User:Godtone: Difference between revisions

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* The '''tonic''', notated '''0'''. The number 0 is meant to represent an origin/reference point. This is always [[1/1]] - and due to assuming octave-equivalence throughout this analysis of function - is also always equivalent to [[2/1]], [[4/1]], etc.

* The '''dominant''', notated '''6'''. The number 6 looks visually somewhat like a reversed delta/d symbol, for "dominant". The 6th harmonic up to octave-equivalence is [[3/2]], the perfect fifth. The dominant should ideally always be a(n approximation of a) rooted generator.

* The '''serviant''', notated '''4'''. The number 4 should remind you that the perfect '''fourth'' is [[4/3|'''4'''/3]]. The serviant is always the octave-complement (AKA period-complement) of the dominant, with the only distinguishing feature between the two being which is a (up to octave-equivalence) a (rooted) harmonic and which is a subharmonic. The serviant should (ideally) always be a subharmonic of the tonic.

* The '''serviant''', notated '''4'''. The number 4 should remind you that the perfect '''fourth''' is [[4/3|'''4'''/3]]. The serviant is always the octave-complement (AKA period-complement) of the dominant, with the only distinguishing feature between the two being which is a (up to octave-equivalence) a (rooted) harmonic and which is a subharmonic. The serviant should (ideally) always be a subharmonic of the tonic.

* The '''supertonic''', notated '''2''', is a stack of two dominants upwards (and octave-reduced if needed). The '''2''' should remind you that it is '''two''' dominants.

* The '''subtonic''', notated '''8''', is the octave-complement of the '''supertonic''', and is thus a stack of two serviants upwards (and octave-reduced if needed). The '''8''' should remind you that it is two (2) serviants (4) as 2 * 4 = 8.

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* The '''lead''', notated '''9''', is a small step (s) or chroma (L-s) below the tonic. The number 9 represents being 1 step (chroma or small step) below the tonic (1**0** - **1** = **9**). The name is derived from the term "leading tone". Diatonically, this is a semitone.

* The '''contralead''', notated '''1''', is a small step (s) or chroma (L-s) above the tonic. The name is derived from contra+lead. The number 1 represents being 1 step (chroma or small step) above the tonic.

Finally, there are a pair of functions which act as 'gap fillers' in the MOSS, which can be assigned to the result of stacking 3 to 5 generators (whichever amounts avoid the regions corresponding to other functions and fit the restrictions). These are the '''mediant''' and the '''contramediant'''. Before defining the mediant, an analysis of the order of the functions so far is required. The basic order is:

tonic < contralead < antitonic < lead < tonic (or in numeric notation: '''0''' < '''1''' < '''5''' < '''9''' < 1'''0''')

Then the position of the dominant, serviant, supertonic and subtonic depend on whether the dominant is above or below the antitonic.

If it is above the antitonic, then it is also below the lead. This means that two dominants exceed an octave and therefore that the supertonic is always strictly between the contralead and the lead. Thus the subtonic is also strictly between the contralead and the lead. We will examine this first as there are more low-complexity rooted generators in this case, namely [[3/2]], [[7/4]], [[13/8]] and [[15/8]], with [[5/4]] and [[11/8]] being the main exceptions, and with [[9/8]] contained in the MOSS for [[3/2]] and [[5/4]] reachable indirectly through [[meantone]] or [[schismic]]. It is [[3/2]], [[5/4]] and [[7/4] that are of most interest though, as in those cases the generator is simple enough that stacking it twice still constitutes a reasonable consonance, while [[11/8]] requires sometihng as complex as [[121/64]]. Finally, [[15/8]] stacked twice is [[225/128]] (up to octave-equivalence) which can be equated with [[7/4]] if you [[Marvel family|temper]] [[225/224]]. The result is:

tonic < contralead < serviant < antitonic < dominant < lead < tonic AND contralead < supertonic/subtonic < lead

Can we deduce more than this? We can answer this question by considering the extreme cases. The lowest dominant above the antitonic would result in a supertonic just above the contralead (such as [[5L 2s]] in [[12edo]]). The highest dominant above the antitonic would result in a supertonic just below the contralead, but also below the dominant. Therefore, the supertonic must be below the dominant and the subtonic must be above the serviant.

contralead < supertonic < dominant < lead AND contralead < serviant < subtonic < lead

As the dominant is between the antitonic

== Intervals ==

@@ Line 141: / Line 141: @@
 * The '''tonic''', notated '''0'''. The number 0 is meant to represent an origin/reference point. This is always [[1/1]] - and due to assuming octave-equivalence throughout this analysis of function - is also always equivalent to [[2/1]], [[4/1]], etc.
 * The '''dominant''', notated '''6'''. The number 6 looks visually somewhat like a reversed delta/d symbol, for "dominant". The 6th harmonic up to octave-equivalence is [[3/2]], the perfect fifth. The dominant should ideally always be a(n approximation of a) rooted generator.
-* The '''serviant''', notated '''4'''. The number 4 should remind you that the perfect '''fourth'' is [[4/3|'''4'''/3]]. The serviant is always the octave-complement (AKA period-complement) of the dominant, with the only distinguishing feature between the two being which is a (up to octave-equivalence) a (rooted) harmonic and which is a subharmonic. The serviant should (ideally) always be a subharmonic of the tonic.
+* The '''serviant''', notated '''4'''. The number 4 should remind you that the perfect '''fourth''' is [[4/3|'''4'''/3]]. The serviant is always the octave-complement (AKA period-complement) of the dominant, with the only distinguishing feature between the two being which is a (up to octave-equivalence) a (rooted) harmonic and which is a subharmonic. The serviant should (ideally) always be a subharmonic of the tonic.
 * The '''supertonic''', notated '''2''', is a stack of two dominants upwards (and octave-reduced if needed). The '''2''' should remind you that it is '''two''' dominants.
 * The '''subtonic''', notated '''8''', is the octave-complement of the '''supertonic''', and is thus a stack of two serviants upwards (and octave-reduced if needed). The '''8''' should remind you that it is two (2) serviants (4) as 2 * 4 = 8.
@@ Line 149: / Line 149: @@
 * The '''lead''', notated '''9''', is a small step (s) or chroma (L-s) below the tonic. The number 9 represents being 1 step (chroma or small step) below the tonic (1**0** - **1** = **9**). The name is derived from the term "leading tone". Diatonically, this is a semitone.
 * The '''contralead''', notated '''1''', is a small step (s) or chroma (L-s) above the tonic. The name is derived from contra+lead. The number 1 represents being 1 step (chroma or small step) above the tonic.
+Finally, there are a pair of functions which act as 'gap fillers' in the MOSS, which can be assigned to the result of stacking 3 to 5 generators (whichever amounts avoid the regions corresponding to other functions and fit the restrictions). These are the '''mediant''' and the '''contramediant'''. Before defining the mediant, an analysis of the order of the functions so far is required. The basic order is:
+tonic < contralead < antitonic < lead < tonic (or in numeric notation: '''0''' < '''1''' < '''5''' < '''9''' < 1'''0''')
+Then the position of the dominant, serviant, supertonic and subtonic depend on whether the dominant is above or below the antitonic.
+If it is above the antitonic, then it is also below the lead. This means that two dominants exceed an octave and therefore that the supertonic is always strictly between the contralead and the lead. Thus the subtonic is also strictly between the contralead and the lead. We will examine this first as there are more low-complexity rooted generators in this case, namely [[3/2]], [[7/4]], [[13/8]] and [[15/8]], with [[5/4]] and [[11/8]] being the main exceptions, and with [[9/8]] contained in the MOSS for [[3/2]] and [[5/4]] reachable indirectly through [[meantone]] or [[schismic]]. It is [[3/2]], [[5/4]] and [[7/4] that are of most interest though, as in those cases the generator is simple enough that stacking it twice still constitutes a reasonable consonance, while [[11/8]] requires sometihng as complex as [[121/64]]. Finally, [[15/8]] stacked twice is [[225/128]] (up to octave-equivalence) which can be equated with [[7/4]] if you [[Marvel family|temper]] [[225/224]]. The result is:
+tonic < contralead < serviant < antitonic < dominant < lead < tonic AND contralead < supertonic/subtonic < lead
+Can we deduce more than this? We can answer this question by considering the extreme cases. The lowest dominant above the antitonic would result in a supertonic just above the contralead (such as [[5L 2s]] in [[12edo]]). The highest dominant above the antitonic would result in a supertonic just below the contralead, but also below the dominant. Therefore, the supertonic must be below the dominant and the subtonic must be above the serviant.
+contralead < supertonic < dominant < lead AND contralead < serviant < subtonic < lead
+As the dominant is between the antitonic
 == Intervals ==