Kite's thoughts on pergens: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-22 14:58:53 UTC</tt>.<br>

: The original revision id was <tt>626778259</tt>.<br>

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further%20Discussion-Various%20proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If n = 1, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

||= (P8/3, P4/2) ||= third-8ve half-4th ||= third-8ve splits, half-4th splits, M6/6, m10/6, A11/12 ||

||= (P8/3, P5/2) ||= third-8ve half-5th ||= third-8ve splits, half-5th splits, m3/6, d5/12 ||

||= (P8/2, P4/3) ||= half-8ve third-4th ||= half-8ve splits, third-4th splits, A4/6, M10/6 ||

||= (P8/2, P5/3) ||= half-8ve third-5th ||= half-8ve splits, third-5th splits, m6/6, M9/6, A12/6 ||

||= (P8/2, P11/3) ||= half-8ve third-11th ||= half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24 ||

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.&lt;br /&gt;

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Various%20proofs"&gt;below&lt;/a&gt;). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If n = 1, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. &amp;quot;Every&amp;quot; means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.&lt;br /&gt;

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under &amp;quot;all pergens&amp;quot;. (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;third-8ve splits, half-4th splits, M6/6, m10/6, A11/12&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;third-8ve splits, half-5th splits, m3/6, d5/12&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;half-8ve splits, third-4th splits, A4/6, M10/6&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;half-8ve splits, third-5th splits, m6/6, M9/6, A12/6&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24&lt;br /&gt;

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