34edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 575536419 - Original comment: ** |
Wikispaces>PiotrGrochowski **Imported revision 589911620 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-22 07:01:50 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>589911620</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup. | 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 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup. | ||
= Approximations to Just Intonation = | =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 one somewhere upon the cycle of seventeen fifths. | 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 one somewhere upon the cycle of seventeen fifths. | ||
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Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are accustom to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless [[68edo]] (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just. | Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are accustom to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless [[68edo]] (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just. | ||
= 34edo and phi = | =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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]]. | 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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]]. | ||
=Rank two temperaments= | =Rank two temperaments= | ||
[[xenharmonic/List of 34edo rank two temperaments by badness|List of 34edo rank two temperaments by badness]] | [[xenharmonic/List of 34edo rank two temperaments by badness|List of 34edo rank two temperaments by badness]] | ||
||~ Periods | ||~ Periods | ||
| Line 45: | Line 45: | ||
|| 17 || 1\34 || 35.294 || || | || 17 || 1\34 || 35.294 || || | ||
= Intervals = | =Intervals= | ||
|| degrees of 34edo || solfege || cents | || degrees of 34edo || solfege || cents | ||
DMS || approx. ratios of | DMS || approx. ratios of | ||
[[tel/2.3.5.13.17|2.3.5.13.17]] [[xenharmonic/subgroup|subgroup]] || additional ratios | [[tel/2.3.5.13.17|2.3.5.13.17]] [[xenharmonic/subgroup|subgroup]] || additional ratios | ||
of | of 7 and 11 || pseudo-traditional | ||
notation || | notation || | ||
|| 0 || do || 0.0 || 1/1 || || C = B^^ = A## || | || 0 || do || 0.0 || 1/1 || || C = B^^ = A## || | ||
|| 1 || di || 35.294 | || 1 || di || 35.294 | ||
10°35'18" || | 10°35'18" || 128/125 ([[128_125|diesis]]) || 50/49 || C ^ || | ||
|| 2 || rih || 70.588 | || 2 || rih || 70.588 | ||
21°10'35" || 25/24 || || Db = C ^^ = B# || | 21°10'35" || 25/24, 648/625 ([[648_625|large diesis]]) || || Db = C ^^ = B# || | ||
|| 3 || ra || 105.882 | || 3 || ra || 105.882 | ||
31°35'18" || 17/16, 18/17, 16/15 || 15/14 || C#v = Db^ || | 31°35'18" || 17/16, 18/17, 16/15 || 15/14 || C#v = Db^ || | ||
| Line 119: | Line 119: | ||
349°24'42" || || 49/25 || B^ = A##v || | 349°24'42" || || 49/25 || B^ = A##v || | ||
== Selected just intervals by error == | ==Selected just intervals by error== | ||
The following table shows how [[Just-24|some prominent just intervals]] are represented in 34edo (ordered by absolute error). | The following table shows how [[Just-24|some prominent just intervals]] are represented in 34edo (ordered by absolute error). | ||
|| **Interval, complement** || **Error (abs., in [[cent|cents]])** || | || **Interval, complement** || **Error (abs., in [[cent|cents]])** || | ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= | ||= [[15_13|15/13]], [[26_15|26/15]] ||= 0.682 || | ||
||= [[18_13|18/13]], [[13_9|13/9]] | ||= [[18_13|18/13]], [[13_9|13/9]] ||= 1.324 || | ||
||= [[5_4|5/4]], | ||= [[5_4|5/4]], [[8_5|8/5]] ||= 1.922 || | ||
||= [[6_5|6/5]], | ||= [[6_5|6/5]], [[5_3|5/3]] ||= 2.006 || | ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= | ||= [[13_12|13/12]], [[24_13|24/13]] ||= 2.604 || | ||
||= [[4_3|4/3]], | ||= [[4_3|4/3]], [[3_2|3/2]] ||= 3.927 || | ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= | ||= [[13_10|13/10]], [[20_13|20/13]] ||= 4.610 || | ||
||= [[11_9|11/9]], | ||= [[11_9|11/9]], [[18_11|18/11]] ||= 5.533 || | ||
||= [[16_15|16/15]], [[15_8|15/8]] | ||= [[16_15|16/15]], [[15_8|15/8]] ||= 5.849 || | ||
||= [[10_9|10/9]], | ||= [[10_9|10/9]], [[9_5|9/5]] ||= 5.933 || | ||
||= [[14_11|14/11]], [[11_7|11/7]] | ||= [[14_11|14/11]], [[11_7|11/7]] ||= 6.021 || | ||
||= [[16_13|16/13]], [[13_8|13/8]] | ||= [[16_13|16/13]], [[13_8|13/8]] ||= 6.531 || | ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= | ||= [[13_11|13/11]], [[22_13|22/13]] ||= 6.857 || | ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= | ||= [[15_11|15/11]], [[22_15|22/15]] ||= 7.539 || | ||
||= [[9_8|9/8]], | ||= [[9_8|9/8]], [[16_9|16/9]] ||= 7.855 || | ||
||= [[12_11|12/11]], [[11_6|11/6]] | ||= [[12_11|12/11]], [[11_6|11/6]] ||= 9.461 || | ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= 11.466 || | ||= [[11_10|11/10]], [[20_11|20/11]] ||= 11.466 || | ||
||= [[9_7|9/7]], | ||= [[9_7|9/7]], [[14_9|14/9]] ||= 11.555 || | ||
||= [[14_13|14/13]], [[13_7|13/7]] | ||= [[14_13|14/13]], [[13_7|13/7]] ||= 12.878 || | ||
||= [[11_8|11/8]], | ||= [[11_8|11/8]], [[16_11|16/11]] ||= 13.388 || | ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= 13.560 || | ||= [[15_14|15/14]], [[28_15|28/15]] ||= 13.560 || | ||
||= [[7_6|7/6]], | ||= [[7_6|7/6]], [[12_7|12/7]] ||= 15.482 || | ||
||= [[8_7|8/7]], | ||= [[8_7|8/7]], [[7_4|7/4]] ||= 15.885 || | ||
||= [[7_5|7/5]], | ||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 || | ||
= Notations = | =Notations= | ||
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 means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which //may not be most efficient expressions of some musical purposes.// The reader can construct his own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C ... F, instead of seven, might be waste - of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended. | 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 means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which //may not be most efficient expressions of some musical purposes.// The reader can construct his own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C ... F, instead of seven, might be waste - of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended. | ||
= Commas = | =Commas= | ||
34-EDO [[tempering out|tempers out]] the following [[ | 34-EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] < [[tel/34 54 79 95 118 126|34 54 79 95 118 126]] |.) | ||
||= **Comma** ||= **Monzo** ||= **Value (Cents)** ||= **Names** || | ||= **Comma** ||= **Monzo** ||= **Value (Cents)** ||= **Names** || | ||
||= 134217728/129140163 || | 27 -17 > ||> 66.765 ||= 17-comma | ||= 134217728/129140163 || | 27 -17 > ||> 66.765 ||= 17-comma || | ||
||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis, Tetracot Comma | ||= 20000/19683 || | 5 -9 4 > ||> 27.660 ||= Minimal Diesis, Tetracot Comma || | ||
||= 2048/2025 || | 11 -4 -2 > ||> 19.553 ||= Diaschisma | ||= 2048/2025 || | 11 -4 -2 > ||> 19.553 ||= Diaschisma || | ||
||= 393216/390625 || | 17 1 -8 > ||> 11.445 ||= Würschmidt comma | ||= 393216/390625 || | 17 1 -8 > ||> 11.445 ||= Würschmidt comma || | ||
||= 15625/15552 || | -6 -5 6 > ||> 8.1073 ||= Kleisma, Semicomma Majeur | ||= 15625/15552 || | -6 -5 6 > ||> 8.1073 ||= Kleisma, Semicomma Majeur || | ||
||= 1212717/1210381 || | 23 6 -14 > ||> 3.338 ||= Vishnuzma, Semisuper | ||= 1212717/1210381 || | 23 6 -14 > ||> 3.338 ||= Vishnuzma, Semisuper || | ||
||= 1029/1000 || | -3 1 -3 3 > ||> 49.492 ||= Keega | ||= 1029/1000 || | -3 1 -3 3 > ||> 49.492 ||= Keega || | ||
||= [[49_48|49/48]] || | -4 -1 0 2 > ||> 35.697 ||= Slendro Diesis | ||= [[49_48|49/48]] || | -4 -1 0 2 > ||> 35.697 ||= Slendro Diesis || | ||
||= 875/864 || | -5 -3 3 1 > ||> 21.902 ||= Keema | ||= 875/864 || | -5 -3 3 1 > ||> 21.902 ||= Keema || | ||
||= 126/125 || | 1 2 -3 1 > ||> 13.795 ||= Starling comma, Septimal semicomma | ||= 126/125 || | 1 2 -3 1 > ||> 13.795 ||= Starling comma, Septimal semicomma || | ||
||= 100/99 || | 2 -2 2 0 -1> ||> 17.399 ||= Ptolemisma, Ptolemy's comma | ||= 100/99 || | 2 -2 2 0 -1> ||> 17.399 ||= Ptolemisma, Ptolemy's comma || | ||
||= 243/242 || | -1 5 0 0 -2 > ||> 7.1391 ||= Rastma, Neutral third comma | ||= 243/242 || | -1 5 0 0 -2 > ||> 7.1391 ||= Rastma, Neutral third comma || | ||
||= 385/384 || | -7 -1 1 1 1 > ||> 4.5026 ||= Keenanisma | ||= 385/384 || | -7 -1 1 1 1 > ||> 4.5026 ||= Keenanisma || | ||
||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.120 ||= Superleap | ||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.120 ||= Superleap || | ||
= Listen = | =Listen= | ||
* [[@http://www.archive.org/details/Ascension_105|Ascension]] | * [[@http://www.archive.org/details/Ascension_105|Ascension]] | ||
* [[@https://www.youtube.com/watch?v=FXTM0HeuExk|Uncomfortable In Crowds (extended)]] by Robin Perry | * [[@https://www.youtube.com/watch?v=FXTM0HeuExk|Uncomfortable In Crowds (extended)]] by Robin Perry | ||
= Links = | =Links= | ||
* [[http://www.microstick.net/34guitararticle.htm|34 Equal Guitar]] by [[xenharmonic/Larry Hanson|Larry Hanson]] | * [[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://microstick.net|http://microstick.net/]] websites of Neil Haverstick | ||
* [[https://myspace.com/microstick]]</pre></div> | * [[https://myspace.com/microstick]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>34edo</title></head><body><!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Approximations to Just Intonation"> Approximations to Just Intonation </a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#x34edo and phi"> 34edo and phi </a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Intervals"> Intervals </a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Notations"> Notations </a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Commas"> Commas </a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Listen"> Listen </a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Links"> Links </a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>34edo</title></head><body><!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Approximations to Just Intonation">Approximations to Just Intonation</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#x34edo and phi">34edo and phi</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Notations">Notations</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Listen">Listen</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | ||
<!-- ws:end:WikiTextTocRule:28 --><hr /> | <!-- ws:end:WikiTextTocRule:28 --><hr /> | ||
<br /> | <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 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup.<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 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Approximations to Just Intonation"></a><!-- ws:end:WikiTextHeadingRule:0 --> Approximations to Just Intonation </h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Approximations to Just Intonation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Approximations to Just Intonation</h1> | ||
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 &quot;syntonic comma&quot; 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 one somewhere upon the cycle of seventeen fifths.<br /> | 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 &quot;syntonic comma&quot; 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 one somewhere upon the cycle of seventeen fifths.<br /> | ||
<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 /> | <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 /> | ||
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Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper &quot;dominant seventh&quot; found in jazz - which some listeners are accustom to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless <a class="wiki_link" href="/68edo">68edo</a> (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just.<br /> | Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper &quot;dominant seventh&quot; found in jazz - which some listeners are accustom to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless <a class="wiki_link" href="/68edo">68edo</a> (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:2 --> 34edo and phi </h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x34edo and phi"></a><!-- ws:end:WikiTextHeadingRule:2 -->34edo and phi</h1> | ||
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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and <a class="wiki_link" href="/36edo">36edo</a>.<br /> | 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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and <a class="wiki_link" href="/36edo">36edo</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h1> | ||
<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 /> | <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 /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:6 --> Intervals </h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:6 -->Intervals</h1> | ||
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</td> | </td> | ||
<td>additional ratios<br /> | <td>additional ratios<br /> | ||
of | of 7 and 11<br /> | ||
</td> | </td> | ||
<td>pseudo-traditional<br /> | <td>pseudo-traditional<br /> | ||
| Line 426: | Line 426: | ||
10°35'18&quot;<br /> | 10°35'18&quot;<br /> | ||
</td> | </td> | ||
<td><br /> | <td>128/125 (<a class="wiki_link" href="/128_125">diesis</a>)<br /> | ||
</td> | </td> | ||
<td>50/49<br /> | <td>50/49<br /> | ||
| Line 441: | Line 441: | ||
21°10'35&quot;<br /> | 21°10'35&quot;<br /> | ||
</td> | </td> | ||
<td>25/24<br /> | <td>25/24, 648/625 (<a class="wiki_link" href="/648_625">large diesis</a>)<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 916: | Line 916: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:8 --> Selected just intervals by error </h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:8 -->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 34edo (ordered by absolute error).<br /> | The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 34edo (ordered by absolute error).<br /> | ||
| Line 940: | Line 940: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, | <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> | ||
<td style="text-align: center;">1.922<br /> | <td style="text-align: center;">1.922<br /> | ||
| Line 946: | Line 946: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, | <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> | ||
<td style="text-align: center;">2.006<br /> | <td style="text-align: center;">2.006<br /> | ||
| Line 958: | Line 958: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, | <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> | ||
<td style="text-align: center;">3.927<br /> | <td style="text-align: center;">3.927<br /> | ||
| Line 970: | Line 970: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, | <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> | ||
<td style="text-align: center;">5.533<br /> | <td style="text-align: center;">5.533<br /> | ||
| Line 982: | Line 982: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, | <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> | ||
<td style="text-align: center;">5.933<br /> | <td style="text-align: center;">5.933<br /> | ||
| Line 1,012: | Line 1,012: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, | <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> | ||
<td style="text-align: center;">7.855<br /> | <td style="text-align: center;">7.855<br /> | ||
| Line 1,030: | Line 1,030: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, | <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> | ||
<td style="text-align: center;">11.555<br /> | <td style="text-align: center;">11.555<br /> | ||
| Line 1,042: | Line 1,042: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, | <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> | ||
<td style="text-align: center;">13.388<br /> | <td style="text-align: center;">13.388<br /> | ||
| Line 1,054: | Line 1,054: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, | <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> | ||
<td style="text-align: center;">15.482<br /> | <td style="text-align: center;">15.482<br /> | ||
| Line 1,060: | Line 1,060: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, | <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> | ||
<td style="text-align: center;">15.885<br /> | <td style="text-align: center;">15.885<br /> | ||
| Line 1,066: | Line 1,066: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, | <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> | ||
<td style="text-align: center;">17.488<br /> | <td style="text-align: center;">17.488<br /> | ||
| Line 1,074: | Line 1,074: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Notations"></a><!-- ws:end:WikiTextHeadingRule:10 --> Notations </h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Notations"></a><!-- ws:end:WikiTextHeadingRule:10 -->Notations</h1> | ||
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 means a minor third away from the natural. This has led certain &quot;complainers&quot;, in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which <em>may not be most efficient expressions of some musical purposes.</em> The reader can construct his own notation to the needs of the music and performer. As an example, a system with 15 &quot;nominals&quot; like A, B, C ... F, instead of seven, might be waste - of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for &quot;up&quot; and &quot;down&quot;, but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended. <br /> | 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 means a minor third away from the natural. This has led certain &quot;complainers&quot;, in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which <em>may not be most efficient expressions of some musical purposes.</em> The reader can construct his own notation to the needs of the music and performer. As an example, a system with 15 &quot;nominals&quot; like A, B, C ... F, instead of seven, might be waste - of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for &quot;up&quot; and &quot;down&quot;, but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:12 --> Commas </h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:12 -->Commas</h1> | ||
34-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 the <a class="wiki_link" href="/val">val</a> &lt; <a class="wiki_link" href="http://tel.wikispaces.com/34%2054%2079%2095%20118%20126">34 54 79 95 118 126</a> |.)<br /> | 34-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 the <a class="wiki_link" href="/val">val</a> &lt; <a class="wiki_link" href="http://tel.wikispaces.com/34%2054%2079%2095%20118%20126">34 54 79 95 118 126</a> |.)<br /> | ||
| Line 1,235: | Line 1,235: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Listen"></a><!-- ws:end:WikiTextHeadingRule:14 --> Listen </h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Listen"></a><!-- ws:end:WikiTextHeadingRule:14 -->Listen</h1> | ||
<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><br /> | <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><br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:16 --> Links </h1> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:16 -->Links</h1> | ||
<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></pre></div> | <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></pre></div> | ||