User:Godtone: Difference between revisions

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* The duality (2) of finality (9), given that judgement requires an axis along which to judge.

* The mix of 5 and 6, where 5 represents a stable foundation and 6 represents corelessness, and where 5 represents grace and harmony and 6 represents sin and disharmony. Thus, it takes on a "neutral" nature imbetween the two, necessary for judgement, which is fitting considering that the lowest complexity intervals involving this harmonic are neutral. Even 11/8 is a mixture of an antitonic and a serviant, thus representing an ambiguous region which is nonetheless rooted (contrasting it with 4/3).

~~<br/>~~

'''12''' : The number of divine, perfect order, as it is 7+5 and 4*3. Means pure grace/graceful purity stabilised on the earth and is the stabilisation of death such that it becomes under control for good. This is also partly because it has a large number of factors, thus integrating many simpler orders into a perfect (7) and graceful (5) combined one.

<br/>

'''13''' : Represents rebellion against the first divine order (12+1), but also represents the attempted progression upwards/further and thus also represents new beginnings. Thus represents the smokeless flame in its raw and partially sinful (in the sense of imperfect) desire to blaze new trails and fly higher. It is 7+6 and thus can also be a number representing pure evil, although using it for this meaning only would be shallow, as mistakes (sin; 6) give further opportunity for enhanced perfection (7). It is 8+5 and thus can be the graceful side of power, again representing a sinful counterpart to divine order, hence the potential association with evil. It is 9+4 and thus is supreme/complete stability applied to the laws of earth and applied to death without necessarily giving sufficient concern to other principles/values, hence again fitting all the themes mentioned prior. Finally, it is 10+3, representing the stabilisation of the order of man, and thus the attempted solidification of imperfection into an essence. Thus is associated with fire, as misguided transcendence may have steep consequences and as fire can be used for good for burning off imperfection by first drawing strong attention to it by illuminating it. Note that [[13/8]] is a rooted rational approximation of phi (the golden ratio) which is mathematically in some sense "the most irrational number" due to having a continued fraction consisting only of 1's, thus [[13/8]] takes on some of this mysterious quality while being perfectly just; a sort of ambiguous smoothness emerges in at least some of the low complexity intervals involving this prime essence, analogous to the beauty and smoothness of a smokeless flame.

== Diatonic functions ==

This section is largely inspired by [[User:Aura/Aura's_Ideas_on_Functional_Harmony|Aura's Ideas on Functional Harmony]] combined with my own observations, intuitions and simplifications, as I admittedly do not understand nearly as much as Aura on this subject, but I do intuitively understand my musical sense, which I wish to approximately translate here. This will also focus on generalisation from the [[5L 2s|diatonic]] MOSS to other MOSSes.

We begin with a rooted generator, meaning an interval of the form a/2^n where 2^(n+1) > a > 2^n. This is taken as the most important interval, other than the period, which we will assume and define to be an octave as I believe this is the most natural period for a MOSS, especially in the case where the generator or its octave-complement is rooted. We also begin with a tonic, meaning a 'centre' of the scale from which the scale is constructed by stacking periods and generators up or down. The position of the tonic depends on the mode of the MOSS used. Note that we will start with the [[Pythagorean tuning]] of the [[5L 2s|diatonic]] MOSS and generalise from there. Now we are ready to start defining the most fundamental functions:

* 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 '''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.

* The '''antitonic''', notated '''5''', is approximately half of an octave, being important due to both being a less strong consonance or a dissonance and due to being the mirror line when performing octave-complementing.

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 '''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.

== Intervals ==

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 * The duality (2) of finality (9), given that judgement requires an axis along which to judge.
 * The mix of 5 and 6, where 5 represents a stable foundation and 6 represents corelessness, and where 5 represents grace and harmony and 6 represents sin and disharmony. Thus, it takes on a "neutral" nature imbetween the two, necessary for judgement, which is fitting considering that the lowest complexity intervals involving this harmonic are neutral. Even 11/8 is a mixture of an antitonic and a serviant, thus representing an ambiguous region which is nonetheless rooted (contrasting it with 4/3).
-<br/>
 '''12''' : The number of divine, perfect order, as it is 7+5 and 4*3. Means pure grace/graceful purity stabilised on the earth and is the stabilisation of death such that it becomes under control for good. This is also partly because it has a large number of factors, thus integrating many simpler orders into a perfect (7) and graceful (5) combined one.
 <br/>
 '''13''' : Represents rebellion against the first divine order (12+1), but also represents the attempted progression upwards/further and thus also represents new beginnings. Thus represents the smokeless flame in its raw and partially sinful (in the sense of imperfect) desire to blaze new trails and fly higher. It is 7+6 and thus can also be a number representing pure evil, although using it for this meaning only would be shallow, as mistakes (sin; 6) give further opportunity for enhanced perfection (7). It is 8+5 and thus can be the graceful side of power, again representing a sinful counterpart to divine order, hence the potential association with evil. It is 9+4 and thus is supreme/complete stability applied to the laws of earth and applied to death without necessarily giving sufficient concern to other principles/values, hence again fitting all the themes mentioned prior. Finally, it is 10+3, representing the stabilisation of the order of man, and thus the attempted solidification of imperfection into an essence. Thus is associated with fire, as misguided transcendence may have steep consequences and as fire can be used for good for burning off imperfection by first drawing strong attention to it by illuminating it. Note that [[13/8]] is a rooted rational approximation of phi (the golden ratio) which is mathematically in some sense "the most irrational number" due to having a continued fraction consisting only of 1's, thus [[13/8]] takes on some of this mysterious quality while being perfectly just; a sort of ambiguous smoothness emerges in at least some of the low complexity intervals involving this prime essence, analogous to the beauty and smoothness of a smokeless flame.
+== Diatonic functions ==
+This section is largely inspired by [[User:Aura/Aura's_Ideas_on_Functional_Harmony|Aura's Ideas on Functional Harmony]] combined with my own observations, intuitions and simplifications, as I admittedly do not understand nearly as much as Aura on this subject, but I do intuitively understand my musical sense, which I wish to approximately translate here. This will also focus on generalisation from the [[5L 2s|diatonic]] MOSS to other MOSSes.
+We begin with a rooted generator, meaning an interval of the form a/2^n where 2^(n+1) > a > 2^n. This is taken as the most important interval, other than the period, which we will assume and define to be an octave as I believe this is the most natural period for a MOSS, especially in the case where the generator or its octave-complement is rooted. We also begin with a tonic, meaning a 'centre' of the scale from which the scale is constructed by stacking periods and generators up or down. The position of the tonic depends on the mode of the MOSS used. Note that we will start with the [[Pythagorean tuning]] of the [[5L 2s|diatonic]] MOSS and generalise from there.  Now we are ready to start defining the most fundamental functions:
+* 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 '''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.
+* The '''antitonic''', notated '''5''', is approximately half of an octave, being important due to both being a less strong consonance or a dissonance and due to being the mirror line when performing octave-complementing.
+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 '''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.
 == Intervals ==