User:Godtone: Difference between revisions

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'''0''' (tonic) < '''1''' (contralead) < '''2''' (supertonic) ? '''4''' (serviant) < '''5''' (antitonic) < '''6''' (dominant) ? '''8''' (subtonic) < '''9''' (lead) < (1)'''0''' (tonic)

Note however that the supertonic may in some cases be above the antitonic (and correspondingly the subtonic may in some cases be below the antitonic). This leaves '''3''' (the mediant) and '''7''' (the contramediant) as gap-fillers for below and above '''5''' (the antitonic) respectively.

Note however that the supertonic may in some cases be above the antitonic (and correspondingly the subtonic may in some cases be below the antitonic). This leaves '''3''' (the mediant) and '''7''' (the contramediant) as gap-fillers for below and above '''5''' (the antitonic) respectively. This does however point to the next place to investigate: when is the supertonic above the antitonic? This would imply that twice the dominant is above 1.5 octaves, or that the dominant is above 900 cents while being below the contralead, thus implying that an EDO approximation must be 9 or above to support such a MOSS (as there must be space for the contralead to be distinct).

Another interesting observation is that this system suggests an at-least-7-note-per-period MOSS because in order for all of the functions '''0, 1, 2, 4, 6, 8, 9''' to be present and distinct from each-other, there must be at least 7 notes. (Note that the only digits absent are '''3, 5, 7''' where '''3, 7''' are context-dependent gap-filler functions while '''5''' is a function unrelated to the generator and instead related to the period-complement symmetry of the functions suggesting a midpoint between adjacent instances of the tonic.) However, yet more interestingly, if you have a 5-note MOSS, it naturally implies using the 7-, 8- or 9-note child MOSS as the basis for analysing a 5-note MOSS with a small step that is too large to serve as a true (contra)lead as that child MOSS will instead have the contralead be the chroma of that MOSS. Meanwhile, if the small step is small enough to serve as a (contra)lead, it will necessarily be confused with either a supertonic or a subtonic, implying an interesting musical reality for such MOSSes. This system thus also suggests that modes which have at least two generators from the tonic in each direction have a unique musical capability and thus modes that do not satisfy this restriction have emergent musical properties based on their very lack of some of those functions.

== Examples of functions ==

* If [[5/4]] is taken as the generator and dominant, then [[25/16]] is a supertonic. Thus [[8/5]] is a serviant and [[32/25]] is a subtonic. if [[32/25]] is equated with [[9/7]] by tempering [[225/224]], then [[9/7]] is a subtonic and [[14/9]] is a supertonic. This forms a [[3L 7s]] scale.

* if [[13/8]] is taken as the generator and dominant, then [[169/128]] is a supertonic, which seems rather complex, but this can neatly be equated with [[21/16]] by tempering [[169/168]], which is convenient as we want to preserve the rootedness of the generator when stacking it. Thus [[16/13]] is a serviant and [[32/21]] is a subtonic. This also forms a [[3L 7s]] scale, but the generator is now above the antitonic rather than below.

* If [[3/2]] is taken as the generator and dominant, then [[9/8]] is a supertonic. Thus [[4/3]] is a serviant and [[16/9]] is a subtonic. This is a uniquely very low complexity instance.

* If [[7/4]] is taken as the generator and dominant, then [[49/32]] is a supertonic. Thus [[8/7]] is a serviant and [[64/49]] is a subtonic.

* If [[11/8]] is taken as the generator and dominant, then [[121/64]] is a supertonic. Thus [[16/11]] is a serviant and [[128/121]] is a subtonic. This is a peculiar case where the supertonic can reasonably be equated with the lead (and correspondingly the subtonic can reasonably be equated with the contralead). Given this, it doesn't seem necessary to introduce any temperings, as leads and contraleads are a more dissonant type of function.

* If [[15/8]] is taken as the generator and dominant, then [[225/128]] is a supertonic. Thus [[16/15]] is a serviant and [[256/225]] is a subtonic. This is a peculiar case where the dominant can reasonably be equated with the lead (and correspondingly the serviant can reasonably be equated with the contralead). Furthermore, if we temper [[225/224]] then the supertonic is [[7/4]] and the subtonic is [[8/7]], meaning the supertonic and subtonic are noticeably more consonant and stable both harmonically and functionally than the dominant and serviant.

== Intervals ==

@@ Line 166: / Line 166: @@
 '''0''' (tonic) < '''1''' (contralead) < '''2''' (supertonic) ? '''4''' (serviant) < '''5''' (antitonic) < '''6''' (dominant) ? '''8''' (subtonic) < '''9''' (lead) < (1)'''0''' (tonic)
-Note however that the supertonic may in some cases be above the antitonic (and correspondingly the subtonic may in some cases be below the antitonic). This leaves '''3''' (the mediant) and '''7''' (the contramediant) as gap-fillers for below and above '''5''' (the antitonic) respectively.
+Note however that the supertonic may in some cases be above the antitonic (and correspondingly the subtonic may in some cases be below the antitonic). This leaves '''3''' (the mediant) and '''7''' (the contramediant) as gap-fillers for below and above '''5''' (the antitonic) respectively. This does however point to the next place to investigate: when is the supertonic above the antitonic? This would imply that twice the dominant is above 1.5 octaves, or that the dominant is above 900 cents while being below the contralead, thus implying that an EDO approximation must be 9 or above to support such a MOSS (as there must be space for the contralead to be distinct).
+Another interesting observation is that this system suggests an at-least-7-note-per-period MOSS because in order for all of the functions '''0, 1, 2, 4, 6, 8, 9''' to be present and distinct from each-other, there must be at least 7 notes. (Note that the only digits absent are '''3, 5, 7''' where '''3, 7''' are context-dependent gap-filler functions while '''5''' is a function unrelated to the generator and instead related to the period-complement symmetry of the functions suggesting a midpoint between adjacent instances of the tonic.) However, yet more interestingly, if you have a 5-note MOSS, it naturally implies using the 7-, 8- or 9-note child MOSS as the basis for analysing a 5-note MOSS with a small step that is too large to serve as a true (contra)lead as that child MOSS will instead have the contralead be the chroma of that MOSS. Meanwhile, if the small step is small enough to serve as a (contra)lead, it will necessarily be confused with either a supertonic or a subtonic, implying an interesting musical reality for such MOSSes. This system thus also suggests that modes which have at least two generators from the tonic in each direction have a unique musical capability and thus modes that do not satisfy this restriction have emergent musical properties based on their very lack of some of those functions.
+== Examples of functions ==
+* If [[5/4]] is taken as the generator and dominant, then [[25/16]] is a supertonic. Thus [[8/5]] is a serviant and [[32/25]] is a subtonic. if [[32/25]] is equated with [[9/7]] by tempering [[225/224]], then [[9/7]] is a subtonic and [[14/9]] is a supertonic. This forms a [[3L 7s]] scale.
+* if [[13/8]] is taken as the generator and dominant, then [[169/128]] is a supertonic, which seems rather complex, but this can neatly be equated with [[21/16]] by tempering [[169/168]], which is convenient as we want to preserve the rootedness of the generator when stacking it. Thus [[16/13]] is a serviant and [[32/21]] is a subtonic. This also forms a [[3L 7s]] scale, but the generator is now above the antitonic rather than below.
+* If [[3/2]] is taken as the generator and dominant, then [[9/8]] is a supertonic. Thus [[4/3]] is a serviant and [[16/9]] is a subtonic. This is a uniquely very low complexity instance.
+* If [[7/4]] is taken as the generator and dominant, then [[49/32]] is a supertonic. Thus [[8/7]] is a serviant and [[64/49]] is a subtonic.
+* If [[11/8]] is taken as the generator and dominant, then [[121/64]] is a supertonic. Thus [[16/11]] is a serviant and [[128/121]] is a subtonic. This is a peculiar case where the supertonic can reasonably be equated with the lead (and correspondingly the subtonic can reasonably be equated with the contralead). Given this, it doesn't seem necessary to introduce any temperings, as leads and contraleads are a more dissonant type of function.
+* If [[15/8]] is taken as the generator and dominant, then [[225/128]] is a supertonic. Thus [[16/15]] is a serviant and [[256/225]] is a subtonic. This is a peculiar case where the dominant can reasonably be equated with the lead (and correspondingly the serviant can reasonably be equated with the contralead). Furthermore, if we temper [[225/224]] then the supertonic is [[7/4]] and the subtonic is [[8/7]], meaning the supertonic and subtonic are noticeably more consonant and stable both harmonically and functionally than the dominant and serviant.
 == Intervals ==