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== Mediant Subtypes ==
== Mediant Subtypes ==
<todo>


It is worth noting
There are five basic types of Mediant, although only three of them are distinct in [[24edo]]- you'd need to go to [[41edo]] or [[53edo]] to see all five.


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, though it should be noted that 81/64 also has the function of "'''Tetradominant'''" while 32/27 also has the function of "'''Triserviant'''".  Furthermore, since there's such a large range of Mediants, there are types of Mediant that don't have diatonic function.
Proximomediant - This type of Mediant is mapped to 2\7 and 6\24 and is so named due to being located on the end of the Mediant range closest to the Tonic.  Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Mesoproximomediant in finer tuning systems where it is usually more dissonant.  One type of Proximomediant is the "'''Triserviant'''", which is the function specifically of [[32/27]].
 
Mesoproximomediant - This type of Mediant is mapped to both 2\7 and 6\24 and is so named due to being located between the proximal end of the Mediant range and the midline between the Tonic and the Dominant.  Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Proximomediant in finer tuning systems where it is usually more consonant.  A primary example of an interval with this type of function is the interval [[6/5]].
 
Mesomediant - This type of Mediant is mapped to both 2\7 and 7\24 and is so named due to being located along the midline between the Tonic and the Dominant.  This type of Mediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases.  Intervals with this type of function include [[11/9]] and [[27/22]].
 
Mesodistomediant - This type of Mediant is mapped to both 2\7 and 8\24 and is so named due to being located between the distal end of the Mediant range and the midline between the Tonic and the Dominant.  Conspicuously, it is one of the possible the types of Mediant seen in Major keys and is only distinct from the Distomediant in finer tuning systems where it is usually more consonant.  A primary example of an interval with this type of function is the interval [[5/4]].
 
Distomediant - This type of Mediant is mapped to both 2\7 and 8\24 and is so named due to being located on the end of the Mediant range furthest from the Tonic.  Conspicuously, it is one of the possible the types of Mediant seen in Major keys and is only distinct from the Mesodistomediant in finer tuning systems where it is usually more dissonant.  One type of Distomediant is the "'''Quadridominant'''", which is the function specifically of [[81/64]].


== Contramediant Subtypes ==
== Contramediant Subtypes ==
<todo>


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, though it should be noted that 27/16 also has the function of "'''Tridominant'''" while 128/81 also has the function of "'''Tetraserviant'''".
There are five basic types of Contraediant, although only three of them are distinct in [[24edo]]- you'd need to go to [[41edo]] or [[53edo]] to see all five.
 
Proximocontramediant - This type of Contramediant is mapped to 5\7 and 18\24 and is so named due to being located on the end of the Contramediant range closest to the Tonic.  Conspicuously, it is one of the possible the types of Contraediant seen in Major keys and is only distinct from the Mesoproximocontramediant in finer tuning systems where it is usually more dissonant.  One type of Proximocontramediant is the "'''Tridominant'''", which is the function specifically of [[27/16]].
 
Mesoproximocontramediant - This type of Contramediant is mapped to both 5\7 and 18\24 and is so named due to being located between the proximal end of the Mediant range and the midline between the Tonic and the Serviant.  Conspicuously, it is one of the possible the types of Contramediant seen in Major keys and is only distinct from the Proximocontramediant in finer tuning systems where it is usually more consonant.  A primary example of an interval with this type of function is the interval [[5/3]].
 
Mesocontramediant - This type of Contramediant is mapped to both 5\7 and 17\24 and is so named due to being located along the midline between the Tonic and the Serviant.  This type of Contramediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases.  Intervals with this type of function include [[18/11]] and [[44/27]].
 
Mesodistocontramediant - This type of Contramediant is mapped to both 5\7 and 16\24 and is so named due to being located between the distal end of the Contramediant range and the midline between the Tonic and the Serviant.  Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Distocontramediant in finer tuning systems where it is usually more consonant.  A primary example of an interval with this type of function is the interval [[8/5]].
 
Distocontramediant - This type of Contramediant is mapped to both 5\7 and 16\24 and is so named due to being located on the end of the Contramediant range furthest from the Tonic.  Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesodistocontramediant in finer tuning systems where it is usually more dissonant.  One type of Distomediant is the "'''Quadriserviant'''", which is the function specifically of [[128/81]].  


== Antitonic Subtypes ==
== Antitonic Subtypes ==
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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.  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.  One of the obvious applications of this 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 three or more steps away from the Tonic along the Circle of Fifths.
'''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.  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.  One of the obvious applications of this 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 three or more steps away from the Tonic along the Circle of Fifths.


== Going Beyond the 5-limit ==
== Paradiantonic and Parachromatic Functional Harmony ==


In addition to all the aforementioned Diatonic and Chromatic functions, as well as the detemperings of diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.
In addition to all the aforementioned Diatonic and Chromatic functions, as well as the detemperings of diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.
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== Functional Harmony in the 7-limit and 11-limit ==
== Functional Harmony in the 7-limit and 11-limit ==


In January of 2022, Quartertone Harmony posted a video in which he grouped together a series of functions he refered to in the video as the "shadow scale"<ref>[https://www.youtube.com/watch?v=P6WJryxB_0Y Quartertone Harmony - Harmonic Functions of Quartertones SD 480p]</ref>, which I will refer to here as a '''paradiatonic scale''' since there are technically two of these, and this in turn led to the separation of Paradiatonic and Parachromatic harmonic functions for me.  This whole concept of a "shadow scale", in addition to everything else discuss in this section, paves the way for the my idea of [[MOS-Shadow theory]], but, aside from how it applies to Diatonic-scale based functional harmony, MOS-Shadow theory is another whole discussion for another time.
In January of 2022, Quartertone Harmony posted a video in which he grouped together a series of functions he refered to in the video as the "shadow scale"<ref>[https://www.youtube.com/watch?v=P6WJryxB_0Y Quartertone Harmony - Harmonic Functions of Quartertones SD 480p]</ref>, which I will refer to here as a '''paradiatonic scale''' since there are technically two of these, and this in turn led to the separation of Paradiatonic and Parachromatic harmonic functions for me.  This whole concept of a "shadow scale", in addition to everything else discuss in this section, paves the way for my idea of [[MOS-Shadow theory]], but, aside from how it applies to Diatonic-scale based functional harmony, MOS-Shadow theory is another whole discussion for another time.


=== The Paradiatonic Scales ===
=== The Paradiatonic Scales ===
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In addition to the six known Diatonic Function-Deriving Operations listed above, there's also one Paradiatonic Function-Deriving Operation known as '''Neutralization''', which, as the name suggests, creates paradiatonic functions from the neutralization and hybridization of Major and Minor Diatonic scale degrees.
In addition to the six known Diatonic Function-Deriving Operations listed above, there's also one Paradiatonic Function-Deriving Operation known as '''Neutralization''', which, as the name suggests, creates paradiatonic functions from the neutralization and hybridization of Major and Minor Diatonic scale degrees.
'''Middle Mediant''' - This is nothing other than a neutralized Mediant, and thus, it has the Tonic Parallel, Serviant Adjacent, Preserviant and Predominant functions that you expect from a Mediant, only, it doesn't serve well at phrase endings, rather, its Tonic Parallel function is only appropriate during the middle of musical phrases.
'''Middle Contramediant''' - This is nothing other than a neutralized Contramediant, and thus, it has the Tonic Parallel, Dominant Adjacent, Preserviant and Predominant functions that you expect from a Contramediant, but, like with the Middle Mediant, it doesn't serve well at phrase endings, rather, its Tonic Parallel function is only appropriate during the middle of musical phrases.


'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Lead- however, there are functional differences between a Subgradient and a Lead in Bass-Up Tonality.  In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Leads, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]].  Furthermore, whereas a Lead can resolve to the Tonic in part through a strong harmonic connection, a Subgradient is much more likely to do so through sheer brute force, and even these cases require a proper set-up, as otherwise, the awkward tonal disconnect between the Subdietic and the Tonic is likely to result in the Subgradient resolving back down to either the Lead or the Semilead.  As if that weren't enough, the Subgradient also has the Antintersubiant function.  This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).
'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Lead- however, there are functional differences between a Subgradient and a Lead in Bass-Up Tonality.  In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Leads, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]].  Furthermore, whereas a Lead can resolve to the Tonic in part through a strong harmonic connection, a Subgradient is much more likely to do so through sheer brute force, and even these cases require a proper set-up, as otherwise, the awkward tonal disconnect between the Subdietic and the Tonic is likely to result in the Subgradient resolving back down to either the Lead or the Semilead.  As if that weren't enough, the Subgradient also has the Antintersubiant function.  This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).
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