Pentatonic Functional Just System: Difference between revisions

Line 362:

A lot of interesting things show up here. First of all, we finally have just representations for "[[neutral]]" intervals, which are between the minor and major intervals in their category. Here, [[15/13]], which is beteeen [[8/7]] and [[7/6]], can be considered a neutral 5second (especially if [[676/675]] is tempered out), [[13/10]] a semi-sub 5third, [[20/13]] a semi-super 5fourth, and [[26/15]] a neutral 5fifth. Intervals which are neutral here are considered [[interseptimal]] by diatonic classification, as they fall right between two diatonic interval categories.

Now, there are intervals between the pentatonic 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|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentatonic, 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 5subfourth and 16/11 being a 5superthird. 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 pentatonic, since these regions are interpental, they are ambiguously between 5thirds and 5fourths, 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 (~40 cents wide).

Now, there are intervals between the pentatonic 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|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentatonic, 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 5subfourth and 16/11 being a 5superthird. 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 pentatonic, since these regions are interpental, they are ambiguously between 5thirds and 5fourths, 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"

|+ Interpental regions

|-

! Region !! Is between !! Name (Diatonic) !! Name (Pentatonic)

|-

| 123–171{{c}} || S1–m2 || Neutral 2nd || Terric (Earth)

|-

| 327–375{{c}} || M2–s3 || Neutral 3rd || Argic (Silver)

|-

| 531–579{{c}} || P3–s4 || Superfourth || Pyric (Fire)

|-

| 621–669{{c}} || S3–P4 || Subfifth || Hydric (Water)

|-

| 825–873{{c}} || S4–m5 || Neutral 6th || Auric (Gold)

|-

| 1029–1077{{c}} || M5–s6 || Neutral 7th || Aeric (Air)

|}

In [[13-limit]] [[superpyth]], [[11/8]] is a sub-sub-sub-5fourth, and [[13/8]] is a sub-sub-5fifth.

This system could be extended to higher limits, but we'll leave it here for now.

@@ Line 362: / Line 362: @@
 A lot of interesting things show up here. First of all, we finally have just representations for "[[neutral]]" intervals, which are between the minor and major intervals in their category. Here, [[15/13]], which is beteeen [[8/7]] and [[7/6]], can be considered a neutral <sub>5</sub>second (especially if [[676/675]] is tempered out), [[13/10]] a semi-sub <sub>5</sub>third, [[20/13]] a semi-super <sub>5</sub>fourth, and [[26/15]] a neutral <sub>5</sub>fifth. Intervals which are neutral here are considered [[interseptimal]] by diatonic classification, as they fall right between two diatonic interval categories.
-Now, there are intervals between the pentatonic 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|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentatonic, 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 pentatonic, 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 (~40 cents wide).
+Now, there are intervals between the pentatonic 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|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentatonic, 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 pentatonic, 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"
+|+ Interpental regions
+|-
+! Region !! Is between !! Name (Diatonic) !! Name (Pentatonic)
+|-
+| 123–171{{c}} || S1–m2 || Neutral 2nd || Terric (Earth)
+|-
+| 327–375{{c}} || M2–s3 || Neutral 3rd || Argic (Silver)
+|-
+| 531–579{{c}} || P3–s4 || Superfourth || Pyric (Fire)
+|-
+| 621–669{{c}} || S3–P4 || Subfifth || Hydric (Water)
+|-
+| 825–873{{c}} || S4–m5 || Neutral 6th || Auric (Gold)
+|-
+| 1029–1077{{c}} || M5–s6 || Neutral 7th || Aeric (Air)
+|}
+In [[13-limit]] [[superpyth]], [[11/8]] is a sub-sub-sub-<sub>5</sub>fourth, and [[13/8]] is a sub-sub-<sub>5</sub>fifth.
+This system could be extended to higher limits, but we'll leave it here for now.