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| '''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, 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. |
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| === Implications of the goverening rules ===
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| 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.
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| '''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.
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| == 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. |
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| == Differences from Traditional Neo-Riemannian Theory ==
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| <todo>
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| 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.
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| '''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.
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| '''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.
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| '''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.
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| '''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.
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| '''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.
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| '''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.
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| '''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.
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| == References == | | == References == |