13edo: Difference between revisions
Wikispaces>spt3125 **Imported revision 615832227 - Original comment: added link (deorphaning)** |
Wikispaces>spt3125 **Imported revision 616004689 - 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:spt3125|spt3125]] and made on <tt>2017-07- | : This revision was by author [[User:spt3125|spt3125]] and made on <tt>2017-07-30 15:10:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>616004689</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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||~ Degree ||~ Cents ||~ Approximated 21-limit Ratios* ||~ Erv Wilson ||~ Archaeotonic ||~ Oneirotonic ||~ [[26edo]] names ||~ Kentaku || | ||~ Degree ||~ Cents ||~ Approximated 21-limit Ratios* ||~ Erv Wilson ||~ Archaeotonic ||~ Oneirotonic ||~ [[26edo]] names ||~ Kentaku || | ||
||= 0 ||> 0.00 ||= 1/1 ||= H ||= C ||= C ||= C ||= J || | ||= 0 ||> 0.00 ||= 1/1 ||= H ||= C ||= C ||= C ||= J || | ||
||= 1 ||> 92.31 ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 ||= | ||= 1 ||> 92.31 ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 ||= β ||= C#/Db ||= C#/Db ||= Cx/Dbb ||= J#/Kb || | ||
||= 2 ||> 184.62 ||= 9/8, 10/9, 11/10, 19/17, 21/19 ||= A ||= D ||= D ||= D ||= K || | ||= 2 ||> 184.62 ||= 9/8, 10/9, 11/10, 19/17, 21/19 ||= A ||= D ||= D ||= D ||= K || | ||
||= 3 ||> 276.92 ||= 7/6, 13/11, 20/17, 19/16, 22/19 ||= | ||= 3 ||> 276.92 ||= 7/6, 13/11, 20/17, 19/16, 22/19 ||= δ ||= D#/Eb ||= D#/Eb ||= Dx/Ebb ||= K#/Lb || | ||
||= 4 ||> 369.23 ||= 5/4, 11/9, 16/13, 26/21 ||= C ||= E ||= E ||= E ||= L || | ||= 4 ||> 369.23 ||= 5/4, 11/9, 16/13, 26/21 ||= C ||= E ||= E ||= E ||= L || | ||
||= 5 ||> 461.54 ||= 13/10, 17/13, 21/16, 22/17 ||= B ||= E#/Fb ||= F ||= Ex/Fb ||= M || | ||= 5 ||> 461.54 ||= 13/10, 17/13, 21/16, 22/17 ||= B ||= E#/Fb ||= F ||= Ex/Fb ||= M || | ||
||= 6 ||> 553.85 ||= 11/8, 18/13, 26/19 ||= | ||= 6 ||> 553.85 ||= 11/8, 18/13, 26/19 ||= ε ||= F ||= F#/Gb ||= F# ||= M#/Nb || | ||
||= 7 ||> 646.15 ||= 16/11, 13/9, 19/13 ||= D ||= F#/Gb ||= G ||= Gb ||= N || | ||= 7 ||> 646.15 ||= 16/11, 13/9, 19/13 ||= D ||= F#/Gb ||= G ||= Gb ||= N || | ||
||= 8 ||> 738.46 ||= 17/11, 20/13, 26/17, 32/21 ||= | ||= 8 ||> 738.46 ||= 17/11, 20/13, 26/17, 32/21 ||= γ ||= G ||= G#/Hb ||= G# ||= N#/Ob || | ||
||= 9 ||> 830.77 ||= 8/5, 13/8, 18/11, 21/13 ||= F ||= G#/Ab ||= H ||= Ab ||= O || | ||= 9 ||> 830.77 ||= 8/5, 13/8, 18/11, 21/13 ||= F ||= G#/Ab ||= H ||= Ab ||= O || | ||
||= 10 ||> 923.08 ||= [[17_10|17/10]], [[12_7|12/7]], [[22_13|22/13]], [[19_11|19/11]] ||= E ||= A ||= A ||= A# ||= P || | ||= 10 ||> 923.08 ||= [[17_10|17/10]], [[12_7|12/7]], [[22_13|22/13]], [[19_11|19/11]] ||= E ||= A ||= A ||= A# ||= P || | ||
||= 11 ||> 1015.38 ||= 9/5, 16/9, 20/11, 34/19, 38/21 ||= | ||= 11 ||> 1015.38 ||= 9/5, 16/9, 20/11, 34/19, 38/21 ||= α ||= A#/Bb ||= A#/Bb ||= Bb ||= P#/Qb || | ||
||= 12 ||> 1107.69 ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 ||= G ||= B/Cb ||= B ||= B#/Cbb ||= Q || | ||= 12 ||> 1107.69 ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 ||= G ||= B/Cb ||= B ||= B#/Cbb ||= Q || | ||
||= 13 ||> 1200.00 ||= 2/1 ||= H ||= C/B# ||= C ||= C ||= J || | ||= 13 ||> 1200.00 ||= 2/1 ||= H ||= C/B# ||= C ||= C ||= J || | ||
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For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic and octotonic fifth-based) and [[Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic second-based). | For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic and octotonic fifth-based) and [[Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic second-based). | ||
[[file:13edo-chromatic-scale.mid|13 edo chromatic ascending and descending scale on C (MIDI)]] | [[file:13edo-chromatic-scale.mid|13 edo chromatic ascending and descending scale on C (MIDI)]] | ||
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Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected. | Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected. | ||
=Harmony in 13edo= | =Harmony in 13edo= | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (__[[13edo#top|degree]]s__ 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (__[[13edo#top|degree]]s__ 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. | ||
By this, we can assume that the major ninth of | By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos.This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo. | ||
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite __[[13edo#top|close]]__ to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads can be assumed to likely be the most consonant base sonorities available in 13 edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are | The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite __[[13edo#top|close]]__ to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads can be assumed to likely be the most consonant base sonorities available in 13 edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well. | ||
Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems. | Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems. | ||
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13edo offers two main candidates for diatonic-like scales: the 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones. | 13edo offers two main candidates for diatonic-like scales: the 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones. | ||
To facilitate discussion of these scales, Cryptic Ruse has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. The 2\13-based heptatonic has been named "archeotonic" after the "Old Ones" that rule the Dreamlands, and the 5\13-based octatonic has been named "oneirotonic" after the Dreamlands themselves. Modes of the archeotonic are named after the individual Old Ones themselves; modes of the oneirotonic are named after cities in the Dreamlands. | |||
===Modes and Harmony in The Archaeotonic Scale=== | ===Modes and Harmony in The Archaeotonic Scale=== | ||
A 7-nominal | A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo. | ||
[[image:Archeotonic.png]] | [[image:Archeotonic.png]] | ||
Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21. | Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21. | ||
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[[image:Oneirotonic.png]] | [[image:Oneirotonic.png]] | ||
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming. | There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming. | ||
==The Kentaku (aka William Lynch) Method for Octatonic Notation== | ==The Kentaku (aka William Lynch) Method for Octatonic Notation== | ||
Normally, 13edo can be notated by adding an accidental between E and F. For some reading the same staff with the same letters but in different places can be mind boggling and lead to confusion. That's why some have recommended different options. | |||
13edo may be better suited to be notated by using the 8 notes of the MOS father[8] as the notes except that similarly to meantone, the 4th mode of Father[8] is the default "Basic" mode of 13edo and is notated with new letters. A new set of letters may benefit by giving 13edo a fresh perspective. [[William Lynch]] proposes using the letters JKLMNOPQ as the notes of the 4th mode of Father[8] with J being an 11/8 from a normal C, in order to give it a more personal identify rather than hearing it as C. The remaining letters of 13 edo are notated with sharps and flats like 12edo. The whole alphabet is written J-J#-K-K#-L-M-M#-N-N#-O-P-P#-Q-J . There are no accidentals between LM, OP, and QJ. The idea behind using the 4th mode rather than the 1st is that this mode contains a warmer sounding shape because it has a flat sixth instead of a natural one so William has chosen this to be the default mode rather than Dylathian. | |||
This new approach simply modifies the way | This new approach simply modifies the way 13edo is systematized and notated but is based on the exact same 8 tone scale that Cryptic Ruse uses. | ||
[[image:13 edo alphabet.PNG]] | [[image:13 edo alphabet.PNG]] | ||
Notation written on the staff looks nearly the same as | Notation written on the staff looks nearly the same as 12edo notation, except that because there are 8 letters instead of 7, the notes offset themselves to where space-to-space and line-to-line are now octaves rather than line-to-space. This makes learning the system much easier because a letter that is on a line is always on a line. | ||
13EDO Chromatic Scale in Octatonic Notation: | |||
[[image:8 tone scale 13 edo.PNG]] | [[image:8 tone scale 13 edo.PNG]] | ||
[[image:13 edo chromatic scale.PNG]] | [[image:13 edo chromatic scale.PNG]] | ||
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|| G || Ab || A || Bb || B || || C || Db || D || Eb || E || F || Gb || G || (if that were to be valuable in any way) || | || G || Ab || A || Bb || B || || C || Db || D || Eb || E || F || Gb || G || (if that were to be valuable in any way) || | ||
The | The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be. | ||
=Commas= | =Commas= | ||
| Line 163: | Line 163: | ||
||= 121/120 || | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || | ||= 121/120 || | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || | ||
||= 441/440 || | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= || | ||= 441/440 || | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= || | ||
=Animism= | |||
The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In | ==Animism== | ||
The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction: | |||
0 4 5 8 9 13 pentatonic | 0 4 5 8 9 13 pentatonic | ||
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<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>13edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/13%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br /> | <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>13edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/13%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br /> | ||
</span><br /> | </span><br /> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:37:&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:37 --><!-- ws:start:WikiTextTocRule:38: --><a href="#Best approximates 2.5.9.11.13.17.19.21">Best approximates 2.5.9.11.13.17.19.21</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#x13edo: 13 equal divisions of the octave">13edo: 13 equal divisions of the octave</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Scales in 13edo">Scales in 13edo</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --> | <a href="#Harmony in 13edo">Harmony in 13edo</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#Notational and Compositional Approaches to 13edo">Notational and Compositional Approaches to 13edo</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --> | <a href="#Mapping to Standard Keyboards">Mapping to Standard Keyboards</a><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --> | <a href="#Guitar">Guitar</a><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --> | ||
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<hr /> | <hr /> | ||
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<td style="text-align: center;">17/16, 18/17, 19/18, 20/19, 21/20, 22/21<br /> | <td style="text-align: center;">17/16, 18/17, 19/18, 20/19, 21/20, 22/21<br /> | ||
</td> | </td> | ||
<td style="text-align: center | <td style="text-align: center;">β<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C#/Db<br /> | <td style="text-align: center;">C#/Db<br /> | ||
| Line 296: | Line 297: | ||
<td style="text-align: center;">7/6, 13/11, 20/17, 19/16, 22/19<br /> | <td style="text-align: center;">7/6, 13/11, 20/17, 19/16, 22/19<br /> | ||
</td> | </td> | ||
<td style="text-align: center | <td style="text-align: center;">δ<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">D#/Eb<br /> | <td style="text-align: center;">D#/Eb<br /> | ||
| Line 350: | Line 351: | ||
<td style="text-align: center;">11/8, 18/13, 26/19<br /> | <td style="text-align: center;">11/8, 18/13, 26/19<br /> | ||
</td> | </td> | ||
<td style="text-align: center | <td style="text-align: center;">ε<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">F<br /> | <td style="text-align: center;">F<br /> | ||
| Line 386: | Line 387: | ||
<td style="text-align: center;">17/11, 20/13, 26/17, 32/21<br /> | <td style="text-align: center;">17/11, 20/13, 26/17, 32/21<br /> | ||
</td> | </td> | ||
<td style="text-align: center | <td style="text-align: center;">γ<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">G<br /> | <td style="text-align: center;">G<br /> | ||
| Line 440: | Line 441: | ||
<td style="text-align: center;">9/5, 16/9, 20/11, 34/19, 38/21<br /> | <td style="text-align: center;">9/5, 16/9, 20/11, 34/19, 38/21<br /> | ||
</td> | </td> | ||
<td style="text-align: center | <td style="text-align: center;">α<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">A#/Bb<br /> | <td style="text-align: center;">A#/Bb<br /> | ||
| Line 766: | Line 767: | ||
<br /> | <br /> | ||
For alternative notations, see <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs">Ups and Downs Notation -&quot;Supersharp&quot; EDOs</a> (pentatonic and octotonic fifth-based) and <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Natural%20Generators">Ups and Downs Notation - Natural Generators</a> (heptatonic second-based).<br /> | For alternative notations, see <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs">Ups and Downs Notation -&quot;Supersharp&quot; EDOs</a> (pentatonic and octotonic fifth-based) and <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Natural%20Generators">Ups and Downs Notation - Natural Generators</a> (heptatonic second-based).<br /> | ||
<br /> | <br /> | ||
<a href="/file/view/13edo-chromatic-scale.mid/389281456/13edo-chromatic-scale.mid" onclick="ws.common.trackFileLink('/file/view/13edo-chromatic-scale.mid/389281456/13edo-chromatic-scale.mid');">13 edo chromatic ascending and descending scale on C (MIDI)</a><br /> | <a href="/file/view/13edo-chromatic-scale.mid/389281456/13edo-chromatic-scale.mid" onclick="ws.common.trackFileLink('/file/view/13edo-chromatic-scale.mid/389281456/13edo-chromatic-scale.mid');">13 edo chromatic ascending and descending scale on C (MIDI)</a><br /> | ||
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<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1130:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/13ed2-001.svg?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@13ed2-001.svg&quot; title=&quot;File: 13ed2-001.svg&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/13ed2-001.svg/480712008/13ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/13ed2-001.svg/480712008/13ed2-001.svg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="13ed2-001.svg" /></a><div><a href="/file/view/13ed2-001.svg/480712008/13ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/13ed2-001.svg/480712008/13ed2-001.svg');" class="filename" title="13ed2-001.svg">13ed2-001.svg</a><br /><ul><li><a href="/file/detail/13ed2-001.svg">Details</a></li><li><a href="/file/view/13ed2-001.svg/480712008/13ed2-001.svg">Download</a></li><li style="color: #666">15 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:1130 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc2"><a name="Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:13 -->Scales in 13edo</h1> | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc2"><a name="Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:13 -->Scales in 13edo</h1> | ||
Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two <u><a class="wiki_link" href="/degree">degree</a>s of</u> 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br /> | Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two <u><a class="wiki_link" href="/degree">degree</a>s of</u> 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br /> | ||
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<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1131:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/13edo%20horograms.pdf?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@13edo horograms.pdf&quot; title=&quot;File: 13edo horograms.pdf&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /></a><div><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf">13edo horograms.pdf</a><br /><ul><li><a href="/file/detail/13edo%20horograms.pdf">Details</a></li><li><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf">Download</a></li><li style="color: #666">242 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:1131 --><br /> | ||
~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson<br /> | ~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson<br /> | ||
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Another neat facet of 13-EDO is the fact that any 12-EDO scale can be &quot;turned into&quot; a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.<br /> | Another neat facet of 13-EDO is the fact that any 12-EDO scale can be &quot;turned into&quot; a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc3"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:15 -->Harmony in 13edo</h1> | <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc3"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:15 -->Harmony in 13edo</h1> | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (<u><a class="wiki_link" href="/13edo#top">degree</a>s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br /> | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (<u><a class="wiki_link" href="/13edo#top">degree</a>s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br /> | ||
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By this, we can assume that the major ninth of | By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos.This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.<br /> | ||
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The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u><a class="wiki_link" href="/13edo#top">close</a></u> to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads can be assumed to likely be the most consonant base sonorities available in 13 edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are | The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u><a class="wiki_link" href="/13edo#top">close</a></u> to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads can be assumed to likely be the most consonant base sonorities available in 13 edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.<br /> | ||
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Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.<br /> | Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.<br /> | ||
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13edo offers two main candidates for diatonic-like scales: the 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are <a class="wiki_link" href="/Rothenberg%20propriety">Rothenberg proper</a>, and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.<br /> | 13edo offers two main candidates for diatonic-like scales: the 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are <a class="wiki_link" href="/Rothenberg%20propriety">Rothenberg proper</a>, and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.<br /> | ||
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To facilitate discussion of these scales, Cryptic Ruse has ascribed them names based on H.P. Lovecraft's &quot;Dream Cycle&quot; mythos. The 2\13-based heptatonic has been named &quot;archeotonic&quot; after the &quot;Old Ones&quot; that rule the Dreamlands, and the 5\13-based octatonic has been named &quot;oneirotonic&quot; after the Dreamlands themselves. Modes of the archeotonic are named after the individual Old Ones themselves; modes of the oneirotonic are named after cities in the Dreamlands.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:21:&lt;h3&gt; --><h3 id="toc6"><a name="Notational and Compositional Approaches to 13edo-The Cryptic Ruse Methods-Modes and Harmony in The Archaeotonic Scale"></a><!-- ws:end:WikiTextHeadingRule:21 -->Modes and Harmony in The Archaeotonic Scale</h3> | <!-- ws:start:WikiTextHeadingRule:21:&lt;h3&gt; --><h3 id="toc6"><a name="Notational and Compositional Approaches to 13edo-The Cryptic Ruse Methods-Modes and Harmony in The Archaeotonic Scale"></a><!-- ws:end:WikiTextHeadingRule:21 -->Modes and Harmony in The Archaeotonic Scale</h3> | ||
A 7-nominal | A 7-nominal notation is proposed, using the letters A-G. The &quot;C natural&quot; scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note &quot;C&quot; tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1123:&lt;img src=&quot;/file/view/Archeotonic.png/252639498/Archeotonic.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Archeotonic.png/252639498/Archeotonic.png" alt="Archeotonic.png" title="Archeotonic.png" /><!-- ws:end:WikiTextLocalImageRule:1123 --><br /> | ||
Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.<br /> | Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:23:&lt;h3&gt; --><h3 id="toc7"><a name="Notational and Compositional Approaches to 13edo-The Cryptic Ruse Methods-Modes and Harmony in the Oneirotonic Scale"></a><!-- ws:end:WikiTextHeadingRule:23 -->Modes and Harmony in the Oneirotonic Scale</h3> | <!-- ws:start:WikiTextHeadingRule:23:&lt;h3&gt; --><h3 id="toc7"><a name="Notational and Compositional Approaches to 13edo-The Cryptic Ruse Methods-Modes and Harmony in the Oneirotonic Scale"></a><!-- ws:end:WikiTextHeadingRule:23 -->Modes and Harmony in the Oneirotonic Scale</h3> | ||
Here an 8-nominal notation is proposed, using letters A-H. The &quot;C natural&quot; scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note &quot;C&quot; tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.<br /> | Here an 8-nominal notation is proposed, using letters A-H. The &quot;C natural&quot; scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note &quot;C&quot; tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1124:&lt;img src=&quot;/file/view/Oneirotonic.png/252639860/Oneirotonic.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Oneirotonic.png/252639860/Oneirotonic.png" alt="Oneirotonic.png" title="Oneirotonic.png" /><!-- ws:end:WikiTextLocalImageRule:1124 --><br /> | ||
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.<br /> | There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.<br /> | ||
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< | <!-- ws:start:WikiTextHeadingRule:25:&lt;h2&gt; --><h2 id="toc8"><a name="Notational and Compositional Approaches to 13edo-The Kentaku (aka William Lynch) Method for Octatonic Notation"></a><!-- ws:end:WikiTextHeadingRule:25 -->The Kentaku (aka William Lynch) Method for Octatonic Notation</h2> | ||
Normally, 13edo can be notated by adding an accidental between E and F. For some reading the same staff with the same letters but in different places can be mind boggling and lead to confusion. That's why some have recommended different options.<br /> | |||
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13edo may be better suited to be notated by using the 8 notes of the MOS father[8] as the notes except that similarly to meantone, the 4th mode of Father[8] is the default &quot;Basic&quot; mode of 13edo and is notated with new letters. A new set of letters may benefit by giving 13edo a fresh perspective. <a class="wiki_link" href="/William%20Lynch">William Lynch</a> proposes using the letters JKLMNOPQ as the notes of the 4th mode of Father[8] with J being an 11/8 from a normal C, in order to give it a more personal identify rather than hearing it as C. The remaining letters of 13 edo are notated with sharps and flats like 12edo. The whole alphabet is written J-J#-K-K#-L-M-M#-N-N#-O-P-P#-Q-J . There are no accidentals between LM, OP, and QJ. The idea behind using the 4th mode rather than the 1st is that this mode contains a warmer sounding shape because it has a flat sixth instead of a natural one so William has chosen this to be the default mode rather than Dylathian.<br /> | |||
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This new approach simply modifies the way | This new approach simply modifies the way 13edo is systematized and notated but is based on the exact same 8 tone scale that Cryptic Ruse uses.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1125:&lt;img src=&quot;/file/view/13%20edo%20alphabet.PNG/506083616/13%20edo%20alphabet.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13%20edo%20alphabet.PNG/506083616/13%20edo%20alphabet.PNG" alt="13 edo alphabet.PNG" title="13 edo alphabet.PNG" /><!-- ws:end:WikiTextLocalImageRule:1125 --><br /> | ||
Notation written on the staff looks nearly the same as | Notation written on the staff looks nearly the same as 12edo notation, except that because there are 8 letters instead of 7, the notes offset themselves to where space-to-space and line-to-line are now octaves rather than line-to-space. This makes learning the system much easier because a letter that is on a line is always on a line.<br /> | ||
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13EDO Chromatic Scale in Octatonic Notation:<br /> | |||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1126:&lt;img src=&quot;/file/view/8%20tone%20scale%2013%20edo.PNG/506382886/8%20tone%20scale%2013%20edo.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/8%20tone%20scale%2013%20edo.PNG/506382886/8%20tone%20scale%2013%20edo.PNG" alt="8 tone scale 13 edo.PNG" title="8 tone scale 13 edo.PNG" /><!-- ws:end:WikiTextLocalImageRule:1126 --><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1127:&lt;img src=&quot;/file/view/13%20edo%20chromatic%20scale.PNG/506089524/13%20edo%20chromatic%20scale.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13%20edo%20chromatic%20scale.PNG/506089524/13%20edo%20chromatic%20scale.PNG" alt="13 edo chromatic scale.PNG" title="13 edo chromatic scale.PNG" /><!-- ws:end:WikiTextLocalImageRule:1127 --><br /> | ||
Also, this approach allows for a circle of fourths to be produced:<br /> | Also, this approach allows for a circle of fourths to be produced:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1128:&lt;img src=&quot;http://xenharmonic.wikispaces.com/file/view/13%20ET%20circle%20of%20fourths.PNG/506093224/13%20ET%20circle%20of%20fourths.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://xenharmonic.wikispaces.com/file/view/13%20ET%20circle%20of%20fourths.PNG/506093224/13%20ET%20circle%20of%20fourths.PNG" alt="13 ET circle of fourths.PNG" title="13 ET circle of fourths.PNG" /><!-- ws:end:WikiTextLocalImageRule:1128 --><br /> | ||
Listen to Ryonian on J (F#):<br /> | Listen to Ryonian on J (F#):<br /> | ||
<!-- ws:start:WikiTextMediaRule:7:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/13%20edo%20Ryo%20mode.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;13 edo Ryo mode.mp3&amp;quot; width=&amp;quot;240&amp;quot; height=&amp;quot;20&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><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%252F13%252Bedo%252BRyo%252Bmode.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:7 --><br /> | <!-- ws:start:WikiTextMediaRule:7:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/13%20edo%20Ryo%20mode.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;13 edo Ryo mode.mp3&amp;quot; width=&amp;quot;240&amp;quot; height=&amp;quot;20&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><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%252F13%252Bedo%252BRyo%252Bmode.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:7 --><br /> | ||
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<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kentaku%27s%20Approach%20to%2013EDO">More on William Lynch's 13 EDO octatonic approach</a><br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kentaku%27s%20Approach%20to%2013EDO">More on William Lynch's 13 EDO octatonic approach</a><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:27:&lt;h1&gt; --><h1 id="toc9"><a name="Mapping to Standard Keyboards"></a><!-- ws:end:WikiTextHeadingRule:27 -->Mapping to Standard Keyboards</h1> | ||
The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.<br /> | The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.<br /> | ||
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The | The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:29:&lt;h1&gt; --><h1 id="toc10"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:29 -->Commas</h1> | ||
13 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 val &lt; 13 21 30 36 45 48 |.)<br /> | 13 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 val &lt; 13 21 30 36 45 48 |.)<br /> | ||
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</table> | </table> | ||
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The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In | <!-- ws:start:WikiTextHeadingRule:31:&lt;h2&gt; --><h2 id="toc11"><a name="Commas-Animism"></a><!-- ws:end:WikiTextHeadingRule:31 -->Animism</h2> | ||
The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:<br /> | |||
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0 4 5 8 9 13 pentatonic<br /> | 0 4 5 8 9 13 pentatonic<br /> | ||
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0 1 3 4 5 8 9 10 12 13 nonatonic<br /> | 0 1 3 4 5 8 9 10 12 13 nonatonic<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:33:&lt;h1&gt; --><h1 id="toc12"><a name="Guitar"></a><!-- ws:end:WikiTextHeadingRule:33 -->Guitar</h1> | ||
<ul><li><a class="wiki_link" href="/13EDO%20Scales%20and%20Chords%20for%20Guitar">13EDO Scales and Chords for Guitar</a></li></ul><br /> | <ul><li><a class="wiki_link" href="/13EDO%20Scales%20and%20Chords%20for%20Guitar">13EDO Scales and Chords for Guitar</a></li></ul><br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:35:&lt;h1&gt; --><h1 id="toc13"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:35 --><strong>Compositions</strong></h1> | ||
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a></span> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br /> | <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a></span> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br /> | ||
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow">Prelude in 13ET</a></span> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt </a>- <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2449" rel="nofollow" target="_blank">Organ version</a><br /> | <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow">Prelude in 13ET</a></span> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt </a>- <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2449" rel="nofollow" target="_blank">Organ version</a><br /> | ||