User:Godtone: Difference between revisions

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Note that in the system, there is always the rule that the octave-complement (AKA period-complement) of a function's number is 10 minus that number. This is because in the final system, 10 functions are derived for diatonic. These functions' numbers are picked in the order they appear in [[5L 2s|diatonic]], as this notation is intended for [[5L 2s|diatonic]]. For other MOSSes, the names of the functions remain the same but some of the notation may change. However, the numbers for the functions '''0, 1, 5, 9''' are generalisable to all octave-period MOSSes. This leads us to our next pair of functions: '''1''' and '''9'''.

* 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 '''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:

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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.

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]]) and therefore below the dominant. The highest dominant above the antitonic would result in a supertonic just below the contralead, but also below the dominant, as by stacking the dominant twice, the distance from the tonic is doubled. Therefore, the supertonic must be below the dominant and the subtonic must be above the serviant. Therefore:

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

This justifies the following numeric notation for when dominant > antitonic:

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

~~As the~~ dominant ~~is between the antitonic~~

== Intervals ==

@@ Line 147: / Line 147: @@
 Note that in the system, there is always the rule that the octave-complement (AKA period-complement) of a function's number is 10 minus that number. This is because in the final system, 10 functions are derived for diatonic. These functions' numbers are picked in the order they appear in [[5L 2s|diatonic]], as this notation is intended for [[5L 2s|diatonic]]. For other MOSSes, the names of the functions remain the same but some of the notation may change. However, the numbers for the functions '''0, 1, 5, 9''' are generalisable to all octave-period MOSSes. This leads us to our next pair of functions: '''1''' and '''9'''.
-* 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 '''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:
@@ Line 158: / Line 158: @@
 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.
+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]]) and therefore below the dominant. The highest dominant above the antitonic would result in a supertonic just below the contralead, but also below the dominant, as by stacking the dominant twice, the distance from the tonic is doubled. Therefore, the supertonic must be below the dominant and the subtonic must be above the serviant. Therefore:
 contralead < supertonic < dominant < lead AND contralead < serviant < subtonic < lead
+This justifies the following numeric notation for when dominant > antitonic:
+'''0''' (tonic) < '''1''' (contralead) < '''2''' (supertonic) ? '''4''' (serviant) < '''5''' (antitonic) < '''6''' (dominant) ? '''8''' (subtonic) < '''9''' (lead) < (1)'''0''' (tonic)
-As the dominant is between the antitonic
 == Intervals ==