17edt
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Original Wikitext content:
=17 Tone Equal Divided Tritave= 17edt is closely related to the Bohlen-Pierce scale, and I might go as far as calling it an 11-limit version of it. Both have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in BP is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step! Nifty right? If the major chord is defined by degrees 0,8,13 (fibonacci numbers, coincidence?), and a major chord is built atop the two chord tones, an hexatonic set of 0,4,8,9,13,16,17 results. Excluding #16 produces a LLsLL pentatonic scale. Continuing the trend by building major chords on the three new chord tones (4, 9, and 16) gives you a set of the notes 0,3,4,7,8,9,12,13,15,16,17. Excluding #15 produces a LsLssLsLs moment of symmetry, which is called "Moll I" or "Delta", if I'm not mistaken. ===Intervals=== || degree of 17edt || cents value || cents value octave reduced || || 0 || 0 || || || 1 || 111.9 || || || 2 || 223.8 || || || 3 || 335.6 || || || 4 || 447.5 || || || 5 || 559.4 || || || 6 || 671.3 || || || 7 || 783.2 || || || 8 || 895.1 || || || 9 || 1006.9 || || || 10 || 1118.8 || || || 11 || 1230.7 || 30.7 || || 12 || 1342.6 || 142.6 || || 13 || 1454.5 || 254.5 || || 14 || 1566.3 || 366.3 || || 15 || 1678.2 || 478.2 || || 16 || 1790.1 || 590.1 || || 17 || 1902.0 || 702.0 || || 18 || 2013.9 || 813.9 || || 19 || 2125.8 || 925.8 || || 20 || 2237.6 || 1037.6 || || 21 || 2349..5 || 1149.5 || || 22 || 2461.4 || 61.4 || || 23 || 2573.2 || 173.2 || || 24 || 2685.2 || 285.2 || || 25 || 2797.1 || 397.1 || || 26 || 2908.9 || 508.9 || || 27 || 3020.8 || 620.8 || || 28 || 3132.7 || 732.7 || || 29 || 3244.6 || 844.6 || || 30 || 3356.5 || 956.5 || || 31 || 3468.3 || 1068.3 || || 32 || 3580.2 || 1180.2 || || 33 || 3692.1 || 92.1 || || 34 || 3804.0 || 204.0 || ===Other BP-esque equal divisions of the Tritave=== The first two MOS arrived at through the 5th and 7th harmonics have four and nine notes each. In a way then, 4edt and 9 edt are like the 5edo and 7edo of Bohlen-Pierce harmony. Adding any number of these together, or following Freud and Fibonacci breeding them, one finds many other equal divisions that have at least the basic skeleton. Among which are 4+4+9 17edt, 4+9+9 22edt, (4+9)+(4+4+9) 30edt, and (4+9)+(4+9+9) 35edt. 39edt, the triple of 13edt, approximates the 11th and 13th harmonics ontop of the framework. And don't forget that there are "equivalent" formations for each, using the 5th or 7th harmonic as frame interval, and the 3rd and 7th, or 3rd and 5th as harmonies, which may be relevant to BP melodic organization.
Original HTML content:
<html><head><title>17edt</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x17 Tone Equal Divided Tritave"></a><!-- ws:end:WikiTextHeadingRule:0 -->17 Tone Equal Divided Tritave</h1>
<br />
<br />
17edt is closely related to the Bohlen-Pierce scale, and I might go as far as calling it an 11-limit version of it. Both have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in BP is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step! Nifty right?<br />
<br />
If the major chord is defined by degrees 0,8,13 (fibonacci numbers, coincidence?), and a major chord is built atop the two chord tones, an hexatonic set of 0,4,8,9,13,16,17 results. Excluding #16 produces a LLsLL pentatonic scale. Continuing the trend by building major chords on the three new chord tones (4, 9, and 16) gives you a set of the notes 0,3,4,7,8,9,12,13,15,16,17. Excluding #15 produces a LsLssLsLs moment of symmetry, which is called "Moll I" or "Delta", if I'm not mistaken.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x17 Tone Equal Divided Tritave--Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h3>
<br />
<table class="wiki_table">
<tr>
<td>degree of 17edt<br />
</td>
<td>cents value<br />
</td>
<td>cents value octave reduced<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>111.9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>223.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>335.6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>447.5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>559.4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>671.3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>783.2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>895.1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>1006.9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>1118.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>1230.7<br />
</td>
<td>30.7<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>1342.6<br />
</td>
<td>142.6<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>1454.5<br />
</td>
<td>254.5<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>1566.3<br />
</td>
<td>366.3<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1678.2<br />
</td>
<td>478.2<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>1790.1<br />
</td>
<td>590.1<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>1902.0<br />
</td>
<td>702.0<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>2013.9<br />
</td>
<td>813.9<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>2125.8<br />
</td>
<td>925.8<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>2237.6<br />
</td>
<td>1037.6<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>2349..5<br />
</td>
<td>1149.5<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>2461.4<br />
</td>
<td>61.4<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>2573.2<br />
</td>
<td>173.2<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>2685.2<br />
</td>
<td>285.2<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>2797.1<br />
</td>
<td>397.1<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>2908.9<br />
</td>
<td>508.9<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>3020.8<br />
</td>
<td>620.8<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>3132.7<br />
</td>
<td>732.7<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>3244.6<br />
</td>
<td>844.6<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>3356.5<br />
</td>
<td>956.5<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>3468.3<br />
</td>
<td>1068.3<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>3580.2<br />
</td>
<td>1180.2<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>3692.1<br />
</td>
<td>92.1<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>3804.0<br />
</td>
<td>204.0<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x17 Tone Equal Divided Tritave--Other BP-esque equal divisions of the Tritave"></a><!-- ws:end:WikiTextHeadingRule:4 -->Other BP-esque equal divisions of the Tritave</h3>
<br />
<br />
The first two MOS arrived at through the 5th and 7th harmonics have four and nine notes each. In a way then, 4edt and 9 edt are like the 5edo and 7edo of Bohlen-Pierce harmony. Adding any number of these together, or following Freud and Fibonacci breeding them, one finds many other equal divisions that have at least the basic skeleton. Among which are 4+4+9 17edt, 4+9+9 22edt, (4+9)+(4+4+9) 30edt, and (4+9)+(4+9+9) 35edt. 39edt, the triple of 13edt, approximates the 11th and 13th harmonics ontop of the framework. And don't forget that there are "equivalent" formations for each, using the 5th or 7th harmonic as frame interval, and the 3rd and 7th, or 3rd and 5th as harmonies, which may be relevant to BP melodic organization.</body></html>