17<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0-8-13<br />
</td>
<td style="text-align: center;">0-457-743<br />
</td>
<td style="text-align: center;">13:17:20<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0-5-15<br />
</td>
<td style="text-align: center;">0-286-857<br />
</td>
<td style="text-align: center;">11:13:18<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:6 -->Moment-of-Symmetry Scales in 21-EDO:</h2>
<br />
Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.<br />
For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.<br />
<br />
For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x21 equal divisions of the octave-Tetrachordal Scales in 21-EDO"></a><!-- ws:end:WikiTextHeadingRule:8 -->Tetrachordal Scales in 21-EDO</h2>
While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:<br />
<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;">Step Pattern<br />
</td>
<td style="text-align: center;">Cents<br />
</td>
<td style="text-align: center;">Name*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3, 3, 3<br />
</td>
<td style="text-align: center;">0-171-343-514<br />
</td>
<td style="text-align: center;">Equal diatonic<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4, 3, 2<br />
</td>
<td style="text-align: center;">0-229-400-514<br />
</td>
<td style="text-align: center;">Hard diatonic<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4, 4, 1<br />
</td>
<td style="text-align: center;">0-229-457-514<br />
</td>
<td style="text-align: center;">Soft diatonic<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5, 3, 1<br />
</td>
<td style="text-align: center;">0-286-457-514<br />
</td>
<td style="text-align: center;">Soft chromatic<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5, 2, 2<br />
</td>
<td style="text-align: center;">0-286-400-514<br />
</td>
<td style="text-align: center;">Hard chromatic<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6, 2, 1<br />
</td>
<td style="text-align: center;">0-343-457-514<br />
</td>
<td style="text-align: center;">Hard enharmonic<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7, 1, 1<br />
</td>
<td style="text-align: center;">0-400-457-514<br />
</td>
<td style="text-align: center;">Soft enharmonic<br />
</td>
</tr>
</table>
*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!<br />
<br />
The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x21 equal divisions of the octave-Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->Rank two temperaments</h2>
<a class="wiki_link" href="/List%20of%2021edo%20rank%20two%20temperaments%20by%20badness">List of 21edo rank two temperaments by badness</a><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\21<br />
</td>
<td><a class="wiki_link" href="/Escapade%20family#Escapade">Escapade</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>2\21<br />
</td>
<td><a class="wiki_link" href="/Gamelismic%20clan#Miracle">Miracle</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>4\21<br />
</td>
<td><a class="wiki_link" href="/Slendric">Slendric</a>/<a class="wiki_link" href="/Gamelismic%20clan#Gorgo">Gorgo</a>/<a class="wiki_link" href="/Gamelismic%20clan#Gidorah">Gidorah</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>5\21<br />
</td>
<td><a class="wiki_link" href="/Mint%20temperaments#Subklei">Subklei</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>8\21<br />
</td>
<td><a class="wiki_link" href="/Chromatic%20pairs#Tridec">Tridec</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>10\21<br />
</td>
<td><a class="wiki_link" href="/Marvel%20temperaments#Triton">Triton</a><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>1\21<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>2\21<br />
</td>
<td><a class="wiki_link" href="/Augmented%20family">Augmented</a>/<a class="wiki_link" href="/August">August</a><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>3\21<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>1\21<br />
</td>
<td><a class="wiki_link" href="/Apotome%20family">Whitewood</a><br />
</td>
</tr>
</table>
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x21 equal divisions of the octave-13-limit Commas"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Commas</h2>
21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)<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>
</tr>
<tr>
<td style="text-align: center;">2187/2048<br />
</td>
<td style="text-align: left;">| -11 7 ><br />
</td>
<td style="text-align: right;">113.69<br />
</td>
<td style="text-align: center;">Apotome<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">128/125<br />
</td>
<td style="text-align: left;">| 7 0 -3 ><br />
</td>
<td style="text-align: right;">41.06<br />
</td>
<td style="text-align: center;">Diesis<br />
</td>
<td style="text-align: center;">Augmented Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">9931568/9752117<br />
</td>
<td style="text-align: left;">| -25 7 6 ><br />
</td>
<td style="text-align: right;">31.57<br />
</td>
<td style="text-align: center;">Ampersand's Comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9193891/9143623<br />
</td>
<td style="text-align: left;">| 32 -7 -9 ><br />
</td>
<td style="text-align: right;">9.49<br />
</td>
<td style="text-align: center;">Escapade Comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1029/1000<br />
</td>
<td style="text-align: left;">| -3 1 -3 3 ><br />
</td>
<td style="text-align: right;">49.49<br />
</td>
<td style="text-align: center;">Keega<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><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9859966/9733137<br />
</td>
<td style="text-align: left;">| -10 7 8 -7 ><br />
</td>
<td style="text-align: right;">22.41<br />
</td>
<td style="text-align: center;">Blackjackisma<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>
</tr>
<tr>
<td style="text-align: center;">225/224<br />
</td>
<td style="text-align: left;">| -5 2 2 -1 ><br />
</td>
<td style="text-align: right;">7.71<br />
</td>
<td style="text-align: center;">Septimal Kleisma<br />
</td>
<td style="text-align: center;">Marvel Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">16875/16807<br />
</td>
<td style="text-align: left;">| 0 3 4 -5 ><br />
</td>
<td style="text-align: right;">6.99<br />
</td>
<td style="text-align: center;">Mirkwai<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2401/2400<br />
</td>
<td style="text-align: left;">| -5 -1 -2 4 ><br />
</td>
<td style="text-align: right;">0.72<br />
</td>
<td style="text-align: center;">Breedsma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">394839/394762<br />
</td>
<td style="text-align: left;">| 47 -7 -7 -7 ><br />
</td>
<td style="text-align: right;">0.34<br />
</td>
<td style="text-align: center;">Akjaysma<br />
</td>
<td style="text-align: center;">5\7 Octave Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">99/98<br />
</td>
<td style="text-align: left;">| -1 2 0 -2 1 ><br />
</td>
<td style="text-align: right;">17.58<br />
</td>
<td style="text-align: center;">Mothwellsma<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>
</tr>
<tr>
<td style="text-align: center;">4000/3993<br />
</td>
<td style="text-align: left;">| 5 -1 3 0 -3 ><br />
</td>
<td style="text-align: right;">3.03<br />
</td>
<td style="text-align: center;">Wizardharry<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Books / Literature:"></a><!-- ws:end:WikiTextHeadingRule:14 --><strong>Books / Literature:</strong></h1>
Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.<br />
<!-- ws:start:WikiTextRemoteImageRule:842:<img src="http://www.ronsword.com/images/ron1.jpg" alt="" title="" style="height: 188px; width: 254px;" /> --><img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><!-- ws:end:WikiTextRemoteImageRule:842 --><!-- ws:start:WikiTextRemoteImageRule:843:<img src="http://www.swordguitars.com/21tetsm.JPG" alt="" title="" style="height: 191px; width: 363px;" /> --><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /><!-- ws:end:WikiTextRemoteImageRule:843 --><br />
<strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Compositions/Listening:"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong>Compositions/Listening:</strong></h1>
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank">Short Clip of 21-edo Acoustic</a> by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a><br />
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank">Open tuning Drone Improvisation in 21-edo</a> by Ron Sword<br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715" rel="nofollow">Anomalous Readings</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3" rel="nofollow">Comets Over Flatland 15</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3" rel="nofollow">Comets Over Flatland 18</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
<a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco" rel="nofollow" target="_blank">L'esatonale ubriaco (the drunk hexatonal)</a>, ALIENAMENTE by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Fabrizio%20Fiale">Fabrizio Fulvio Fausto Fiale</a></body></html>