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Found a better follow-up to a sycophant chord that stays in key.
 
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'''Antipodism''' - Notes that are either opposite in pitch hue or nearly so due to being approximately half an octave away from the starting point are harmonically opposed to the starting point.  Non-tonic notes related through this process tend to have the opposite function relative to the Tonic.  For the notes related to the Tonic by this process, see Antitonic below.
'''Antipodism''' - Notes that are either opposite in pitch hue or nearly so due to being approximately half an octave away from the starting point are harmonically opposed to the starting point.  Non-tonic notes related through this process tend to have the opposite function relative to the Tonic.  For the notes related to the Tonic by this process, see Antitonic below.


'''Preparation''' - Notes that "prepare the way" for either a Dominant or a Serviant through any of the above operations, or through some other mechanism, relate to said notes by this process.  Functions which have this kind of role relative are denoted with a "pre-" prefix here.
'''Preparation''' - Notes that "prepare the way" for either a Dominant or a Serviant through any of the above operations, or through some other mechanism, relate to said notes by this process.  Functions which have this kind of role relative to some other function are denoted with a "pre-" prefix here.


'''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Serviant, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions.  Often, though not always, these notes fall within the uncanny valleys of the three primary functions.
'''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Serviant, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions.  Often, though not always, these notes fall within the uncanny valleys of the three primary functions.
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The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth.
The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth.


'''Sycophant''' - Named as such on account of it having a tendency to "kiss up to" and tonicize the Dominant- that is, to cause the Dominant to become a new Tonic- unless followed up by a different note such as a Subcollocant or Supercollocant, this type of Antitonic is mapped to both 3\7 and 12\24.  A prototypical example of this type of Antitonic is 45/32.
'''Sycophant''' - This type of Antitonic is mapped to both 3\7 and 12\24, and is named as such on account of it having a tendency to "kiss up to" and tonicize the Dominant- that is, to cause the Dominant to become a new Tonic- unless followed up by a different note such as some type of Mediant.  A prototypical example of this type of Antitonic is 45/32.


'''Tyrant''' - If the Antitonic is mapped to both 4\7 and 12\24, it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature, and indeed these brute force Dominant-esque tendencies are the source the name "Tyrant".  For example, if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature.  Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards some other type of substitute for a Dominant, often bypassing this type of Antitonic, though on rare occasions, a Tyrant will act as a leading tone to the Serviant.  A prototypical example of this type of Antitonic is 64/45.
'''Tyrant''' - This type of Antitonic is mapped to both 4\7 and 12\24, and it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature, and indeed these brute force Dominant-esque tendencies are the source of the name "Tyrant".  For example, if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature.  Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards some other type of substitute for a Dominant, often bypassing this type of Antitonic, though on rare occasions, a Tyrant will act as a leading tone to the Serviant.  A prototypical example of this type of Antitonic is 64/45.


== Chromatic Functions ==
== Chromatic Functions ==
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However, in non-meantone settings, it is possible to have a chord which shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor.  Since such an arrangement results in a wolf fifth, and since the wolf fifth is a dissonance requiring resolution, one could argue that such a chord has a function distinct from the more traditional options- specifically one which is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj.  This particular function is what I call the ''Major Contramediant Tensive''.
However, in non-meantone settings, it is possible to have a chord which shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor.  Since such an arrangement results in a wolf fifth, and since the wolf fifth is a dissonance requiring resolution, one could argue that such a chord has a function distinct from the more traditional options- specifically one which is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj.  This particular function is what I call the ''Major Contramediant Tensive''.


'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Serviant along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, however, chord roots that are located three or more steps away from the Tonic along the circle of fifths are not subject to the aforementioned restriction due to their distance.  This means that the Tonic, Dominant, Serviant, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards, but wolf fifths can occur easily as the fifth of chords which are located three or more steps away from the Tonic.
'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Serviant along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, though in the case of a chord that has a root located at two steps away from the Tonic along the along the Circle of Fifths, the fifth of the chord can be deliberately subjected to the process of Displacement as mentioned before and not run afoul of this restriction.  However, chord roots that are located three or more steps away from the Tonic along the circle of fifths are not subject to the aforementioned restriction due to their distance.  This means that the Tonic, Dominant, Serviant, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards, but wolf fifths can occur easily as the fifth of chords which are located three or more steps away from the Tonic.
 
=== Implications of the goverening rules ===
 
One of the obvious applications of the two governing rules listed above is that chords built with wolf fifths must have roots located three or more steps away from the Tonic along the Circle of Fifths, and that when two notes within a given Diatonic system are separated by a wolf fifth, they both must likewise be located two or more steps away from the Tonic along the Circle of Fifths.  This in turn gives rise to a series of restrictions on chord progressions.
 
'''Whole Tone Progression''' - This rule has several different facets.  In Bass-Up tonality, a given chord with a third located at 5/4 above the root cannot be followed up by a second chord with a third located at 6/5 above the root if the second chord's root is at 9/8 above the first chord's root.  Conversely, in Treble-Down tonality, a given chord with a third located at 5/4 below the root cannot be followed up by a second chord with a third located at 6/5 below the root if the second chord's root is at 9/8 below the first chord's root.
 
 


== Beyond Diatonic and Chromatic Functional Harmony ==
== Beyond Diatonic and Chromatic Functional Harmony ==
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'''Paraserviant''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic.  Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function.  Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit.  When considered alongside the actual Serviant function of 4/3, however, Paraserviants take on the function of Preserviant since the Serviant function of 4/3 is stronger.
'''Paraserviant''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic.  Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function.  Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit.  When considered alongside the actual Serviant function of 4/3, however, Paraserviants take on the function of Preserviant since the Serviant function of 4/3 is stronger.
== Differences from Traditional Neo-Riemannian Theory ==
<todo>
The lack of [[Meantone]] temperament has some pretty significant implications for triadic transformations and oblique voice-leading in my theory- namely in that while all three basic [[Wikipedia: Neo-Riemannian theory|Neo-Riemannian]] transformations are available, there must be additional specifiers to distinguish the original triadic transformations from new types of triadic transformations, and, as a result of my own theories on functional harmony, certain triadic transformations need to be renamed.  One must also keep in mind that the direction of transformation indicated doesn't necessarily result in resolution, as a number of transformations indicated can be executed one way for an increase in tension and in the other direction for a decrease in tension.
'''Pythagorean Parallel''' - This transformation involves exchanging a Pythagorean triad for its Pythagorean parallel.  For a Pythagorean Major triad, move the third down by an apotome, and for a Pythagorean Minor triad move the third up by an apotome.
'''Ptolemaic Parallel''' - This transformation involves exchanging a typical Ptolemaic triad for its Ptolemaic parallel.  For a Ptolemaic Major triad, move the third down by a Diptolemaic Chromatic Semitone, and for a Ptolemaic Minor triad move the third up by a Diptolemaic Chromatic Semitone.
'''Pythagoran-Ptolemaic Picardy''' - This transformation involves exchanging a Pythagorean triad for its Ptolemaic parallel.  For a Pythagorean Major triad move the third down by a Ptolemaic Chromatic Semitone, and for a Pythagorean Minor triad move the third up by a Ptolemaic Chromatic Semitone.
'''Pythagorean Relative''' - This transformation involves exchanging a Pythagorean triad for its relative of the same kind.  For a Pythagorean Major triad, move the fifth up by a Pythagorean Whole Tone, and for a Pythagorean Minor triad, move the root down by a Pythagorean Whole Tone.
'''Ptolemaic Relative''' - This transformation involves exchanging a Ptolemaic triad for its relative of the same kind.  For a Ptolemaic Major triad, move the fifth up by a Ptolemaic Whole Tone, and for a Ptolemaic Minor triad, move the root down by a Ptolemaic Whole Tone.
'''Pythagorean Limma Exchange''' - This transformation involves exchanging a Pythagorean triad for a mostly similar triad with one note differing in pitch class by a Pythagorean Diatonic Semitone.  For a Pythagorean Major triad, move the root down by a Pythagorean Diatonic Semitone, and for and Pythagorean Minor triad, move the fifth up by a Pythagorean Diatonic Semitone.
'''Ptolemaic Limma Exchange''' - This transformation involves exchanging a Ptolemaic triad for a mostly similar triad with one note differing in pitch class by a Ptolemaic Diatonic Semitone.  For a Ptolemaic Major triad, move the root down by a Ptolemaic Diatonic Semitone, and for and Ptolemaic Minor triad, move the fifth up by a Ptolemaic Diatonic Semitone.


== References ==
== References ==
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