User:TallKite/Midpoints: Difference between revisions
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== Background == | == Background == | ||
Any interval roughly midway between a M2 and a m3 is an interordinal interval. As is one between a M3 and a P4, and so forth. While it's easy to label or categorize most intervals as 2nds, 3rds, etc., interordinals are difficult to label. Kite Giedraitis proposes labeling them "plus-second", "plus-third", etc., written +2nd or +2. See Kite's Rationales for terminology below. | Any interval roughly midway between a M2 and a m3 is an interordinal interval. As is one between a M3 and a P4, and so forth. While it's easy to label or categorize most intervals as 2nds, 3rds, etc., interordinals are difficult to label. [[Kite Giedraitis]] proposes labeling them "plus-second", "plus-third", etc., written +2nd or +2. See Kite's Rationales for terminology below. | ||
"Neutral" refers to a range of interval sizes. Anything in roughly the 340-360c range can be called a neutral 3rd. But mid refers to a single specific interval midway between major and minor. A mid interval is neutral, but neutral intervals are very often not mid. | "Neutral" refers to a range of interval sizes. Anything in roughly the 340-360c range can be called a neutral 3rd. But mid refers to a single specific interval midway between major and minor. A mid interval is neutral, but neutral intervals are very often not mid. | ||
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Mid and outside are qualities like perfect, major and minor. Neutral and interordinal are also qualities, although slightly vague ones. Perfect, major and minor are in theory precise qualities but in practice they are slightly vague. e.g. 5/4. They can be diverge from their exact by adding super- and sub- serve to . Likewise supermid, subneutral and superoutside, but subinterordinal is going too far. -ish serves to | Mid and outside are qualities like perfect, major and minor. Neutral and interordinal are also qualities, although slightly vague ones. Perfect, major and minor are in theory precise qualities but in practice they are slightly vague. e.g. 5/4. They can be diverge from their exact by adding super- and sub- serve to . Likewise supermid, subneutral and superoutside, but subinterordinal is going too far. -ish serves to | ||
Mid can be generalized to "middish" to include all neutral intervals. Likewise "outside-ish" for all interordinals. | Mid can be generalized to "middish" to include all neutral intervals. Likewise "outside-ish" for all interordinals. | ||
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or 15/13 is "a" +2nd, but 5\19 is "the" +2nd?<br> | or 15/13 is "a" +2nd, but 5\19 is "the" +2nd?<br> | ||
The exact cents of the midpoints are defined by the size of the 5th. (As well as the octave, if it's stretched or compressed.) Assume unstretched 8ves and a 5th of 700+c cents. Thus c = one twelfth of a (possibly tempered) pyth comma. | |||
{| class="wikitable" | |||
+1sn | |+Midpoints | ||
!quality | |||
+2nd | !degree | ||
!fraction | |||
+3rd | !cents | ||
!notes | |||
+4th | |- | ||
|outside | |||
+5th | | +1sn | ||
|m2/2 | |||
+6th | |50¢ - 2.5c | ||
|quartertone | |||
+7th | |- | ||
|mid | |||
|2nd | |||
|m3/2 | |||
|150¢ - 1.5c | |||
| | |||
|- | |||
|outside | |||
| +2nd | |||
|P4/2 | |||
|250¢ - 0.5c | |||
| | |||
|- | |||
|mid | |||
|3rd | |||
|P5/2 | |||
|350¢ + 0.5c | |||
| | |||
|- | |||
|outside | |||
| +3rd | |||
|M6/2 | |||
|450¢ + 1.5c | |||
| | |||
|- | |||
|mid | |||
|4th | |||
|M7/2 | |||
|550¢ + 2.5c | |||
| | |||
|- | |||
|outside | |||
| +4th | |||
|P8/2 | |||
|600¢ | |||
|the half-octave | |||
|- | |||
|mid | |||
|5th | |||
|m9/2 | |||
|650¢ - 2.5c | |||
|P8/2 + o+1sn | |||
|- | |||
|outside | |||
| +5th | |||
|m10/2 | |||
|750¢ - 1.5c | |||
|P8/2 + o+2nd | |||
|- | |||
|mid | |||
|6th | |||
|P11/2 | |||
|850¢ - 0.5c | |||
|etc. | |||
|- | |||
|outside | |||
| +6th | |||
|P12/2 | |||
|950¢ + 0.5c | |||
| | |||
|- | |||
|mid | |||
|7th | |||
|M13/2 | |||
|1050¢ + 1.5c | |||
| | |||
|- | |||
|outside | |||
| +7th | |||
|M14/2 | |||
|1150¢ + 2.5c | |||
|P8 - o+1sn | |||
|} | |||
There is symmetry not only within the octave but also within the half-octave. | |||
There is no mid 1sn or mid 8ve. The +8ve is as expected, a +1sn plus an 8ve. | There is no mid 1sn or mid 8ve. The +8ve is as expected, a +1sn plus an 8ve. | ||
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One could label various interordinals as perfect, major, minor, etc. But outsiders aren't major or minor, they're just outside. Analogous to how mids are just mid. IOW mid and outside are themselves qualities. Neutral is also a quality, although a vague one, whereas mid is exact. Major, minor, etc. are in theory exact, but in practice vague. If they were in practice exact, we would describe e.g. 5/4 as "majorish". | One could label various interordinals as perfect, major, minor, etc. But outsiders aren't major or minor, they're just outside. Analogous to how mids are just mid. IOW mid and outside are themselves qualities. Neutral is also a quality, although a vague one, whereas mid is exact. Major, minor, etc. are in theory exact, but in practice vague. If they were in practice exact, we would describe e.g. 5/4 as "majorish". | ||
Except for the +4th, a plus interval is always half a m2 above the major or perfect version of that interval. | Except for the +4th, a plus interval is always half a m2 above the major or perfect version of that interval. | ||
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The gamut of imperfect degrees runs from minor to major. Extending that to include aug and dim wouldn't change the location of the +2nd or +6th, and would mess up (i.e. give non-intuitive results for) the +1sn and +7th. | The gamut of imperfect degrees runs from minor to major. Extending that to include aug and dim wouldn't change the location of the +2nd or +6th, and would mess up (i.e. give non-intuitive results for) the +1sn and +7th. | ||
The 4th's gamut runs perfect-aug and the 5th's runs dim-perfect. Limiting each to perfect only | The 4th's gamut runs perfect-aug and the 5th's runs dim-perfect. Limiting each gamut to perfect only would mess up the ~4th and the ~5th. Expanding each to dim-aug additionally messes up the +3rd and +5th. Note that often A4 > d5, and the outside +4th isn't actually outside of the nearest gamuts. | ||
The unison's gamut is very small, just P1, to avoid messing up the +1sn. Likewise the octave's gamut is P8, to avoid messing up the +7th. | The unison's gamut is very small, just P1, to avoid messing up the +1sn. Likewise the octave's gamut is P8, to avoid messing up the +7th. | ||
Each gamut is a m2 wide, except for the 1sn and the 8ve. Each gap between gamuts is a m2, except for at the half-8ve. The gaps plus the widths run m2 A1 m2 A1 m2 A1 pyth A1 m2 A1 m2 A1 m2. | Each gamut is a m2 wide, except for the 1sn and the 8ve. Each gap between gamuts is a m2, except for at the half-8ve. The gaps plus the widths run m2 A1 m2 A1 m2 A1 pyth A1 m2 A1 m2 A1 m2. | ||