Heptatonic notation: Difference between revisions
Wikispaces>jake.huryn **Imported revision 612807115 - Original comment: ** |
Wikispaces>jake.huryn **Imported revision 612807123 - 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:jake.huryn|jake.huryn]] and made on <tt>2017-05-14 00:30: | : This revision was by author [[User:jake.huryn|jake.huryn]] and made on <tt>2017-05-14 00:30:34 UTC</tt>.<br> | ||
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* 6: bb dbv db^ b dv d^ n ‡v ‡^ # ‡#v ‡#^ x | * 6: bb dbv db^ b dv d^ n ‡v ‡^ # ‡#v ‡#^ x | ||
* 7: bb dbv db bv b^ d d^ n ‡v ‡ #v #^ ‡# ‡#^ x | * 7: bb dbv db bv b^ d d^ n ‡v ‡ #v #^ ‡# ‡#^ x | ||
* 8: bb dbv db bv b b^ d d^ n ‡v ‡ #v # #^ ‡# ‡#^ x (Because this results in values exactly in between twelfths, some discretion was used to determine these to be the most intuitive.) | * 8: bb dbv db bv b b^ d d^ n ‡v ‡ #v # #^ ‡# ‡#^ x (Because this results in values exactly in between twelfths, some discretion in was used in rounding to determine these to be the most intuitive.) | ||
* 9: bb bb^ db db^ bv b^ dv d nv n n^ ‡ ‡^ #v #^ ‡#v ‡# xv x | * 9: bb bb^ db db^ bv b^ dv d nv n n^ ‡ ‡^ #v #^ ‡#v ‡# xv x | ||
* 10: bb bb^ dbv db^ bv b b^ dv d^ nv n n^ ‡v ‡^ #v # #^ ‡#v ‡#^ xv x | * 10: bb bb^ dbv db^ bv b b^ dv d^ nv n n^ ‡v ‡^ #v # #^ ‡#v ‡#^ xv x | ||
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where bb is a double flat, db is a three-half flat, b is a flat, d is a half-flat, n is a natural, ‡ is a half-sharp, # is a sharp, ‡# is a three-half sharp, and x is a double sharp. The v and ^ are arrows which are attached to the bottom and top of the accidentals, respectively. Because there are twelve flats and twelve sharps, these accidentals allow notation up to 84edo with full enharmonicity, as described above. In order to notate edos which use fewer accidentals, the accidentals used are chosen to best approximate the required modification value, assuming that the above accidentals are in increments of perfect twelfth-accidentals. For example, suppose we wish to notate 45edo; 45 = 7*6+3, so seven sharp and seven flat accidentals are required. To determine, say, the three-sevenths sharp we multiply that value by twelve: 12*3/7 = 5.14 ≈ 5, so we use the five-twelfths sharp, or #v. Doing this for all sets of fewer than or equal to twelve accidentals per sharp/flat class, we find the following:<br /> | where bb is a double flat, db is a three-half flat, b is a flat, d is a half-flat, n is a natural, ‡ is a half-sharp, # is a sharp, ‡# is a three-half sharp, and x is a double sharp. The v and ^ are arrows which are attached to the bottom and top of the accidentals, respectively. Because there are twelve flats and twelve sharps, these accidentals allow notation up to 84edo with full enharmonicity, as described above. In order to notate edos which use fewer accidentals, the accidentals used are chosen to best approximate the required modification value, assuming that the above accidentals are in increments of perfect twelfth-accidentals. For example, suppose we wish to notate 45edo; 45 = 7*6+3, so seven sharp and seven flat accidentals are required. To determine, say, the three-sevenths sharp we multiply that value by twelve: 12*3/7 = 5.14 ≈ 5, so we use the five-twelfths sharp, or #v. Doing this for all sets of fewer than or equal to twelve accidentals per sharp/flat class, we find the following:<br /> | ||
<ul><li>For two accidentals: bb b n # x</li><li>3: bb db^ dv n ‡^ ‡#v x</li><li>4: bb db b d n ‡ # ‡# x</li><li>5: bb dbv bv b^ d^ n ‡v #v #^ ‡#^ x</li><li>6: bb dbv db^ b dv d^ n ‡v ‡^ # ‡#v ‡#^ x</li><li>7: bb dbv db bv b^ d d^ n ‡v ‡ #v #^ ‡# ‡#^ x</li><li>8: bb dbv db bv b b^ d d^ n ‡v ‡ #v # #^ ‡# ‡#^ x (Because this results in values exactly in between twelfths, some discretion was used to determine these to be the most intuitive.)</li><li>9: bb bb^ db db^ bv b^ dv d nv n n^ ‡ ‡^ #v #^ ‡#v ‡# xv x</li><li>10: bb bb^ dbv db^ bv b b^ dv d^ nv n n^ ‡v ‡^ #v # #^ ‡#v ‡#^ xv x</li><li>11: bb bb^ dbv db db^ bv b b^ dv d d^ nv n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x</li><li>12: bb bb^ dbv db db^ bv b b^ dv d d^ nv n n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x</li></ul><br /> | <ul><li>For two accidentals: bb b n # x</li><li>3: bb db^ dv n ‡^ ‡#v x</li><li>4: bb db b d n ‡ # ‡# x</li><li>5: bb dbv bv b^ d^ n ‡v #v #^ ‡#^ x</li><li>6: bb dbv db^ b dv d^ n ‡v ‡^ # ‡#v ‡#^ x</li><li>7: bb dbv db bv b^ d d^ n ‡v ‡ #v #^ ‡# ‡#^ x</li><li>8: bb dbv db bv b b^ d d^ n ‡v ‡ #v # #^ ‡# ‡#^ x (Because this results in values exactly in between twelfths, some discretion in was used in rounding to determine these to be the most intuitive.)</li><li>9: bb bb^ db db^ bv b^ dv d nv n n^ ‡ ‡^ #v #^ ‡#v ‡# xv x</li><li>10: bb bb^ dbv db^ bv b b^ dv d^ nv n n^ ‡v ‡^ #v # #^ ‡#v ‡#^ xv x</li><li>11: bb bb^ dbv db db^ bv b b^ dv d d^ nv n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x</li><li>12: bb bb^ dbv db db^ bv b b^ dv d d^ nv n n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x</li></ul><br /> | ||
This would clearly require a significant amount of memorization to be at all efficient. However, two &quot;shorthand&quot; systems can be made, for three and five accidentals per class:<br /> | This would clearly require a significant amount of memorization to be at all efficient. However, two &quot;shorthand&quot; systems can be made, for three and five accidentals per class:<br /> | ||
<ul><li>3: bb db d n ‡ ‡# x</li><li>5: bb db bv b^ d n ‡ #v #^ ‡# x</li></ul><br /> | <ul><li>3: bb db d n ‡ ‡# x</li><li>5: bb db bv b^ d n ‡ #v #^ ‡# x</li></ul><br /> | ||