User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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The three basic functions have their roots in [[LCJI]], and each has an uncanny valley of sorts around it. The functions themselves are labeled as follows: | The three basic functions have their roots in [[LCJI]], and each has an uncanny valley of sorts around it. The functions themselves are labeled as follows: | ||
'''Tonic''' - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the Tonic). This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as ''Nontonic'', thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 [[cent]]s away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through [[fudging]], or simply [[ | '''Tonic''' - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the Tonic). This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as ''Nontonic'', thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 [[cent]]s away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through [[fudging]], or simply [[Tempering out|tempered out]]. Furthermore, the uncanny valley around the Tonic, unlike those around the other two functions, is relatively large, strong, and, most importantly, unconditional, and as a result, wolf intervals around the Tonic only being tolerable in melody and ornamentation but not harmony. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible. | ||
'''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, and, to use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Secondly, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in the direction of tonality construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "'''Paradominants'''". On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that no Dominant or Paradominant sees the Tonic occurring in the proximal structure of its chord- that is, as a third or fifth. | '''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, and, to use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Secondly, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in the direction of tonality construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "'''Paradominants'''". On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that no Dominant or Paradominant sees the Tonic occurring in the proximal structure of its chord- that is, as a third or fifth. | ||
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'''Supertonic''' - This function is easily derivable through the Tonic Adjacent function, the Serviant Parallel function, and the Didominant function of [[9/8]], or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 1\7 and 4\24 and occurs above the Tonic as the second scale degree. | '''Supertonic''' - This function is easily derivable through the Tonic Adjacent function, the Serviant Parallel function, and the Didominant function of [[9/8]], or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 1\7 and 4\24 and occurs above the Tonic as the second scale degree. | ||
'''Mediant''' - This is the note that maps to 2\7 from the Tonic in the scale's direction of construction, and is named due to being roughly halfway between the Tonic and the Dominant. This is the first of the two diatonic scale degrees | '''Mediant''' - This is the note that maps to 2\7 from the Tonic in the scale's direction of construction, and is named due to being roughly halfway between the Tonic and the Dominant. This is the first of the two diatonic scale degrees that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, not only has the most possibilities for realization and the ability to easily host chords built with wolf fifths, but also a tendency towards stagnation, leading to the historical designation as a "weak harmony". Aside from these, the properties that are central to the Mediant function are all most easily derived through the Tonic Parallel function and the Serviant Adjacent function, and in addition, have both Preserviant and Predominant functions. However, there are other functional aspects of a Mediant that are determined by whether the interval in question is considered consonant or dissonant, with [[5/4]] and [[6/5]] being examples of consonant Mediants, and [[81/64]] and [[32/27]] being examples of dissonant Mediants. | ||
'''Antitonic''' - This is a special case, see the next section for more discussion of this function. | '''Antitonic''' - This is a special case, see the next section for more discussion of this function. | ||
'''Contramediant''' - Compared to the term "''Submediant''" from traditional music theory, the term "Contramediant" may have a slightly different frame of reference, as while a "Submediant" is halfway between the Tonic and a "Subdominant", the "Contramediant" is halfway between the Tonic and the Serviant. The Contramediant is the note that maps to 5\7 from the Tonic in the scale's direction of construction, and is the second of two | '''Contramediant''' - Compared to the term "''Submediant''" from traditional music theory, the term "Contramediant" may have a slightly different frame of reference, as while a "Submediant" is halfway between the Tonic and a "Subdominant", the "Contramediant" is halfway between the Tonic and the Serviant. The Contramediant is the note that maps to 5\7 from the Tonic in the scale's direction of construction, and is the second of two that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, is not only tied with the Mediant for having the most possibilities for realization, but also the ability to easily host chords built with wolf fifths, and lastly, a tendency towards stagnation, leading to the historical designation as a "weak harmony". From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through the Tonic Parallel function and the Dominant Adjacent function, and in addition, have both Preserviant and Predominant functions. However, there are other functional aspects of a Contramediant that are determined by whether the interval in question is considered consonant or dissonant, with [[5/3]] and [[8/5]] being examples of consonant Contramediants, and [[27/16]] and [[128/81]] being examples of dissonant Contramediants. | ||
'''Subtonic''' - This function is easily derivable through the Tonic Adjacent function, the Dominant Parallel function, and the Diserviant function of [[16/9]], with these intervals additionally having Predominant functions, or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 6\7 and 20\24 and occurs above the Tonic as the seventh scale degree. | '''Subtonic''' - This function is easily derivable through the Tonic Adjacent function, the Dominant Parallel function, and the Diserviant function of [[16/9]], with these intervals additionally having Predominant functions, or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 6\7 and 20\24 and occurs above the Tonic as the seventh scale degree. | ||
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=== Circumdominant Regions === | === Circumdominant Regions === | ||
The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny | The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny valleys around the Dominant are not present unconditionally, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | ||
'''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about 30 cents to about 15 cents short of the 3/2 perfect fifth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like [[40/27]], Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preserviant function. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the traditional weak harmonies- the Mediant and the Contramediant- which also has the benefits of strengthening interrupted cadences and creating the sense of impending movement, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweet spot range near the standard issue 3/2. | '''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about 30 cents to about 15 cents short of the 3/2 perfect fifth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like [[40/27]], Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preserviant function. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the traditional weak harmonies- the Mediant and the Contramediant- which also has the benefits of strengthening interrupted cadences and creating the sense of impending movement, but outside of these usages and well supported chords, this kind of thing is best avoided since things repel the similar but not identical. Apart from diatonic contexts, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweet spot range near the standard issue 3/2. | ||
=== Circumserviant Regions === | === Circumserviant Regions === | ||
The circumserviant regions are the two main regions on either side of the Serviant proper, and since the uncanny | The circumserviant regions are the two main regions on either side of the Serviant proper, and since the uncanny valleys around the Dominant are not present unconditionally, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | ||
'''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from about 15 cents to about 30 cents beyond the 4/3 perfect fourth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Serviant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Serviant, hence its name. As typified by intervals like [[27/20]], Geminoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Supertonic, Subtonic, the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweet spot range near the standard issue 4/3. | '''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from about 15 cents to about 30 cents beyond the 4/3 perfect fourth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Serviant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Serviant, hence its name. As typified by intervals like [[27/20]], Geminoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Supertonic, Subtonic, the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweet spot range near the standard issue 4/3. | ||