29edo
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author hstraub and made on 2017-06-10 07:46:47 UTC.
- The original revision id was 614499797.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
<span style="display: block; text-align: right;">[[xenharmonie/29edo|Deutsch]] </span> [[toc|flat]] ---- =<span style="color: #ff4700; font-family: 'Times New Roman',Times,serif; font-size: 113%;">29 tone equal temperament</span>= 29edo divides the 2:1 [[xenharmonic/octave|octave]] into 29 equal steps of approximately 41.37931 [[cent|cents]]. It is the 10th [[prime numbers|prime]] edo, following [[23edo]] and coming before [[31edo]]. 29 is the lowest edo which approximates the [[xenharmonic/3_2|3:2]] just fifth more accurately than [[xenharmonic/12edo|12edo]]: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a [[xenharmonic/positive temperament|positive temperament]] -- a Superpythagorean instead of a Meantone system. The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[xenharmonic/consistent|consistent]]ly represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the [[xenharmonic/5-limit|5-limit]], 49/48 in the [[xenharmonic/7-limit|7-limit]], 55/54 in the [[xenharmonic/11-limit|11-limit]], and 65/64 in the [[xenharmonic/13-limit|13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[xenharmonic/19edo|19edo]] for [[xenharmonic/Marvel temperaments|negri]], as well as an alternative to [[xenharmonic/22edo|22edo]] or [[xenharmonic/15edo|15edo]] for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively). Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[xenharmonic/Schismatic family|garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Chromatic pairs#Edson|edson temperaament]] with essentially perfect accuracy, only 0.034 cents sharp of it. Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[xenharmonic/The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[xenharmonic/petrmic triad|petrmic triad]], a 13-limit [[xenharmonic/Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[xenharmonic/k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[xenharmonic/k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas. 29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively). A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone. =Intervals and linear temperaments= [[List of 29et rank two temperaments by badness]] ||~ Degree ||~ Cents ||~ Approx. ratiosof the [[15-limit]] ||||||~ [[xenharmonic/Ups and Downs Notation|Ups and downs]] notation ||~ Generator for temperaments || ||= 0 ||= 0 ||= 1/1 ||= P1 ||= unison ||= D ||= || ||= 1 ||= 41.379 ||= 25/24~33/32~56/55~81/80 ||= ^1, vm2 ||= up unison, downminor 2nd ||= D^, Ebv ||= || ||= 2 ||= 82.759 ||= 21/20 ||= m2 ||= minor 2nd ||= Eb ||= [[xenharmonic/Nautilus|Nautilus]] || ||= 3 ||= 124.138 ||= 16/15, 15/14, 14/13, 13/12 ||= ^m2 ||= upminor 2nd ||= Eb^ ||= [[xenharmonic/Negri|Negri]]/[[xenharmonic/Negril|Negril]] || ||= 4 ||= 165.517 ||= 12/11, 11/10 ||= vM2 ||= downmajor 2nd ||= Ev ||= [[xenharmonic/Porcupine|Porcupine]]/[[xenharmonic/Porky|Porky]]/[[xenharmonic/Coendou|Coendou]] || ||= 5 ||= 206.897 ||= 9/8 ||= M2 ||= major 2nd ||= E ||= || ||= 6 ||= 248.276 ||= 8/7, 7/6, 15/13 ||= ^M2, vm3 ||= upmajor 2nd, downminor 3rd ||= E^, Fv ||= [[xenharmonic/Chromatic pairs#Bridgetown|Bridgetown]]/[[xenharmonic/Immunity|Immunity]] || ||= 7· ||= 289.655 ||= 13/11 ||= m3 ||= minor 3rd ||= F ||= || ||= 8 ||= 331.035 ||= 6/5, 11/9 ||= ^m3 ||= upminor 3rd ||= F^ ||= || ||= 9 ||= 372.414 ||= 5/4, 16/13 ||= vM3 ||= downmajor 3rd ||= F#v ||= || ||= 10 ||= 413.793 ||= 14/11 ||= M3 ||= major 3rd ||= F# ||= [[xenharmonic/Roman|Roman]] || ||= 11 ||= 455.172 ||= 9/7, 13/10 ||= ^M3, v4 ||= upmajor 3rd down 4th ||= F#^, Gv ||= [[xenharmonic/Ammonite|Ammonite]] || ||= 12· ||= 496.552 ||= 4/3 ||= P4 ||= 4th ||= G ||= [[xenharmonic/Cassandra|Cassandra]] [[Chromatic pairs#Edson|Edson]] [[Chromatic pairs#Pepperoni|Pepperoni]] || ||= 13 ||= 537.931 ||= 11/8, 15/11 ||= ^4 ||= up 4th ||= G^ ||= [[Wilsec]] || ||= 14 ||= 579.310 ||= 7/5, 18/13 ||= vA4, d5 ||= downaug 4th, dim 5th ||= G#v, Ab ||= [[xenharmonic/Tritonic|Tritonic]] || ||= 15 ||= 620.690 ||= 10/7, 13/9 ||= A4, ^d5 ||= aug 4th, updim 5th ||= G#, Ab^ ||= || ||= 16 ||= 662.069 ||= 16/11, 22/15 ||= v5 ||= down 5th ||= Av ||= || ||= 17· ||= 703.448 ||= 3/2 ||= P5 ||= 5th ||= A ||= || ||= 18 ||= 744.828 ||= 14/9, 20/13 ||= ^5, vm6 ||= up 5th, downminor 6th ||= A^, Bbv ||= || ||= 19 ||= 786.207 ||= 11/7 ||= m6 ||= minor 6th ||= Bb ||= || ||= 20 ||= 827.586 ||= 8/5, 13/8 ||= ^m6 ||= upminor 6th ||= Bb^ ||= || ||= 21 ||= 868.965 ||= 5/3, 18/11 ||= vM6 ||= downmajor 6th ||= Bv ||= || ||= 22· ||= 910.345 ||= 22/13 ||= M6 ||= major 6th ||= B ||= || ||= 23 ||= 951.724 ||= 7/4, 12/7, 26/15 ||= ^M6, vm7 ||= upmajor 6th, downminor 7th ||= B^, Cv ||= || ||= 24 ||= 993.103 ||= 16/9 ||= m7 ||= minor 7th ||= C ||= || ||= 25 ||= 1034.483 ||= 11/6, 20/11 ||= ^m7 ||= upminor 7th ||= C^ ||= || ||= 26 ||= 1075.862 ||= 15/8, 28/15, 13/7, 24/13 ||= vM7 ||= downmajor 7th ||= C#v ||= || ||= 27 ||= 1117.241 ||= 40/21 ||= M7 ||= major 7th ||= C# ||= || ||= 28 ||= 1158.621 ||= 48/25~64/33~55/28 ~160/81 ||= ^M7, v8 ||= upmajor 7th, down 8ve ||= C#^, Dv ||= || ||= 29 ||= 1200 ||= 2/1 ||= P8 ||= 8ve ||= D ||= || See also: [[29edo solfege]] Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]]. [[image:29edothumb.png caption="this example in Sagittal notation shows 29-edo as a fifth-tone system."]] ==Selected just intervals by error== The following table shows how [[Just-24|some prominent just intervals]] are represented in 29edo (ordered by absolute error). || **Interval, complement** || **Error (abs., in [[cent|cents]])** || ||= [[13_11|13/11]], [[22_13|22/13]] ||= 0.445 || ||= [[11_10|11/10]], [[20_11|20/11]] ||= 0.513 || ||= [[15_13|15/13]], [[26_15|26/15]] ||= 0.535 || ||= [[13_10|13/10]], [[20_13|20/13]] ||= 0.958 || ||= [[15_11|15/11]], [[22_15|22/15]] ||= 0.980 || ||= [[4_3|4/3]], [[3_2|3/2]] ||= 1.493 || ||= [[9_8|9/8]], [[16_9|16/9]] ||= 2.987 || ||= [[7_5|7/5]], [[10_7|10/7]] ||= 3.202 || ||= [[14_11|14/11]], [[11_7|11/7]] ||= 3.715 || ||= [[14_13|14/13]], [[13_7|13/7]] ||= 4.160 || ||= [[15_14|15/14]], [[28_15|28/15]] ||= 4.695 || ||= [[16_15|16/15]], [[15_8|15/8]] ||= 12.407 || ||= [[16_13|16/13]], [[13_8|13/8]] ||= 12.941 || ||= [[11_8|11/8]], [[16_11|16/11]] ||= 13.387 || ||= [[5_4|5/4]], [[8_5|8/5]] ||= 13.900 || ||= [[13_12|13/12]], [[24_13|24/13]] ||= 14.435 || ||= [[12_11|12/11]], [[11_6|11/6]] ||= 14.880 || ||= [[6_5|6/5]], [[5_3|5/3]] ||= 15.393 || ||= [[18_13|18/13]], [[13_9|13/9]] ||= 15.928 || ||= [[11_9|11/9]], [[18_11|18/11]] ||= 16.373 || ||= [[10_9|10/9]], [[9_5|9/5]] ||= 16.886 || ||= [[8_7|8/7]], [[7_4|7/4]] ||= 17.102 || ||= [[7_6|7/6]], [[12_7|12/7]] ||= 18.595 || ||= [[9_7|9/7]], [[14_9|14/9]] ||= 20.088 || =Commas= 29 EDO tempers out the following commas. (Note: This assumes the val < [[tel/29 46 67 81 100 107|29 46 67 81 100 107]] |, cent values rounded to 5 digits.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 || ||= 16875/16384 || | -14 3 4 > ||> 51.120 ||= Negri Comma ||= Double Augmentation Diesis || ||= 250/243 || | 1 -5 3 > ||> 49.166 ||= Maximal Diesis ||= Porcupine Comma || ||= 32805/32768 || | -15 8 1 > ||> 1.9537 ||= Schisma ||= || ||= 525/512 || | -9 1 2 1 > ||> 43.408 ||= Avicennma ||= Avicenna's Enharmonic Diesis || ||= 49/48 || | -4 -1 0 2 > ||> 35.697 ||= Slendro Diesis ||= || ||= 686/675 || | 1 -3 -2 3 > ||> 27.985 ||= Senga ||= || ||= 64827/64000 || | -9 3 -3 4 > ||> 22.227 ||= Squalentine ||= || ||= 3125/3087 || | 0 -2 5 -3 > ||> 21.181 ||= Gariboh ||= || ||= 50421/50000 || | -4 1 -5 5 > ||> 14.516 ||= Trimyna ||= || ||= 4000/3969 || | 5 -4 3 -2 > ||> 13.469 ||= Octagar ||= || ||= 225/224 || | -5 2 2 -1 > ||> 7.7115 ||= Septimal Kleisma ||= Marvel Comma || ||= 5120/5103 || | 10 -6 1 -1 > ||> 5.7578 ||= Hemifamity ||= || ||= || | 25 -14 0 -1 > ||> 3.8041 ||= Garischisma ||= || ||= 100/99 || | 2 -2 2 0 -1 > ||> 17.399 ||= Ptolemisma ||= || ||= 121/120 || | -3 -1 -1 0 2 > ||> 14.367 ||= Biyatisma ||= || ||= 896/891 || | 7 -4 0 1 -1 > ||> 9.6880 ||= Pentacircle ||= || ||= 441/440 || | -3 2 -1 2 -1 > ||> 3.9302 ||= Werckisma ||= || ||= 4000/3993 || | 5 -1 3 0 -3 > ||> 3.0323 ||= Wizardharry ||= || ||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0.17665 ||= Kalisma ||= Gauss' Comma || ||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.130 ||= Superleap ||= || =The Tetradecatonic System= A variant of porcupine supported in 29edo is [[xenharmonic/nautilus|nautilus]], which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine. The MOS nautilus[14] contains both "even" tetrads (approximating 4:5:6:7 or its inverse) as well as "odd" tetrads (approximating the "Bohlen-Pierce-like" chord 9:11:13:15, or its inverse). Both types are recognizable and consonant, if somewhat heavily tempered. Moreover, one of the four types of tetrads may be built on **each** scale degree of nautilus[14], thus there are as many chords as there are notes, so nautilus[14] has a "circulating" quality to it with as much freedom of modulation as possible. To be exact, there are 4 "major-even", 4 "minor-even", 3 "major-odd", and 3 "minor-odd" chords. Fourteen-note MOSes are worth looking at because taking every other note of them gives a heptatonic, and in many cases diatonic-like, scale. Nautilus[14] is no exception; although the resulting porcupine "diatonic" scale sounds somewhat different from diatonic scales generated from fifths, it can still provide some degree of familiarity. Furthermore, every diatonic chord progression will have at least one loose analogue in nautilus[14], although the chord types might change (for instance, it is possible to have a I-IV-V chord progression where the I is major-odd, and the IV and V are both major-even; the V in this case being on a narrow or "odd" fifth rather than a perfect or "even" fifth). The fact that the generator size is also a step size means that nautilus makes a good candidate for a [[https://en.wikipedia.org/wiki/Generalized_keyboard|generalized keyboard]]; the fingering of nautilus[14] becomes very simple as a result, perhaps even simpler than with traditional keyboards, despite there being more notes. If one can tolerate the tuning error (which is roughly equal to that of 12edo, albeit in the opposite direction for the 5- and 7-limits), this tetradecatonic scale is worth exploring. 29edo is often neglected since it falls so close to the much more popular and well-studied 31edo, but 29 does have its own advantages, and this is one of them. [[media type="file" key="Nautilus14_29edo.mp3"]] Nautilus[14] scale (Lsssssssssssss) in 29edo =Nicetone= 29edo is not a meantone system, but it could nonetheless be used as a basis for common-practice music if one considers the superfourth as a consonant, alternative type of fourth, and the 11:13:16 as an alternative type of consonant "doubly minor" triad. We can then use a diatonic scale such as 5435453 (which resembles Didymus' 5-limit JI diatonic scale, but with the syntonic comma being exaggerated in size). This scale has a very similar harmonic structure to a meantone diatonic scale, except that one of its minor triads is doubly-minor. Such a scale could be called "nicetone" as a play on meantone. Since it preserves most of the same 5-limit relationships, nicetone is only slightly xenharmonic (in contrast to [[superpyth]], which is quite blatantly so). The fact that 29edo's superfourth is within a cent of 15:11, and its 13:11 is within half a cent of a just 13:11, are both happy accidents. One just has to make that one is using a timbre that allows these higher-limit harmonic relationships to sound apparent and consonant enough to substitute for their simpler counterparts. =Scales= [[xenharmonic/bridgetown9|bridgetown9]] [[xenharmonic/bridgetown14|bridgetown14]] [[http://www.youtube.com/watch?v=uP2Z4Gy8lds|Escala Tonal de 17 tonos - Charles Loli]] =Music= [[http://www.microtonalismo.com/el-teclado-29-edo|Mp3 29EDO - Escala tonal de 17 notas]]by [[http://musicool.us/musicool/armonia.htm|Charles Loli A.]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/Paint%20in%20the%20Water%2029.mp3|Paint in the Water 29]] by [[xenharmonic/IgliashonJones|Igliashon Jones]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3|Nautilus Reverie]] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Howling%20of%20the%20Holy.mp3|Howling of the Holy]] by [[xenharmonic/IgliashonJones|Igliashon Jones]] [[http://micro.soonlabel.com/tuning-survey/daily20111026-bridgetown-14.mp3|Route 14 in Bridgetown]] by [[xenharmonic/Chris Vaisvil|Chris Vaisvil]] [[http://www.angelfire.com/mo/oljare/images/crowning.mid|The Crowning Song]] by Mats Öljare [[http://www.angelfire.com/mo/oljare/images/ninedays.mid|Nine Days Later]] by Mats Öljare [[http://www.angelfire.com/mo/oljare/images/stranded.mid|Stranded at Sea]] by Mats Öljare ==Instruments== [[@http://www.microtonalismo.com/|Guitar 29EDO]] * ====**[[http://www.microtonalismo.com/proyecto-xvii|Guitar 29EDO from Peruvian - Charles Loli and Antonio Huamani]]**==== > [[image:http://content.pimp-my-profile.com/i116/2/10/29/f_735065b21747.jpg width="321" height="891"]] [[@http://www.microtonalismo.com/|Bass 29EDO]] * ====**[[http://www.microtonalismo.com/proyecto-xvii|Bass 29EDO from Peruvian - Charles Loli and Antonio Huamani]]**==== > [[image:https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-prn1/r90/550502_538613626155939_2005925977_n.jpg width="305" height="936"]]
Original HTML content:
<html><head><title>29edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/29edo">Deutsch</a><br />
</span><br />
<!-- ws:start:WikiTextTocRule:23:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><a href="#x29 tone equal temperament">29 tone equal temperament</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Intervals and linear temperaments">Intervals and linear temperaments</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#The Tetradecatonic System">The Tetradecatonic System</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Nicetone">Nicetone</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: -->
<!-- ws:end:WikiTextTocRule:35 --><hr />
<br />
<!-- ws:start:WikiTextHeadingRule:1:<h1> --><h1 id="toc0"><a name="x29 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:1 --><span style="color: #ff4700; font-family: 'Times New Roman',Times,serif; font-size: 113%;">29 tone equal temperament</span></h1>
<br />
29edo divides the 2:1 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/octave">octave</a> into 29 equal steps of approximately 41.37931 <a class="wiki_link" href="/cent">cents</a>. It is the 10th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/23edo">23edo</a> and coming before <a class="wiki_link" href="/31edo">31edo</a>.<br />
<br />
29 is the lowest edo which approximates the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/3_2">3:2</a> just fifth more accurately than <a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo">12edo</a>: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/positive%20temperament">positive temperament</a> -- a Superpythagorean instead of a Meantone system.<br />
<br />
The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which <a class="wiki_link" href="http://xenharmonic.wikispaces.com/consistent">consistent</a>ly represents the 15 odd limit. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit">5-limit</a>, 49/48 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit">7-limit</a>, 55/54 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/11-limit">11-limit</a>, and 65/64 in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/13-limit">13-limit</a>. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19edo">19edo</a> for <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments">negri</a>, as well as an alternative to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/22edo">22edo</a> or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/15edo">15edo</a> for porcupine. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).<br />
<br />
Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Schismatic%20family">garibaldi temperament</a> which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for <a class="wiki_link" href="/Chromatic%20pairs#Edson">edson temperaament</a> with essentially perfect accuracy, only 0.034 cents sharp of it.<br />
<br />
Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Archipelago">barbados triad</a> 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/petrmic%20triad">petrmic triad</a>, a 13-limit <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Dyadic%20chord">essentially tempered dyadic chord</a>. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">3*29 subgroup</a> 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/k%2AN%20subgroups">2*29 subgroup</a> 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br />
<br />
29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).<br />
<br />
A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc1"><a name="Intervals and linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:3 -->Intervals and linear temperaments</h1>
<a class="wiki_link" href="/List%20of%2029et%20rank%20two%20temperaments%20by%20badness">List of 29et rank two temperaments by badness</a><br />
<br />
<table class="wiki_table">
<tr>
<th>Degree<br />
</th>
<th>Cents<br />
</th>
<th>Approx. ratiosof the <a class="wiki_link" href="/15-limit">15-limit</a><br />
</th>
<th colspan="3"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">Ups and downs</a> notation<br />
</th>
<th>Generator for temperaments<br />
</th>
</tr>
<tr>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
<td style="text-align: center;">P1<br />
</td>
<td style="text-align: center;">unison<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">41.379<br />
</td>
<td style="text-align: center;">25/24~33/32~56/55~81/80<br />
</td>
<td style="text-align: center;">^1, vm2<br />
</td>
<td style="text-align: center;">up unison,<br />
downminor 2nd<br />
</td>
<td style="text-align: center;">D^, Ebv<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">82.759<br />
</td>
<td style="text-align: center;">21/20<br />
</td>
<td style="text-align: center;">m2<br />
</td>
<td style="text-align: center;">minor 2nd<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Nautilus">Nautilus</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3<br />
</td>
<td style="text-align: center;">124.138<br />
</td>
<td style="text-align: center;">16/15, 15/14, 14/13, 13/12<br />
</td>
<td style="text-align: center;">^m2<br />
</td>
<td style="text-align: center;">upminor 2nd<br />
</td>
<td style="text-align: center;">Eb^<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Negri">Negri</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Negril">Negril</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">165.517<br />
</td>
<td style="text-align: center;">12/11, 11/10<br />
</td>
<td style="text-align: center;">vM2<br />
</td>
<td style="text-align: center;">downmajor 2nd<br />
</td>
<td style="text-align: center;">Ev<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Porcupine">Porcupine</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Porky">Porky</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Coendou">Coendou</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">5<br />
</td>
<td style="text-align: center;">206.897<br />
</td>
<td style="text-align: center;">9/8<br />
</td>
<td style="text-align: center;">M2<br />
</td>
<td style="text-align: center;">major 2nd<br />
</td>
<td style="text-align: center;">E<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">6<br />
</td>
<td style="text-align: center;">248.276<br />
</td>
<td style="text-align: center;">8/7, 7/6, 15/13<br />
</td>
<td style="text-align: center;">^M2, vm3<br />
</td>
<td style="text-align: center;">upmajor 2nd,<br />
downminor 3rd<br />
</td>
<td style="text-align: center;">E^, Fv<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chromatic%20pairs#Bridgetown">Bridgetown</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Immunity">Immunity</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">7·<br />
</td>
<td style="text-align: center;">289.655<br />
</td>
<td style="text-align: center;">13/11<br />
</td>
<td style="text-align: center;">m3<br />
</td>
<td style="text-align: center;">minor 3rd<br />
</td>
<td style="text-align: center;">F<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">8<br />
</td>
<td style="text-align: center;">331.035<br />
</td>
<td style="text-align: center;">6/5, 11/9<br />
</td>
<td style="text-align: center;">^m3<br />
</td>
<td style="text-align: center;">upminor 3rd<br />
</td>
<td style="text-align: center;">F^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9<br />
</td>
<td style="text-align: center;">372.414<br />
</td>
<td style="text-align: center;">5/4, 16/13<br />
</td>
<td style="text-align: center;">vM3<br />
</td>
<td style="text-align: center;">downmajor 3rd<br />
</td>
<td style="text-align: center;">F#v<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">10<br />
</td>
<td style="text-align: center;">413.793<br />
</td>
<td style="text-align: center;">14/11<br />
</td>
<td style="text-align: center;">M3<br />
</td>
<td style="text-align: center;">major 3rd<br />
</td>
<td style="text-align: center;">F#<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Roman">Roman</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">11<br />
</td>
<td style="text-align: center;">455.172<br />
</td>
<td style="text-align: center;">9/7, 13/10<br />
</td>
<td style="text-align: center;">^M3, v4<br />
</td>
<td style="text-align: center;">upmajor 3rd<br />
down 4th<br />
</td>
<td style="text-align: center;">F#^, Gv<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ammonite">Ammonite</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">12·<br />
</td>
<td style="text-align: center;">496.552<br />
</td>
<td style="text-align: center;">4/3<br />
</td>
<td style="text-align: center;">P4<br />
</td>
<td style="text-align: center;">4th<br />
</td>
<td style="text-align: center;">G<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Cassandra">Cassandra</a> <a class="wiki_link" href="/Chromatic%20pairs#Edson">Edson</a> <a class="wiki_link" href="/Chromatic%20pairs#Pepperoni">Pepperoni</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">13<br />
</td>
<td style="text-align: center;">537.931<br />
</td>
<td style="text-align: center;">11/8, 15/11<br />
</td>
<td style="text-align: center;">^4<br />
</td>
<td style="text-align: center;">up 4th<br />
</td>
<td style="text-align: center;">G^<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/Wilsec">Wilsec</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">14<br />
</td>
<td style="text-align: center;">579.310<br />
</td>
<td style="text-align: center;">7/5, 18/13<br />
</td>
<td style="text-align: center;">vA4, d5<br />
</td>
<td style="text-align: center;">downaug 4th,<br />
dim 5th<br />
</td>
<td style="text-align: center;">G#v, Ab<br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Tritonic">Tritonic</a><br />
</td>
</tr>
<tr>
<td style="text-align: center;">15<br />
</td>
<td style="text-align: center;">620.690<br />
</td>
<td style="text-align: center;">10/7, 13/9<br />
</td>
<td style="text-align: center;">A4, ^d5<br />
</td>
<td style="text-align: center;">aug 4th,<br />
updim 5th<br />
</td>
<td style="text-align: center;">G#, Ab^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">16<br />
</td>
<td style="text-align: center;">662.069<br />
</td>
<td style="text-align: center;">16/11, 22/15<br />
</td>
<td style="text-align: center;">v5<br />
</td>
<td style="text-align: center;">down 5th<br />
</td>
<td style="text-align: center;">Av<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">17·<br />
</td>
<td style="text-align: center;">703.448<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td style="text-align: center;">P5<br />
</td>
<td style="text-align: center;">5th<br />
</td>
<td style="text-align: center;">A<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">18<br />
</td>
<td style="text-align: center;">744.828<br />
</td>
<td style="text-align: center;">14/9, 20/13<br />
</td>
<td style="text-align: center;">^5, vm6<br />
</td>
<td style="text-align: center;">up 5th,<br />
downminor 6th<br />
</td>
<td style="text-align: center;">A^, Bbv<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">19<br />
</td>
<td style="text-align: center;">786.207<br />
</td>
<td style="text-align: center;">11/7<br />
</td>
<td style="text-align: center;">m6<br />
</td>
<td style="text-align: center;">minor 6th<br />
</td>
<td style="text-align: center;">Bb<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">20<br />
</td>
<td style="text-align: center;">827.586<br />
</td>
<td style="text-align: center;">8/5, 13/8<br />
</td>
<td style="text-align: center;">^m6<br />
</td>
<td style="text-align: center;">upminor 6th<br />
</td>
<td style="text-align: center;">Bb^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">21<br />
</td>
<td style="text-align: center;">868.965<br />
</td>
<td style="text-align: center;">5/3, 18/11<br />
</td>
<td style="text-align: center;">vM6<br />
</td>
<td style="text-align: center;">downmajor 6th<br />
</td>
<td style="text-align: center;">Bv<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">22·<br />
</td>
<td style="text-align: center;">910.345<br />
</td>
<td style="text-align: center;">22/13<br />
</td>
<td style="text-align: center;">M6<br />
</td>
<td style="text-align: center;">major 6th<br />
</td>
<td style="text-align: center;">B<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">23<br />
</td>
<td style="text-align: center;">951.724<br />
</td>
<td style="text-align: center;">7/4, 12/7, 26/15<br />
</td>
<td style="text-align: center;">^M6, vm7<br />
</td>
<td style="text-align: center;">upmajor 6th,<br />
downminor 7th<br />
</td>
<td style="text-align: center;">B^, Cv<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">24<br />
</td>
<td style="text-align: center;">993.103<br />
</td>
<td style="text-align: center;">16/9<br />
</td>
<td style="text-align: center;">m7<br />
</td>
<td style="text-align: center;">minor 7th<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">25<br />
</td>
<td style="text-align: center;">1034.483<br />
</td>
<td style="text-align: center;">11/6, 20/11<br />
</td>
<td style="text-align: center;">^m7<br />
</td>
<td style="text-align: center;">upminor 7th<br />
</td>
<td style="text-align: center;">C^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">26<br />
</td>
<td style="text-align: center;">1075.862<br />
</td>
<td style="text-align: center;">15/8, 28/15, 13/7, 24/13<br />
</td>
<td style="text-align: center;">vM7<br />
</td>
<td style="text-align: center;">downmajor 7th<br />
</td>
<td style="text-align: center;">C#v<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">27<br />
</td>
<td style="text-align: center;">1117.241<br />
</td>
<td style="text-align: center;">40/21<br />
</td>
<td style="text-align: center;">M7<br />
</td>
<td style="text-align: center;">major 7th<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">28<br />
</td>
<td style="text-align: center;">1158.621<br />
</td>
<td style="text-align: center;">48/25~64/33~55/28 ~160/81<br />
</td>
<td style="text-align: center;">^M7, v8<br />
</td>
<td style="text-align: center;">upmajor 7th,<br />
down 8ve<br />
</td>
<td style="text-align: center;">C#^, Dv<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">29<br />
</td>
<td style="text-align: center;">1200<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
<td style="text-align: center;">P8<br />
</td>
<td style="text-align: center;">8ve<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
See also: <a class="wiki_link" href="/29edo%20solfege">29edo solfege</a><br />
<br />
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs">Ups and Downs Notation - Chord names in other EDOs</a>.<br />
<!-- ws:start:WikiTextLocalImageRule:947:<img src="/file/view/29edothumb.png/277524658/29edothumb.png" alt="this example in Sagittal notation shows 29-edo as a fifth-tone system." title="this example in Sagittal notation shows 29-edo as a fifth-tone system." /> --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/29edothumb.png/277524658/29edothumb.png" alt="29edothumb.png" title="29edothumb.png" /></td></tr><tr><td class="imageCaption">this example in Sagittal notation shows 29-edo as a fifth-tone system.</td></tr></table><!-- ws:end:WikiTextLocalImageRule:947 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h2> --><h2 id="toc2"><a name="Intervals and linear temperaments-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:5 -->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 29edo (ordered by absolute error).<br />
<table class="wiki_table">
<tr>
<td><strong>Interval, complement</strong><br />
</td>
<td><strong>Error (abs., in <a class="wiki_link" href="/cent">cents</a>)</strong><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_11">13/11</a>, <a class="wiki_link" href="/22_13">22/13</a><br />
</td>
<td style="text-align: center;">0.445<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_10">11/10</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
<td style="text-align: center;">0.513<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_13">15/13</a>, <a class="wiki_link" href="/26_15">26/15</a><br />
</td>
<td style="text-align: center;">0.535<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_10">13/10</a>, <a class="wiki_link" href="/20_13">20/13</a><br />
</td>
<td style="text-align: center;">0.958<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/22_15">22/15</a><br />
</td>
<td style="text-align: center;">0.980<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br />
</td>
<td style="text-align: center;">1.493<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br />
</td>
<td style="text-align: center;">2.987<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br />
</td>
<td style="text-align: center;">3.202<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/11_7">11/7</a><br />
</td>
<td style="text-align: center;">3.715<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/13_7">13/7</a><br />
</td>
<td style="text-align: center;">4.160<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/28_15">28/15</a><br />
</td>
<td style="text-align: center;">4.695<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_8">15/8</a><br />
</td>
<td style="text-align: center;">12.407<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_13">16/13</a>, <a class="wiki_link" href="/13_8">13/8</a><br />
</td>
<td style="text-align: center;">12.941<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br />
</td>
<td style="text-align: center;">13.387<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br />
</td>
<td style="text-align: center;">13.900<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_12">13/12</a>, <a class="wiki_link" href="/24_13">24/13</a><br />
</td>
<td style="text-align: center;">14.435<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a>, <a class="wiki_link" href="/11_6">11/6</a><br />
</td>
<td style="text-align: center;">14.880<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br />
</td>
<td style="text-align: center;">15.393<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/18_13">18/13</a>, <a class="wiki_link" href="/13_9">13/9</a><br />
</td>
<td style="text-align: center;">15.928<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br />
</td>
<td style="text-align: center;">16.373<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br />
</td>
<td style="text-align: center;">16.886<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
<td style="text-align: center;">17.102<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br />
</td>
<td style="text-align: center;">18.595<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br />
</td>
<td style="text-align: center;">20.088<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:7:<h1> --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:7 -->Commas</h1>
29 EDO tempers out the following commas. (Note: This assumes the val < <a class="wiki_link" href="http://tel.wikispaces.com/29%2046%2067%2081%20100%20107">29 46 67 81 100 107</a> |, cent values rounded to 5 digits.)<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;">16875/16384<br />
</td>
<td>| -14 3 4 ><br />
</td>
<td style="text-align: right;">51.120<br />
</td>
<td style="text-align: center;">Negri Comma<br />
</td>
<td style="text-align: center;">Double Augmentation Diesis<br />
</td>
</tr>
<tr>
<td style="text-align: center;">250/243<br />
</td>
<td>| 1 -5 3 ><br />
</td>
<td style="text-align: right;">49.166<br />
</td>
<td style="text-align: center;">Maximal Diesis<br />
</td>
<td style="text-align: center;">Porcupine Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">32805/32768<br />
</td>
<td>| -15 8 1 ><br />
</td>
<td style="text-align: right;">1.9537<br />
</td>
<td style="text-align: center;">Schisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">525/512<br />
</td>
<td>| -9 1 2 1 ><br />
</td>
<td style="text-align: right;">43.408<br />
</td>
<td style="text-align: center;">Avicennma<br />
</td>
<td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
</td>
</tr>
<tr>
<td style="text-align: center;">49/48<br />
</td>
<td>| -4 -1 0 2 ><br />
</td>
<td style="text-align: right;">35.697<br />
</td>
<td style="text-align: center;">Slendro Diesis<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">686/675<br />
</td>
<td>| 1 -3 -2 3 ><br />
</td>
<td style="text-align: right;">27.985<br />
</td>
<td style="text-align: center;">Senga<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">64827/64000<br />
</td>
<td>| -9 3 -3 4 ><br />
</td>
<td style="text-align: right;">22.227<br />
</td>
<td style="text-align: center;">Squalentine<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3125/3087<br />
</td>
<td>| 0 -2 5 -3 ><br />
</td>
<td style="text-align: right;">21.181<br />
</td>
<td style="text-align: center;">Gariboh<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">50421/50000<br />
</td>
<td>| -4 1 -5 5 ><br />
</td>
<td style="text-align: right;">14.516<br />
</td>
<td style="text-align: center;">Trimyna<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3969<br />
</td>
<td>| 5 -4 3 -2 ><br />
</td>
<td style="text-align: right;">13.469<br />
</td>
<td style="text-align: center;">Octagar<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">225/224<br />
</td>
<td>| -5 2 2 -1 ><br />
</td>
<td style="text-align: right;">7.7115<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;">5120/5103<br />
</td>
<td>| 10 -6 1 -1 ><br />
</td>
<td style="text-align: right;">5.7578<br />
</td>
<td style="text-align: center;">Hemifamity<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td>| 25 -14 0 -1 ><br />
</td>
<td style="text-align: right;">3.8041<br />
</td>
<td style="text-align: center;">Garischisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">100/99<br />
</td>
<td>| 2 -2 2 0 -1 ><br />
</td>
<td style="text-align: right;">17.399<br />
</td>
<td style="text-align: center;">Ptolemisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">121/120<br />
</td>
<td>| -3 -1 -1 0 2 ><br />
</td>
<td style="text-align: right;">14.367<br />
</td>
<td style="text-align: center;">Biyatisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">896/891<br />
</td>
<td>| 7 -4 0 1 -1 ><br />
</td>
<td style="text-align: right;">9.6880<br />
</td>
<td style="text-align: center;">Pentacircle<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">441/440<br />
</td>
<td>| -3 2 -1 2 -1 ><br />
</td>
<td style="text-align: right;">3.9302<br />
</td>
<td style="text-align: center;">Werckisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3993<br />
</td>
<td>| 5 -1 3 0 -3 ><br />
</td>
<td style="text-align: right;">3.0323<br />
</td>
<td style="text-align: center;">Wizardharry<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9801/9800<br />
</td>
<td>| -3 4 -2 -2 2 ><br />
</td>
<td style="text-align: right;">0.17665<br />
</td>
<td style="text-align: center;">Kalisma<br />
</td>
<td style="text-align: center;">Gauss' Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">91/90<br />
</td>
<td>| -1 -2 -1 1 0 1 ><br />
</td>
<td style="text-align: right;">19.130<br />
</td>
<td style="text-align: center;">Superleap<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:9:<h1> --><h1 id="toc4"><a name="The Tetradecatonic System"></a><!-- ws:end:WikiTextHeadingRule:9 -->The Tetradecatonic System</h1>
<br />
A variant of porcupine supported in 29edo is <a class="wiki_link" href="http://xenharmonic.wikispaces.com/nautilus">nautilus</a>, which splits the porcupine generator in half (tempering out 50:49 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.<br />
<br />
The MOS nautilus[14] contains both "even" tetrads (approximating 4:5:6:7 or its inverse) as well as "odd" tetrads (approximating the "Bohlen-Pierce-like" chord 9:11:13:15, or its inverse). Both types are recognizable and consonant, if somewhat heavily tempered. Moreover, one of the four types of tetrads may be built on <strong>each</strong> scale degree of nautilus[14], thus there are as many chords as there are notes, so nautilus[14] has a "circulating" quality to it with as much freedom of modulation as possible. To be exact, there are 4 "major-even", 4 "minor-even", 3 "major-odd", and 3 "minor-odd" chords.<br />
<br />
Fourteen-note MOSes are worth looking at because taking every other note of them gives a heptatonic, and in many cases diatonic-like, scale. Nautilus[14] is no exception; although the resulting porcupine "diatonic" scale sounds somewhat different from diatonic scales generated from fifths, it can still provide some degree of familiarity. Furthermore, every diatonic chord progression will have at least one loose analogue in nautilus[14], although the chord types might change (for instance, it is possible to have a I-IV-V chord progression where the I is major-odd, and the IV and V are both major-even; the V in this case being on a narrow or "odd" fifth rather than a perfect or "even" fifth).<br />
<br />
The fact that the generator size is also a step size means that nautilus makes a good candidate for a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Generalized_keyboard" rel="nofollow">generalized keyboard</a>; the fingering of nautilus[14] becomes very simple as a result, perhaps even simpler than with traditional keyboards, despite there being more notes.<br />
<br />
If one can tolerate the tuning error (which is roughly equal to that of 12edo, albeit in the opposite direction for the 5- and 7-limits), this tetradecatonic scale is worth exploring. 29edo is often neglected since it falls so close to the much more popular and well-studied 31edo, but 29 does have its own advantages, and this is one of them.<br />
<br />
<!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/file-audio/Nautilus14_29edo.mp3?h=20&w=240" class="WikiMedia WikiMediaFile" id="wikitext@@media@@type=&quot;file&quot; key=&quot;Nautilus14_29edo.mp3&quot;" title="Local Media File"height="20" width="240"/> --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FNautilus14_29edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:0 --><br />
Nautilus[14] scale (Lsssssssssssss) in 29edo<br />
<br />
<!-- ws:start:WikiTextHeadingRule:11:<h1> --><h1 id="toc5"><a name="Nicetone"></a><!-- ws:end:WikiTextHeadingRule:11 -->Nicetone</h1>
<br />
29edo is not a meantone system, but it could nonetheless be used as a basis for common-practice music if one considers the superfourth as a consonant, alternative type of fourth, and the 11:13:16 as an alternative type of consonant "doubly minor" triad. We can then use a diatonic scale such as 5435453 (which resembles Didymus' 5-limit JI diatonic scale, but with the syntonic comma being exaggerated in size). This scale has a very similar harmonic structure to a meantone diatonic scale, except that one of its minor triads is doubly-minor.<br />
<br />
Such a scale could be called "nicetone" as a play on meantone. Since it preserves most of the same 5-limit relationships, nicetone is only slightly xenharmonic (in contrast to <a class="wiki_link" href="/superpyth">superpyth</a>, which is quite blatantly so). The fact that 29edo's superfourth is within a cent of 15:11, and its 13:11 is within half a cent of a just 13:11, are both happy accidents. One just has to make that one is using a timbre that allows these higher-limit harmonic relationships to sound apparent and consonant enough to substitute for their simpler counterparts.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:13:<h1> --><h1 id="toc6"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:13 -->Scales</h1>
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/bridgetown9">bridgetown9</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/bridgetown14">bridgetown14</a><br />
<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=uP2Z4Gy8lds" rel="nofollow">Escala Tonal de 17 tonos - Charles Loli</a><br />
<!-- ws:start:WikiTextHeadingRule:15:<h1> --><h1 id="toc7"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:15 -->Music</h1>
<a class="wiki_link_ext" href="http://www.microtonalismo.com/el-teclado-29-edo" rel="nofollow">Mp3 29EDO - Escala tonal de 17 notas</a>by <a class="wiki_link_ext" href="http://musicool.us/musicool/armonia.htm" rel="nofollow">Charles Loli A.</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/Paint%20in%20the%20Water%2029.mp3" rel="nofollow">Paint in the Water 29</a> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/IgliashonJones">Igliashon Jones</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3" rel="nofollow">Nautilus Reverie</a> by <a class="wiki_link" href="/IgliashonJones">Igliashon Calvin Jones-Coolidge</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Howling%20of%20the%20Holy.mp3" rel="nofollow">Howling of the Holy</a> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/IgliashonJones">Igliashon Jones</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/tuning-survey/daily20111026-bridgetown-14.mp3" rel="nofollow">Route 14 in Bridgetown</a> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://www.angelfire.com/mo/oljare/images/crowning.mid" rel="nofollow">The Crowning Song</a> by Mats Öljare<br />
<a class="wiki_link_ext" href="http://www.angelfire.com/mo/oljare/images/ninedays.mid" rel="nofollow">Nine Days Later</a> by Mats Öljare<br />
<a class="wiki_link_ext" href="http://www.angelfire.com/mo/oljare/images/stranded.mid" rel="nofollow">Stranded at Sea</a> by Mats Öljare<br />
<br />
<!-- ws:start:WikiTextHeadingRule:17:<h2> --><h2 id="toc8"><a name="Music-Instruments"></a><!-- ws:end:WikiTextHeadingRule:17 -->Instruments</h2>
<a class="wiki_link_ext" href="http://www.microtonalismo.com/" rel="nofollow" target="_blank">Guitar 29EDO</a><br />
<ul><li><!-- ws:start:WikiTextHeadingRule:19:<h4> --><h4 id="toc9"><a name="Music-Instruments--Guitar 29EDO from Peruvian - Charles Loli and Antonio Huamani"></a><!-- ws:end:WikiTextHeadingRule:19 --><strong><a class="wiki_link_ext" href="http://www.microtonalismo.com/proyecto-xvii" rel="nofollow">Guitar 29EDO from Peruvian - Charles Loli and Antonio Huamani</a></strong></h4>
<br />
<!-- ws:start:WikiTextRemoteImageRule:948:<img src="http://content.pimp-my-profile.com/i116/2/10/29/f_735065b21747.jpg" alt="" title="" style="height: 891px; width: 321px;" /> --><img src="http://content.pimp-my-profile.com/i116/2/10/29/f_735065b21747.jpg" alt="external image f_735065b21747.jpg" title="external image f_735065b21747.jpg" style="height: 891px; width: 321px;" /><!-- ws:end:WikiTextRemoteImageRule:948 --></li></ul><br />
<a class="wiki_link_ext" href="http://www.microtonalismo.com/" rel="nofollow" target="_blank">Bass 29EDO</a><br />
<ul><li><!-- ws:start:WikiTextHeadingRule:21:<h4> --><h4 id="toc10"><a name="Music-Instruments--Bass 29EDO from Peruvian - Charles Loli and Antonio Huamani"></a><!-- ws:end:WikiTextHeadingRule:21 --><strong><a class="wiki_link_ext" href="http://www.microtonalismo.com/proyecto-xvii" rel="nofollow">Bass 29EDO from Peruvian - Charles Loli and Antonio Huamani</a></strong></h4>
<br />
<!-- ws:start:WikiTextRemoteImageRule:949:<img src="https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-prn1/r90/550502_538613626155939_2005925977_n.jpg" alt="" title="" style="height: 936px; width: 305px;" /> --><img src="https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-prn1/r90/550502_538613626155939_2005925977_n.jpg" alt="external image 550502_538613626155939_2005925977_n.jpg" title="external image 550502_538613626155939_2005925977_n.jpg" style="height: 936px; width: 305px;" /><!-- ws:end:WikiTextRemoteImageRule:949 --></li></ul></body></html>