34edo
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= = 34edo divides the octave into 34 equal steps of approximately 35.29412 [[xenharmonic/cent|cent]]s. 34edo contains two [[xenharmonic/17edo|17edo]]'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than [[31edo]], but with a sharp fifth rather than a flat one, and supports hanson, srutal, tetracot, würschmidt and vishnu temperaments. It does less well in the 7-limit, with two mappings possible for 7/4: a flat one from the patent val, and a sharp one from the 34d val. By way of the patent val 34 supports keemun temperament, and 34d is an excellent alternative to [[22edo]] for pajara temperament. In the 11-limit the 34d val supports pajara, vishnu and würschmidt. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup. ===Approximations to Just Intonation=== Like [[xenharmonic/17edo|17edo]], 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11.* 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 18/17, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the "syntonic comma" of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a [[xenharmonic/meantone|meantone ]]system. In layman's terms while no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds, technically will be the same pitch as something, somewhere upon the cycle of seventeen fifths. //Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.// ([[http://en.wikipedia.org/wiki/34_equal_temperament|Wikipedia]]) *The sharpness of ~13 cents of 11/8 can fit somewhat with the 9/8 and 13/8 which both are about 7 cents sharp. Likewise the 20-cent sharpness or flatness of either approximation to 7/4 isn't impossible. The ability to tolerate these errors may depend on subtle natural changes in mood. [[68edo]], double 34, has both these intervals in more perfect form, the 7 especially so. ===34edo and phi=== As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[xenharmonic/MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. ===Rank two temperaments=== [[xenharmonic/List of 34edo rank two temperaments by badness|List of 34edo rank two temperaments by badness]] ||~ Periods per octave ||~ Generator ||~ Cents ||~ Linear temperaments || || 1 || 1\34 || 35.294 || || || || 3\34 || 105.882 || || || || 5\34 || 176.471 || [[xenharmonic/Tetracot|Tetracot]]/[[xenharmonic/Bunya|Bunya]]/[[xenharmonic/Monkey|Monkey]] || || || 7\34 || 247.059 || [[xenharmonic/Immunity|Immunity]] || || || 9\34 || 317.647 || [[xenharmonic/Hanson|Hanson]]/[[xenharmonic/Keemun|Keemun]] || || || 11\34 || 388.235 || [[xenharmonic/Wuerschmidt|Wuerschmidt]]/[[xenharmonic/Worschmidt|Worschmidt]] || || || 13\34 || 458.824 || || || || 15\34 || 529.412 || || || 2 || 1\34 || 35.294 || || || || 2\34 || 70.588 || [[xenharmonic/Vishnu|Vishnu]] || || || 3\34 || 105.882 || [[xenharmonic/Srutal|Srutal]]/[[xenharmonic/Pajara|Pajara]]/[[xenharmonic/Diaschismic|Diaschismic]] || || || 4\34 || 141.176 || [[xenharmonic/Fifive|Fifive]] || || || 5\34 || 176.471 || || || || 6\34 || 211.765 || || || || 7\34 || 247.059 || || || || 8\34 || 282.353 || || || 17 || 1\34 || 35.294 || || ===Intervals:=== || degrees of 34edo || solfege || cents || approx. ratios of [[tel/2.3.5.13.17|2.3.5.13.17]] [[xenharmonic/subgroup|subgroup]] || additional ratios of the full [[xenharmonic/17-limit|17-limit]] || pseudo-traditional notation || || 0 || do || 0.0 || 1/1 || || C = B^^ = A## || || 1 || di || 35.294 || || || C ^ || || 2 || rih || 70.588 || || || Db = C ^^ = B# || || 3 || ra || 105.882 || 17/16, 18/17, 16/15 || 15/14 || C#v = Db^ || || 4 || ru || 141.176 || 13/12 || 14/13, 12/11 || C# || || 5 || reh || 176.471 || 10/9 || 11/10 || C#^ = Dv || || 6 || re || 211.765 || 9/8, 17/15 || 8/7 || D || || 7 || raw || 247.059 || 15/13 || || D^ || || 8 || meh || 282.353 || 20/17, 75/64 || 7/6, 13/11 || Eb || || 9 || me || 317.647 || 6/5 || 17/14 || D#v || || 10 || mu || 352.941 || 16/13 || 11/9 || D# || || 11 || mi || 388.235 || 5/4 || || || || 12 || maa || 423.529 || 51/40, 32/25 || 14/11, 9/7 || E || || 13 || maw || 458.823 || 13/10, 17/13 || 22/17 || E^ = Fv || || 14 || fa || 494.118 || 4/3 || || F || || 15 || fih || 529.412 || || 15/11 || F^ = E#v || || 16 || fu || 564.706 || 18/13 || 11/8 || Gb || || 17 || fi/se || 600 || 17/12, 24/17 || 7/5, 10/7 || Gb^ || || 18 || su || 635.294 || 13/9 || 16/11 || F# || || 19 || sih || 670.588 || || 22/15 || F#^ || || 20 || sol || 705.882 || 3/2 || || G || || 21 || saw || 741.176 || 20/13, 26/17 || 17/11 || G^ || || 22 || leh || 776.471 || 25/16, 80/51 || 14/9 || Ab || || 23 || le || 811.765 || 8/5 || || Ab^ || || 24 || lu || 847.059 || 13/8 || 18/11 || G# || || 25 || la || 882.353 || 5/3 || 28/17 || Av || || 26 || laa || 917.647 || 17/10 || 12/7, 22/13 || A || || 27 || law || 952.941 || 26/15 || || A^ = Bbv =G## || || 28 || teh || 988.235 || 16/9, 30/17 || 7/4 || Bb || || 29 || te || 1023.529 || 9/5 || 20/11 || Bb^ || || 30 || tu || 1058.823 || 24/13 || 13/7, 11/6 || A# || || 31 || ti || 1094.118 || 32/17, 17/9, 15/8 || 28/15 || A#^ = Bv || || 32 || taa || 1129.412 || || || B || || 33 || da || 1164.706 || || || B^ = A##v || ==<span style="background-color: #ffffff;">Notations</span>== The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away - the double-sharp adds up to a minor third away. This has led certain complainers, aiming to notate 17 edo (which is relatively popular), to want an extra character to raise something a small step of which. The 34 tone equal temperament, however, can be constructed from two equally spaced 17-note scales: a symbol indicating an adjustment of 1/34 up or down also serves the purpose of the previous sentence, by using two of it. This systemology of course emphasizes certain aspects of 34-edo which may not be most efficient expressions of some musical purposes: The reader can easily construct his own notation. One concern is that a system with 15 nominals for example, instead of seven, might be waste - of paper, space, brainmemory etc. if they aren't used consecutively and frequently. The system spelled out here has familiarity as an advantage or disadvantage. Tangentially, while the table uses ^ and v for "up" and "down", Kosmorosky prefers using filled in triangles because that's what I decided on years ago, and he's one of a handful of people worldwide who uses the tuning extensively, as far as I know, but whatever, I can't hit you through a computer if you use alternate symbols! ==<span style="background-color: #ffffff;">Commas</span>== <span style="background-color: #ffffff;">34-EDO [[xenharmonic/tempering out|tempers out]] the following [[xenharmonic/comma|comma]]s. (Note: This assumes the [[xenharmonic/val|val]] < [[tel/34 54 79 95 118 126|34 54 79 95 118 126]] |.)</span> ||~ <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">**Comma**</span></span> ||~ <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">**Monzo**</span></span> ||~ <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">**Value (Cents)**</span></span> ||~ <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">**Name 1**</span></span> ||~ <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">**Name 2**</span></span> ||~ <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">**Name 3**</span></span> || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">134217728/129140163</span></span> || | 27 -17 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">66.765</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">17-comma</span></span> || || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">20000/19683</span></span> || | 5 -9 4 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">27.660</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Minimal Diesis</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Tetracot Comma</span></span> || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">2048/2025</span></span> || | 11 -4 -2 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">19.553</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Diaschisma</span></span> || || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">393216/390625</span></span> || | 17 1 -8 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">11.445</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Würschmidt comma</span></span> || || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">15625/15552</span></span> || | -6 -5 6 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">8.1073</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Kleisma</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Semicomma Majeur</span></span> || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">1212717/1210381</span></span> || | 23 6 -14 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">3.338</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Vishnuzma</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Semisuper</span></span> || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">1029/1000</span></span> || | -3 1 -3 3 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">49.492</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Keega</span></span> || || || || <span style="display: block; text-align: center;"><span style="color: blue; display: block; text-align: center;">[[xenharmonic/49_48|49/48]]</span> </span> || | -4 -1 0 2 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">35.697</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Slendro Diesis</span></span> || || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">875/864</span></span> || | -5 -3 3 1 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">21.902</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Keema</span></span> || || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">126/125</span></span> || | 1 2 -3 1 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">13.795</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Starling comma</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Septimal semicomma</span></span> || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">100/99</span></span> || | 2 -2 2 0 -1> || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">17.399</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Ptolemisma</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Ptolemy's comma</span></span> || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">243/242</span></span> || | -1 5 0 0 -2 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">7.1391</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Rastma</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Neutral third comma</span></span> || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">385/384</span></span> || | -7 -1 1 1 1 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">4.5026</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Keenanisma</span></span> || || || || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">91/90</span></span> || | -1 -2 -1 1 0 1 > || <span style="display: block; text-align: right;"><span style="display: block; text-align: right;">19.120</span></span> || <span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Superleap</span></span> || || || ==Listen== * [[@http://www.archive.org/details/Ascension_105|Ascension]] * [[@https://www.youtube.com/watch?v=FXTM0HeuExk|Uncomfortable In Crowds (extended)]] by Robin Perry ==Links== * [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[xenharmonic/Larry Hanson|Larry Hanson]] * [[https://microstick.net|http://microstick.net/]] websites of Neil Haverstick * [[https://myspace.com/microstick]]
Original HTML content:
<html><head><title>34edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> </h1>
<br />
34edo divides the octave into 34 equal steps of approximately 35.29412 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">cent</a>s. 34edo contains two <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than <a class="wiki_link" href="/31edo">31edo</a>, but with a sharp fifth rather than a flat one, and supports hanson, srutal, tetracot, würschmidt and vishnu temperaments. It does less well in the 7-limit, with two mappings possible for 7/4: a flat one from the patent val, and a sharp one from the 34d val. By way of the patent val 34 supports keemun temperament, and 34d is an excellent alternative to <a class="wiki_link" href="/22edo">22edo</a> for pajara temperament. In the 11-limit the 34d val supports pajara, vishnu and würschmidt. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x--Approximations to Just Intonation"></a><!-- ws:end:WikiTextHeadingRule:2 -->Approximations to Just Intonation</h3>
Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo">17edo</a>, 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11.* 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 18/17, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the "syntonic comma" of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/meantone">meantone </a>system. In layman's terms while no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds, technically will be the same pitch as something, somewhere upon the cycle of seventeen fifths.<br />
<br />
<em>Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.</em> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/34_equal_temperament" rel="nofollow">Wikipedia</a>)<br />
<br />
*The sharpness of ~13 cents of 11/8 can fit somewhat with the 9/8 and 13/8 which both are about 7 cents sharp. Likewise the 20-cent sharpness or flatness of either approximation to 7/4 isn't impossible. The ability to tolerate these errors may depend on subtle natural changes in mood. <a class="wiki_link" href="/68edo">68edo</a>, double 34, has both these intervals in more perfect form, the 7 especially so.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x--34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:4 -->34edo and phi</h3>
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">Moment of Symmetry</a> scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and 140625/140608. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x--Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->Rank two temperaments</h3>
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/List%20of%2034edo%20rank%20two%20temperaments%20by%20badness">List of 34edo rank two temperaments by badness</a><br />
<table class="wiki_table">
<tr>
<th>Periods<br />
per octave<br />
</th>
<th>Generator<br />
</th>
<th>Cents<br />
</th>
<th>Linear temperaments<br />
</th>
</tr>
<tr>
<td>1<br />
</td>
<td>1\34<br />
</td>
<td>35.294<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>3\34<br />
</td>
<td>105.882<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>5\34<br />
</td>
<td>176.471<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tetracot">Tetracot</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Bunya">Bunya</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monkey">Monkey</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>7\34<br />
</td>
<td>247.059<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Immunity">Immunity</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>9\34<br />
</td>
<td>317.647<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hanson">Hanson</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Keemun">Keemun</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>11\34<br />
</td>
<td>388.235<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Wuerschmidt">Wuerschmidt</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Worschmidt">Worschmidt</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>13\34<br />
</td>
<td>458.824<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>15\34<br />
</td>
<td>529.412<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>1\34<br />
</td>
<td>35.294<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>2\34<br />
</td>
<td>70.588<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vishnu">Vishnu</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>3\34<br />
</td>
<td>105.882<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Srutal">Srutal</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Pajara">Pajara</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Diaschismic">Diaschismic</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>4\34<br />
</td>
<td>141.176<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Fifive">Fifive</a><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>5\34<br />
</td>
<td>176.471<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>6\34<br />
</td>
<td>211.765<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>7\34<br />
</td>
<td>247.059<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>8\34<br />
</td>
<td>282.353<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>1\34<br />
</td>
<td>35.294<br />
</td>
<td><br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x--Intervals:"></a><!-- ws:end:WikiTextHeadingRule:8 -->Intervals:</h3>
<table class="wiki_table">
<tr>
<td>degrees of 34edo<br />
</td>
<td>solfege<br />
</td>
<td>cents<br />
</td>
<td>approx. ratios of<br />
<a class="wiki_link" href="http://tel.wikispaces.com/2.3.5.13.17">2.3.5.13.17</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/subgroup">subgroup</a><br />
</td>
<td>additional ratios<br />
of the full <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17-limit">17-limit</a><br />
</td>
<td>pseudo-traditional<br />
notation<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>do<br />
</td>
<td>0.0<br />
</td>
<td>1/1<br />
</td>
<td><br />
</td>
<td>C = B^^ = A##<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>di<br />
</td>
<td>35.294<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>C ^<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>rih<br />
</td>
<td>70.588<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>Db = C ^^ = B#<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>ra<br />
</td>
<td>105.882<br />
</td>
<td>17/16, 18/17, 16/15<br />
</td>
<td>15/14<br />
</td>
<td>C#v = Db^<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>ru<br />
</td>
<td>141.176<br />
</td>
<td>13/12<br />
</td>
<td>14/13, 12/11<br />
</td>
<td>C#<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>reh<br />
</td>
<td>176.471<br />
</td>
<td>10/9<br />
</td>
<td>11/10<br />
</td>
<td>C#^ = Dv<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>re<br />
</td>
<td>211.765<br />
</td>
<td>9/8, 17/15<br />
</td>
<td>8/7<br />
</td>
<td>D<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>raw<br />
</td>
<td>247.059<br />
</td>
<td>15/13<br />
</td>
<td><br />
</td>
<td>D^<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>meh<br />
</td>
<td>282.353<br />
</td>
<td>20/17, 75/64<br />
</td>
<td>7/6, 13/11<br />
</td>
<td>Eb<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>me<br />
</td>
<td>317.647<br />
</td>
<td>6/5<br />
</td>
<td>17/14<br />
</td>
<td>D#v<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>mu<br />
</td>
<td>352.941<br />
</td>
<td>16/13<br />
</td>
<td>11/9<br />
</td>
<td>D#<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>mi<br />
</td>
<td>388.235<br />
</td>
<td>5/4<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>maa<br />
</td>
<td>423.529<br />
</td>
<td>51/40, 32/25<br />
</td>
<td>14/11, 9/7<br />
</td>
<td>E<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>maw<br />
</td>
<td>458.823<br />
</td>
<td>13/10, 17/13<br />
</td>
<td>22/17<br />
</td>
<td>E^ = Fv<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>fa<br />
</td>
<td>494.118<br />
</td>
<td>4/3<br />
</td>
<td><br />
</td>
<td>F<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>fih<br />
</td>
<td>529.412<br />
</td>
<td><br />
</td>
<td>15/11<br />
</td>
<td>F^ = E#v<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>fu<br />
</td>
<td>564.706<br />
</td>
<td>18/13<br />
</td>
<td>11/8<br />
</td>
<td>Gb<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>fi/se<br />
</td>
<td>600<br />
</td>
<td>17/12, 24/17<br />
</td>
<td>7/5, 10/7<br />
</td>
<td>Gb^<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>su<br />
</td>
<td>635.294<br />
</td>
<td>13/9<br />
</td>
<td>16/11<br />
</td>
<td>F#<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>sih<br />
</td>
<td>670.588<br />
</td>
<td><br />
</td>
<td>22/15<br />
</td>
<td>F#^<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>sol<br />
</td>
<td>705.882<br />
</td>
<td>3/2<br />
</td>
<td><br />
</td>
<td>G<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>saw<br />
</td>
<td>741.176<br />
</td>
<td>20/13, 26/17<br />
</td>
<td>17/11<br />
</td>
<td>G^<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>leh<br />
</td>
<td>776.471<br />
</td>
<td>25/16, 80/51<br />
</td>
<td>14/9<br />
</td>
<td>Ab<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>le<br />
</td>
<td>811.765<br />
</td>
<td>8/5<br />
</td>
<td><br />
</td>
<td>Ab^<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>lu<br />
</td>
<td>847.059<br />
</td>
<td>13/8<br />
</td>
<td>18/11<br />
</td>
<td>G#<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>la<br />
</td>
<td>882.353<br />
</td>
<td>5/3<br />
</td>
<td>28/17<br />
</td>
<td>Av<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>laa<br />
</td>
<td>917.647<br />
</td>
<td>17/10<br />
</td>
<td>12/7, 22/13<br />
</td>
<td>A<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>law<br />
</td>
<td>952.941<br />
</td>
<td>26/15<br />
</td>
<td><br />
</td>
<td>A^ = Bbv =G##<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>teh<br />
</td>
<td>988.235<br />
</td>
<td>16/9, 30/17<br />
</td>
<td>7/4<br />
</td>
<td>Bb<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>te<br />
</td>
<td>1023.529<br />
</td>
<td>9/5<br />
</td>
<td>20/11<br />
</td>
<td>Bb^<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>tu<br />
</td>
<td>1058.823<br />
</td>
<td>24/13<br />
</td>
<td>13/7, 11/6<br />
</td>
<td>A#<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>ti<br />
</td>
<td>1094.118<br />
</td>
<td>32/17, 17/9, 15/8<br />
</td>
<td>28/15<br />
</td>
<td>A#^ = Bv<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>taa<br />
</td>
<td>1129.412<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>B<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>da<br />
</td>
<td>1164.706<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>B^ = A##v<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-Notations"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="background-color: #ffffff;">Notations</span></h2>
The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away - the double-sharp adds up to a minor third away. This has led certain complainers, aiming to notate 17 edo (which is relatively popular), to want an extra character to raise something a small step of which. The 34 tone equal temperament, however, can be constructed from two equally spaced 17-note scales: a symbol indicating an adjustment of 1/34 up or down also serves the purpose of the previous sentence, by using two of it. This systemology of course emphasizes certain aspects of 34-edo which may not be most efficient expressions of some musical purposes: The reader can easily construct his own notation. One concern is that a system with 15 nominals for example, instead of seven, might be waste - of paper, space, brainmemory etc. if they aren't used consecutively and frequently. The system spelled out here has familiarity as an advantage or disadvantage. Tangentially, while the table uses ^ and v for "up" and "down", Kosmorosky prefers using filled in triangles because that's what I decided on years ago, and he's one of a handful of people worldwide who uses the tuning extensively, as far as I know, but whatever, I can't hit you through a computer if you use alternate symbols!<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x-Commas"></a><!-- ws:end:WikiTextHeadingRule:12 --><span style="background-color: #ffffff;">Commas</span></h2>
<span style="background-color: #ffffff;">34-EDO <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out">tempers out</a> the following <a class="wiki_link" href="http://xenharmonic.wikispaces.com/comma">comma</a>s. (Note: This assumes the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/val">val</a> < <a class="wiki_link" href="http://tel.wikispaces.com/34%2054%2079%2095%20118%20126">34 54 79 95 118 126</a> |.)</span><br />
<table class="wiki_table">
<tr>
<th><span style="display: block; text-align: center;"><span style="display: block; text-align: center;"><strong>Comma</strong></span></span><br />
</th>
<th><span style="display: block; text-align: center;"><span style="display: block; text-align: center;"><strong>Monzo</strong></span></span><br />
</th>
<th><span style="display: block; text-align: center;"><span style="display: block; text-align: center;"><strong>Value (Cents)</strong></span></span><br />
</th>
<th><span style="display: block; text-align: center;"><span style="display: block; text-align: center;"><strong>Name 1</strong></span></span><br />
</th>
<th><span style="display: block; text-align: center;"><span style="display: block; text-align: center;"><strong>Name 2</strong></span></span><br />
</th>
<th><span style="display: block; text-align: center;"><span style="display: block; text-align: center;"><strong>Name 3</strong></span></span><br />
</th>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">134217728/129140163</span></span><br />
</td>
<td>| 27 -17 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">66.765</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">17-comma</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">20000/19683</span></span><br />
</td>
<td>| 5 -9 4 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">27.660</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Minimal Diesis</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Tetracot Comma</span></span><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">2048/2025</span></span><br />
</td>
<td>| 11 -4 -2 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">19.553</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Diaschisma</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">393216/390625</span></span><br />
</td>
<td>| 17 1 -8 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">11.445</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Würschmidt comma</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">15625/15552</span></span><br />
</td>
<td>| -6 -5 6 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">8.1073</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Kleisma</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Semicomma Majeur</span></span><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">1212717/1210381</span></span><br />
</td>
<td>| 23 6 -14 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">3.338</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Vishnuzma</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Semisuper</span></span><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">1029/1000</span></span><br />
</td>
<td>| -3 1 -3 3 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">49.492</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Keega</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="color: blue; display: block; text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/49_48">49/48</a></span><br />
</span><br />
</td>
<td>| -4 -1 0 2 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">35.697</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Slendro Diesis</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">875/864</span></span><br />
</td>
<td>| -5 -3 3 1 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">21.902</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Keema</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">126/125</span></span><br />
</td>
<td>| 1 2 -3 1 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">13.795</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Starling comma</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Septimal semicomma</span></span><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">100/99</span></span><br />
</td>
<td>| 2 -2 2 0 -1><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">17.399</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Ptolemisma</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Ptolemy's comma</span></span><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">243/242</span></span><br />
</td>
<td>| -1 5 0 0 -2 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">7.1391</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Rastma</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Neutral third comma</span></span><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">385/384</span></span><br />
</td>
<td>| -7 -1 1 1 1 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">4.5026</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Keenanisma</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">91/90</span></span><br />
</td>
<td>| -1 -2 -1 1 0 1 ><br />
</td>
<td><span style="display: block; text-align: right;"><span style="display: block; text-align: right;">19.120</span></span><br />
</td>
<td><span style="display: block; text-align: center;"><span style="display: block; text-align: center;">Superleap</span></span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="x-Listen"></a><!-- ws:end:WikiTextHeadingRule:14 -->Listen</h2>
<ul><li><a class="wiki_link_ext" href="http://www.archive.org/details/Ascension_105" rel="nofollow" target="_blank">Ascension</a></li><li><a class="wiki_link_ext" href="https://www.youtube.com/watch?v=FXTM0HeuExk" rel="nofollow" target="_blank">Uncomfortable In Crowds (extended)</a> by Robin Perry</li></ul><!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:16 -->Links</h2>
<ul><li><a class="wiki_link_ext" href="http://www.microstick.net/34guitararticle.htm" rel="nofollow">34 Equal Guitar</a> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Larry%20Hanson">Larry Hanson</a></li><li><a class="wiki_link_ext" href="https://microstick.net" rel="nofollow">http://microstick.net/</a> websites of Neil Haverstick</li><li><a class="wiki_link_ext" href="https://myspace.com/microstick" rel="nofollow">https://myspace.com/microstick</a></li></ul></body></html>