Pentatonic Functional Just System: Difference between revisions
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We start by examining pythagorean intervals based on [[2L 3s]] classification. Note that the subscript 5 before the interval name means it is pentatonic, and that a factor of [[5/1|5]] in the denominator of a ratio would be a subscript 5 ''after'' the interval name. | We start by examining pythagorean intervals based on [[2L 3s]] classification. Note that the subscript 5 before the interval name means it is pentatonic, and that a factor of [[5/1|5]] in the denominator of a ratio would be a subscript 5 ''after'' the interval name. | ||
{| class="wikitable right-all" | {| class="wikitable right-all left-3" | ||
|+ style="font-size: 105%;" | Pythagorean intervals | |+ style="font-size: 105%;" | Pythagorean intervals | ||
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{| class="wikitable right-all" | {| class="wikitable right-all left-3" | ||
|+ style="font-size: 105%;" | Ratios with a factor of 7 | |+ style="font-size: 105%;" | Ratios with a factor of 7 | ||
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|+ style="font-size: 105%;" | Ratios with two factors of 7 | |+ style="font-size: 105%;" | Ratios with two factors of 7 | ||
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{| class="wikitable right-all" | {| class="wikitable right-all left-3" | ||
|+ style="font-size: 105%;" | Ratios with a factor of 5 | |+ style="font-size: 105%;" | Ratios with a factor of 5 | ||
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{| class="wikitable right-all" | {| class="wikitable right-all left-3" | ||
|+ style="font-size: 105%;" | Ratios with two factors of 5 | |+ style="font-size: 105%;" | Ratios with two factors of 5 | ||
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Now, there are intervals between the pentic categories, such as [[11/9]] and [[12/11]]. The edges of each interval category can be considered the 5-limit intervals (such as [[16/15]], [[10/9]], [[6/5]], and [[5/4]]), thus the regions between interval categories can be termed "interpental" (not to be confused with [[Interpental|interpental temperament]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentic, such as 12/11 being between 16/15 and 10/9, and 11/9 being between 6/5 and 5/4. One may realize that [[11/8]] and [[16/11]] are classified rather out of place, with 11/8 being a <sub>5</sub>subfourth and 16/11 being a <sub>5</sub>superthird. The PFJS is not perfect, and this system was designed to keep [[13/11]] and [[15/13]] in the right category; thus 11/8 must be messed up (though other intervals of 11 are interpental, so are fine). However, 11/8 is in the region between 4/3 and 3/2, where there can be considered to be ''two'' interpental regions: one between [[27/20]] and [[45/32]], and another between [[64/45]] and [[40/27]]. These are the [[superfourth]] and [[subfifth]] regions in diatonic, which can also be considered neutral regions. In pentic, since these regions are interpental, they are ambiguously between <sub>5</sub>thirds and <sub>5</sub>fourths, justifying the otherwise out-of-place classification of 11/8. However, one may not be fond of the fact that [[7/5]] and [[10/7]] are just barely in these ranges; thus, one may prefer to make them narrower (~48 cents wide). | Now, there are intervals between the pentic categories, such as [[11/9]] and [[12/11]]. The edges of each interval category can be considered the 5-limit intervals (such as [[16/15]], [[10/9]], [[6/5]], and [[5/4]]), thus the regions between interval categories can be termed "interpental" (not to be confused with [[Interpental|interpental temperament]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentic, such as 12/11 being between 16/15 and 10/9, and 11/9 being between 6/5 and 5/4. One may realize that [[11/8]] and [[16/11]] are classified rather out of place, with 11/8 being a <sub>5</sub>subfourth and 16/11 being a <sub>5</sub>superthird. The PFJS is not perfect, and this system was designed to keep [[13/11]] and [[15/13]] in the right category; thus 11/8 must be messed up (though other intervals of 11 are interpental, so are fine). However, 11/8 is in the region between 4/3 and 3/2, where there can be considered to be ''two'' interpental regions: one between [[27/20]] and [[45/32]], and another between [[64/45]] and [[40/27]]. These are the [[superfourth]] and [[subfifth]] regions in diatonic, which can also be considered neutral regions. In pentic, since these regions are interpental, they are ambiguously between <sub>5</sub>thirds and <sub>5</sub>fourths, justifying the otherwise out-of-place classification of 11/8. However, one may not be fond of the fact that [[7/5]] and [[10/7]] are just barely in these ranges; thus, one may prefer to make them narrower (~48 cents wide). | ||
{| class="wikitable right-1" | {| class="wikitable right-1 right-2" | ||
|+ style="font-size: 105%;" | Interpental regions | |+ style="font-size: 105%;" | Interpental regions | ||
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