Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 623967575 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624088505 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:TallKite|TallKite]] and made on <tt>2017-12- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-19 23:50:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624088505</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> | ||
| Line 315: | Line 315: | ||
All single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is: | All single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is: | ||
<span style="display: block; text-align: center;">P1 - ^^m2=v<span style="vertical-align: super;">3</span>A1 - vM2 - ^m3 - ^<span style="vertical-align: super;">3</span>d4=vvM3 - P4 </span><span style="display: block; text-align: center;">C - Db^^=C#v<span style="vertical-align: super;">3</span> - Dv - Eb^ - Fb^<span style="vertical-align: super;">3</span>=Evv - F</span> | <span style="display: block; text-align: center;">P1 - ^^m2=v<span style="vertical-align: super;">3</span>A1 - vM2 - ^m3 - ^<span style="vertical-align: super;">3</span>d4=vvM3 - P4</span><span style="display: block; text-align: center;">C - Db^^=C#v<span style="vertical-align: super;">3</span> - Dv - Eb^ - Fb^<span style="vertical-align: super;">3</span>=Evv - F</span> | ||
(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions. | (P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions. | ||
| Line 2,051: | Line 2,051: | ||
All single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is:<br /> | All single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is:<br /> | ||
<br /> | <br /> | ||
<span style="display: block; text-align: center;">P1 - ^^m2=v<span style="vertical-align: super;">3</span>A1 - vM2 - ^m3 - ^<span style="vertical-align: super;">3</span>d4=vvM3 - P4 </span><span style="display: block; text-align: center;">C - Db^^=C#v<span style="vertical-align: super;">3</span> - Dv - Eb^ - Fb^<span style="vertical-align: super;">3</span>=Evv - F</span><br /> | <span style="display: block; text-align: center;">P1 - ^^m2=v<span style="vertical-align: super;">3</span>A1 - vM2 - ^m3 - ^<span style="vertical-align: super;">3</span>d4=vvM3 - P4</span><span style="display: block; text-align: center;">C - Db^^=C#v<span style="vertical-align: super;">3</span> - Dv - Eb^ - Fb^<span style="vertical-align: super;">3</span>=Evv - F</span><br /> | ||
(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | (P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | ||
<br /> | <br /> | ||