13edo: Difference between revisions
Wikispaces>TallKite **Imported revision 602810166 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 602900924 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-12-29 17:22:00 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>602900924</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size. | As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size. | ||
|| | ||~ Degree ||~ Cents ||~ Approximated 21-limit Ratios* ||~ Erv Wilson ||~ Archaeotonic ||~ Oneirotonic ||~ [[26edo]] names ||~ Kentaku || | ||
||= 0 || | ||= 0 ||> 0.00 ||= 1/1 ||= H ||= C ||= C ||= C ||= J || | ||
||= 1 || | ||= 1 ||> 92.31 ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 ||= <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">β</span> ||= C#/Db ||= C#/Db ||= Cx/Dbb ||= J#/Kb || | ||
||= 2 || | ||= 2 ||> 184.62 ||= 9/8, 10/9, 11/10, 19/17, 21/19 ||= A ||= D ||= D ||= D ||= K || | ||
||= 3 || | ||= 3 ||> 276.92 ||= 7/6, 13/11, 20/17, 19/16, 22/19 ||= <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">δ</span> ||= D#/Eb ||= D#/Eb ||= Dx/Ebb ||= K#/Lb || | ||
||= 4 || | ||= 4 ||> 369.23 ||= 5/4, 11/9, 16/13, 26/21 ||= C ||= E ||= E ||= E ||= L || | ||
||= 5 || | ||= 5 ||> 461.54 ||= 13/10, 17/13, 21/16, 22/17 ||= B ||= E#/Fb ||= F ||= Ex/Fb ||= M || | ||
||= 6 || | ||= 6 ||> 553.85 ||= 11/8, 18/13, 26/19 ||= <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">ε</span> ||= F ||= F#/Gb ||= F# ||= M#/Nb || | ||
||= 7 || | ||= 7 ||> 646.15 ||= 16/11, 13/9, 19/13 ||= D ||= F#/Gb ||= G ||= Gb ||= N || | ||
||= 8 || | ||= 8 ||> 738.46 ||= 17/11, 20/13, 26/17, 32/21 ||= <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">γ</span> ||= G ||= G#/Hb ||= G# ||= N#/Ob || | ||
||= 9 || | ||= 9 ||> 830.77 ||= 8/5, 13/8, 18/11, 21/13 ||= F ||= G#/Ab ||= H ||= Ab ||= O || | ||
||= 10 || | ||= 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 || | ||= 11 ||> 1015.38 ||= 9/5, 16/9, 20/11, 34/19, 38/21 ||= <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">α</span> ||= A#/Bb ||= A#/Bb ||= Bb ||= P#/Qb || | ||
||= 12 || | ||= 12 ||> 1107.69 ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 ||= G ||= B/Cb ||= B ||= B#/Cbb ||= Q || | ||
||= 13 || | ||= 13 ||> 1200.00 ||= 2/1 ||= H ||= C/B# ||= C ||= C ||= J || | ||
*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | ||
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The second approach preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly". | The second approach preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly". | ||
|| | ||~ Degree ||~ Cents ||||||~ [[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, | ||
with major wider than minor |||||| | with major wider than minor ||||||~ Up/down notation using the narrow 5th of 7\13, | ||
with major narrower than minor || | with major narrower than minor || | ||
||= 0 || | ||= 0 ||> 0 ||= perfect unison ||= P1 ||= D ||= perfect unison ||= P1 ||= D || | ||
||= 1 || | ||= 1 ||> 92 ||= up unison, minor 2nd ||= ^1, m2 ||= D^, E ||= up unison, major 2nd ||= ^1, M2 ||= D^, E || | ||
||= 2 || | ||= 2 ||> 185 ||= upminor 2nd, minor 3rd ||= ^m2, m3 ||= E^, Fb ||= upmajor 2nd, major 3rd ||= ^M2, M3 ||= E^, F# || | ||
||= 3 || | ||= 3 ||> 277 ||= downmajor 2nd, upminor 3rd ||= vM2, ^m3 ||= E#v, Fb^ ||= downminor 2nd, upmajor 3rd ||= vm2, ^M3 ||= Ebv, F#^ || | ||
||= 4 || | ||= 4 ||> 369 ||= major 2nd, downmajor 3rd ||= M2, vM3 ||= E#, Fv ||= minor 2nd, downminor 3rd ||= m2, vm3 ||= Eb, Fv || | ||
||= 5 || | ||= 5 ||> 462 ||= major 3rd, down 4th ||= M3, v4 ||= F, Gv ||= minor 3rd, down 4th ||= m3, v4 ||= F, Gv || | ||
||= 6 || | ||= 6 ||> 554 ||= perfect 4th, down 5th ||= P4, v5 ||= G, Av ||= perfect 4th, down 5th ||= P4, v5 ||= G, Av || | ||
||= 7 || | ||= 7 ||> 646 ||= up 4th, perfect 5th ||= ^4, P5 ||= G^, A ||= up 4th, perfect 5th ||= ^4, P5 ||= G^, A || | ||
||= 8 || | ||= 8 ||> 738 ||= up 5th, minor 6th ||= ^5, m6 ||= A^, B ||= up 5th, major 6th ||= ^5, M6 ||= A^, B || | ||
||= 9 || | ||= 9 ||> 831 ||= upminor 6th, minor 7th ||= ^m6, m7 ||= B^, Cb ||= upmajor 6th, major 7th ||= ^M6, M7 ||= B^, C# || | ||
||= 10 || | ||= 10 ||> 923 ||= downmajor 6th, upminor 7th ||= vM6, ^m7 ||= B#v, Cb^ ||= downminor 6th, upmajor 7th ||= vm6, ^M7 ||= Bbv, C#^ || | ||
||= 11 || | ||= 11 ||> 1015 ||= major 6th, downmajor 7th ||= M6, vM7 ||= B#, Cv ||= minor 6th, downminor 7th ||= m6, vm7 ||= Bb, Cv || | ||
||= 12 || | ||= 12 ||> 1108 ||= major 7th, down 8ve ||= M7, v8 ||= C, Dv ||= minor 7th, down 8ve ||= m7, v8 ||= C, Dv || | ||
||= 13 || | ||= 13 ||> 1200 ||= perfect 8ve ||= P8 ||= D ||= perfect 8ve ||= P8 ||= D || | ||
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). | ||
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=Commas= | =Commas= | ||
13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < 13 21 30 36 45 48 | 13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < 13 21 30 36 45 48 |.) | ||
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || | ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || | ||
||= | ||= 2109375/2097152 || | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||= || | ||
||= 1029/1000 || | ||= 1029/1000 || | -3 1 -3 3 > ||> 49.49 ||= Keega ||= ||= || | ||
||= 525/512 || | ||= 525/512 || | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= || | ||
||= 64/63 || | ||= 64/63 || | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma || | ||
||= 64827/64000 || | ||= 64827/64000 || | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||= ||= || | ||
||= 3125/3087 || | ||= 3125/3087 || | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||= ||= || | ||
||= 3136/3125 || | ||= 3136/3125 || | 6 0 -5 2 > ||> 6.08 ||= Hemimean ||= ||= || | ||
||= 121/120 || | ||= 121/120 || | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || | ||
||= 441/440 || | ||= 441/440 || | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= || | ||
=Animism= | =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 13 edo, 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 | 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 13 edo, 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 | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
< | <th>Degree<br /> | ||
</ | </th> | ||
< | <th>Cents<br /> | ||
</ | </th> | ||
< | <th>Approximated 21-limit Ratios*<br /> | ||
</ | </th> | ||
< | <th>Erv Wilson<br /> | ||
</ | </th> | ||
< | <th>Archaeotonic<br /> | ||
</ | </th> | ||
< | <th>Oneirotonic<br /> | ||
</ | </th> | ||
< | <th><a class="wiki_link" href="/26edo">26edo</a> names<br /> | ||
</ | </th> | ||
< | <th>Kentaku<br /> | ||
</ | </th> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">0<br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">0.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1/1<br /> | <td style="text-align: center;">1/1<br /> | ||
| Line 251: | Line 250: | ||
<td style="text-align: center;">1<br /> | <td style="text-align: center;">1<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">92.31<br /> | ||
</td> | </td> | ||
<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 /> | ||
| Line 269: | Line 268: | ||
<td style="text-align: center;">2<br /> | <td style="text-align: center;">2<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">184.62<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9/8, 10/9, 11/10, 19/17, 21/19<br /> | <td style="text-align: center;">9/8, 10/9, 11/10, 19/17, 21/19<br /> | ||
| Line 287: | Line 286: | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">276.92<br /> | ||
</td> | </td> | ||
<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 /> | ||
| Line 305: | Line 304: | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">369.23<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">5/4, 11/9, 16/13, 26/21<br /> | <td style="text-align: center;">5/4, 11/9, 16/13, 26/21<br /> | ||
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<td style="text-align: center;">5<br /> | <td style="text-align: center;">5<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">461.54<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">13/10, 17/13, 21/16, 22/17<br /> | <td style="text-align: center;">13/10, 17/13, 21/16, 22/17<br /> | ||
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<td style="text-align: center;">6<br /> | <td style="text-align: center;">6<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">553.85<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">11/8, 18/13, 26/19<br /> | <td style="text-align: center;">11/8, 18/13, 26/19<br /> | ||
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<td style="text-align: center;">7<br /> | <td style="text-align: center;">7<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">646.15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">16/11, 13/9, 19/13<br /> | <td style="text-align: center;">16/11, 13/9, 19/13<br /> | ||
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<td style="text-align: center;">8<br /> | <td style="text-align: center;">8<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">738.46<br /> | ||
</td> | </td> | ||
<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 /> | ||
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<td style="text-align: center;">9<br /> | <td style="text-align: center;">9<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">830.77<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8/5, 13/8, 18/11, 21/13<br /> | <td style="text-align: center;">8/5, 13/8, 18/11, 21/13<br /> | ||
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<td style="text-align: center;">10<br /> | <td style="text-align: center;">10<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">923.08<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/17_10">17/10</a>, <a class="wiki_link" href="/12_7">12/7</a>, <a class="wiki_link" href="/22_13">22/13</a>, <a class="wiki_link" href="/19_11">19/11</a><br /> | <td style="text-align: center;"><a class="wiki_link" href="/17_10">17/10</a>, <a class="wiki_link" href="/12_7">12/7</a>, <a class="wiki_link" href="/22_13">22/13</a>, <a class="wiki_link" href="/19_11">19/11</a><br /> | ||
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<td style="text-align: center;">11<br /> | <td style="text-align: center;">11<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">1015.38<br /> | ||
</td> | </td> | ||
<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 /> | ||
| Line 449: | Line 448: | ||
<td style="text-align: center;">12<br /> | <td style="text-align: center;">12<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">1107.69<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">17/9, 19/10, 21/11, 32/17, 36/19, 40/21<br /> | <td style="text-align: center;">17/9, 19/10, 21/11, 32/17, 36/19, 40/21<br /> | ||
| Line 467: | Line 466: | ||
<td style="text-align: center;">13<br /> | <td style="text-align: center;">13<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">1200.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2/1<br /> | <td style="text-align: center;">2/1<br /> | ||
| Line 480: | Line 479: | ||
</td> | </td> | ||
<td style="text-align: center;">J<br /> | <td style="text-align: center;">J<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 512: | Line 493: | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
< | <th>Degree<br /> | ||
</ | </th> | ||
< | <th>Cents<br /> | ||
</ | </th> | ||
< | <th colspan="3"><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">Up/down notation</a> using the narrow 5th of 7\13,<br /> | ||
with major wider than minor<br /> | with major wider than minor<br /> | ||
</ | </th> | ||
< | <th colspan="3">Up/down notation using the narrow 5th of 7\13,<br /> | ||
with major narrower than minor<br /> | with major narrower than minor<br /> | ||
</ | </th> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">0<br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">perfect unison<br /> | <td style="text-align: center;">perfect unison<br /> | ||
| Line 544: | Line 525: | ||
<td style="text-align: center;">1<br /> | <td style="text-align: center;">1<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">92<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">up unison, minor 2nd<br /> | <td style="text-align: center;">up unison, minor 2nd<br /> | ||
| Line 562: | Line 543: | ||
<td style="text-align: center;">2<br /> | <td style="text-align: center;">2<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">185<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">upminor 2nd, minor 3rd<br /> | <td style="text-align: center;">upminor 2nd, minor 3rd<br /> | ||
| Line 580: | Line 561: | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">277<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">downmajor 2nd, upminor 3rd<br /> | <td style="text-align: center;">downmajor 2nd, upminor 3rd<br /> | ||
| Line 598: | Line 579: | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">369<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">major 2nd, downmajor 3rd<br /> | <td style="text-align: center;">major 2nd, downmajor 3rd<br /> | ||
| Line 616: | Line 597: | ||
<td style="text-align: center;">5<br /> | <td style="text-align: center;">5<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">462<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">major 3rd, down 4th<br /> | <td style="text-align: center;">major 3rd, down 4th<br /> | ||
| Line 634: | Line 615: | ||
<td style="text-align: center;">6<br /> | <td style="text-align: center;">6<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">554<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">perfect 4th, down 5th<br /> | <td style="text-align: center;">perfect 4th, down 5th<br /> | ||
| Line 652: | Line 633: | ||
<td style="text-align: center;">7<br /> | <td style="text-align: center;">7<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">646<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">up 4th, perfect 5th<br /> | <td style="text-align: center;">up 4th, perfect 5th<br /> | ||
| Line 670: | Line 651: | ||
<td style="text-align: center;">8<br /> | <td style="text-align: center;">8<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">738<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">up 5th, minor 6th<br /> | <td style="text-align: center;">up 5th, minor 6th<br /> | ||
| Line 688: | Line 669: | ||
<td style="text-align: center;">9<br /> | <td style="text-align: center;">9<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">831<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">upminor 6th, minor 7th<br /> | <td style="text-align: center;">upminor 6th, minor 7th<br /> | ||
| Line 706: | Line 687: | ||
<td style="text-align: center;">10<br /> | <td style="text-align: center;">10<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">923<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">downmajor 6th, upminor 7th<br /> | <td style="text-align: center;">downmajor 6th, upminor 7th<br /> | ||
| Line 724: | Line 705: | ||
<td style="text-align: center;">11<br /> | <td style="text-align: center;">11<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">1015<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">major 6th, downmajor 7th<br /> | <td style="text-align: center;">major 6th, downmajor 7th<br /> | ||
| Line 742: | Line 723: | ||
<td style="text-align: center;">12<br /> | <td style="text-align: center;">12<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">1108<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">major 7th, down 8ve<br /> | <td style="text-align: center;">major 7th, down 8ve<br /> | ||
| Line 760: | Line 741: | ||
<td style="text-align: center;">13<br /> | <td style="text-align: center;">13<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: right;">1200<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">perfect 8ve<br /> | <td style="text-align: center;">perfect 8ve<br /> | ||
| Line 786: | Line 767: | ||
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<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1126:&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:1126 --><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:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1118:&lt;img src=&quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><!-- ws:end:WikiTextLocalImageRule:1118 --><br /> | ||
<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1127:&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:1127 --><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 /> | ||
<br /> | <br /> | ||
| Line 822: | Line 803: | ||
<!-- 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 notaiton 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 /> | A 7-nominal notaiton 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:1119:&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:1119 --><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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| Line 829: | Line 810: | ||
<!-- 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:1120:&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:1120 --><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 /> | ||
<!-- ws:start:WikiTextHeadingRule:25:&lt;h2&gt; --><h2 id="toc8"><!-- ws:end:WikiTextHeadingRule:25 --> </h2> | <!-- ws:start:WikiTextHeadingRule:25:&lt;h2&gt; --><h2 id="toc8"><!-- ws:end:WikiTextHeadingRule:25 --> </h2> | ||
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This new approach simply modifies the way 13 edo is systematized and notated but is based on the exact same 8 tone scale that cryptic ruse uses.<br /> | This new approach simply modifies the way 13 edo 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:1121:&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:1121 --><br /> | ||
Notation written on the staff looks nearly the same as 12 edo notation except that because there are 8 letters instead of 7, the notes offset themselves to where space to space and line to line is now octaves rather than line to space. This actually makes learning the system much, much easier because a letter that is on a line is always on a line.<br /> | Notation written on the staff looks nearly the same as 12 edo notation except that because there are 8 letters instead of 7, the notes offset themselves to where space to space and line to line is now octaves rather than line to space. This actually makes learning the system much, much easier because a letter that is on a line is always on a line.<br /> | ||
<br /> | <br /> | ||
3 EDO Chromatic Scale in Octatonic Notation:<br /> | 3 EDO Chromatic Scale in Octatonic Notation:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1122:&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:1122 --><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1123:&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:1123 --><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:1124:&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:1124 --><br /> | ||
Listen to Ryonian on J (F#) <!-- ws:start:WikiTextMediaRule:7:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/13%20edo%20Ryo%20mode.wav?h=50&amp;w=300&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;13 edo Ryo mode.wav&amp;quot; width=&amp;quot;300&amp;quot; height=&amp;quot;50&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;50&quot; width=&quot;300&quot;/&gt; --><embed type="audio/wav" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/13+edo+Ryo+mode.wav" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"></embed><!-- ws:end:WikiTextMediaRule:7 --><br /> | Listen to Ryonian on J (F#) <!-- ws:start:WikiTextMediaRule:7:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/13%20edo%20Ryo%20mode.wav?h=50&amp;w=300&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;13 edo Ryo mode.wav&amp;quot; width=&amp;quot;300&amp;quot; height=&amp;quot;50&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;50&quot; width=&quot;300&quot;/&gt; --><embed type="audio/wav" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/13+edo+Ryo+mode.wav" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"></embed><!-- ws:end:WikiTextMediaRule:7 --><br /> | ||
Listen to Dylathian on J: (F#) <!-- ws:start:WikiTextMediaRule:8:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/13%20edo%208%20tone%20mode%201.wav?h=50&amp;w=300&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;13 edo 8 tone mode 1.wav&amp;quot; width=&amp;quot;300&amp;quot; height=&amp;quot;50&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;50&quot; width=&quot;300&quot;/&gt; --><embed type="audio/wav" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/13+edo+8+tone+mode+1.wav" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"></embed><!-- ws:end:WikiTextMediaRule:8 --><br /> | Listen to Dylathian on J: (F#) <!-- ws:start:WikiTextMediaRule:8:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/13%20edo%208%20tone%20mode%201.wav?h=50&amp;w=300&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;13 edo 8 tone mode 1.wav&amp;quot; width=&amp;quot;300&amp;quot; height=&amp;quot;50&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;50&quot; width=&quot;300&quot;/&gt; --><embed type="audio/wav" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/13+edo+8+tone+mode+1.wav" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"></embed><!-- ws:end:WikiTextMediaRule:8 --><br /> | ||
| Line 1,247: | Line 1,228: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:31:&lt;h1&gt; --><h1 id="toc11"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:31 -->Commas</h1> | <!-- ws:start:WikiTextHeadingRule:31:&lt;h1&gt; --><h1 id="toc11"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:31 -->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 | 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 /> | ||
| Line 1,266: | Line 1,247: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">2109375/2097152<br /> | ||
</td> | </td> | ||
<td | <td>| -21 3 7 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">10.06<br /> | <td style="text-align: right;">10.06<br /> | ||
| Line 1,282: | Line 1,263: | ||
<td style="text-align: center;">1029/1000<br /> | <td style="text-align: center;">1029/1000<br /> | ||
</td> | </td> | ||
<td | <td>| -3 1 -3 3 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">49.49<br /> | <td style="text-align: right;">49.49<br /> | ||
| Line 1,296: | Line 1,277: | ||
<td style="text-align: center;">525/512<br /> | <td style="text-align: center;">525/512<br /> | ||
</td> | </td> | ||
<td | <td>| -9 1 2 1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">43.41<br /> | <td style="text-align: right;">43.41<br /> | ||
| Line 1,310: | Line 1,291: | ||
<td style="text-align: center;">64/63<br /> | <td style="text-align: center;">64/63<br /> | ||
</td> | </td> | ||
<td | <td>| 6 -2 0 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">27.26<br /> | <td style="text-align: right;">27.26<br /> | ||
| Line 1,324: | Line 1,305: | ||
<td style="text-align: center;">64827/64000<br /> | <td style="text-align: center;">64827/64000<br /> | ||
</td> | </td> | ||
<td | <td>| -9 3 -3 4 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">22.23<br /> | <td style="text-align: right;">22.23<br /> | ||
| Line 1,338: | Line 1,319: | ||
<td style="text-align: center;">3125/3087<br /> | <td style="text-align: center;">3125/3087<br /> | ||
</td> | </td> | ||
<td | <td>| 0 -2 5 -3 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">21.18<br /> | <td style="text-align: right;">21.18<br /> | ||
| Line 1,352: | Line 1,333: | ||
<td style="text-align: center;">3136/3125<br /> | <td style="text-align: center;">3136/3125<br /> | ||
</td> | </td> | ||
<td | <td>| 6 0 -5 2 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">6.08<br /> | <td style="text-align: right;">6.08<br /> | ||
| Line 1,366: | Line 1,347: | ||
<td style="text-align: center;">121/120<br /> | <td style="text-align: center;">121/120<br /> | ||
</td> | </td> | ||
<td | <td>| -3 -1 -1 0 2 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">14.37<br /> | <td style="text-align: right;">14.37<br /> | ||
| Line 1,380: | Line 1,361: | ||
<td style="text-align: center;">441/440<br /> | <td style="text-align: center;">441/440<br /> | ||
</td> | </td> | ||
<td | <td>| -3 2 -1 2 -1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">3.93<br /> | <td style="text-align: right;">3.93<br /> | ||