User:Aura/Aura's Ideas of Consonance
Just about every musician these days is acquainted with the idea of "Consonance" and "Dissonance". However, as has been noted- there are difference concepts of "Consonance" and "Dissonance". This page is for exploring my particular concepts of "Consonance" and "Dissonance" as defined by something I call "connectivity".
Connectivity is a property that only exists between a given Tonic and its various harmonics and subharmonics- something most clearly indicated when a pure power of 2 is in either the numerator or the denominator of the fraction expressing the frequency ratio- and appears to be one of the factors controlling both the degree of timberal fusion between pitches as well as the appearance of a virtual fundamental the same pitch class as the Tonic, with both of these properties lending themselves well to expressions of tonality. While it can be generally said that the larger the numbers involved, the smaller the degree of connectivity, I should point out that even harmonic entropy minima where the fraction expressing the frequency ratio lacks a pure power of 2 would count as "disconnected". This would explain why a Minor triad built on the Tonic from the bass upwards doesn't have the same degree of timberal fusion, and why a virtual fundamental of the same pitch class as the Tonic fails appear in the bass whatsoever, and why the utonal equivalent of a virtual fundamental that *does* appear for a 5-limit minor triad as a result of all three notes sharing a single harmonic- a property that is characteristic of all utonal chords- also doesn't belong to the Tonic's pitch class.
Inferring from what I know so far, degrees of connectivity may perhaps be quantified by taking consecutive sections of the the minimum Rényi entropy curve corresponding to single octaves of the harmonic series, overlaying them on top of another, and then adding the values of the different sections together. If this type of transformation is performed repeatedly, overlaying more consecutive octave sections, I hypothesize that harmonic entropy minima with high connectivity should become more clearly emphasized as the different sections of the harmonic entropy curve repeatedly hit octave-equivalent harmonics- leading to the values for these areas adding up far less quickly than with other sections of the resulting curve, while other, less well-connected intervals will either be canceled out, or simply won't be emphasized as quickly. That said, I don't know what other effects of this will be.
Given the nature of connectivity as explained in the previous section, I shall have to rank the various types of harmonies based on their odd-limit as well as their prime-limit- assigning ranks as I go.
This family of harmonics and subharmonics- extending from the fundamental out to the 32nd harmonic and 32nd subharmonic- is the only one to be further broken down into various classes due to these harmonies having the most intimate relationships with the fundamental- which most naturally doubles as the Tonic.
Pitch Class Prime - This is the class to which the Tonic itself- as well as all pitches related to it by 2-limit harmonies- belong. This is the case in light of how pitches related to the Tonic by powers of two naturally seem to our hearing to be the same as the Tonic in ways that other primes don't.
Diatonic Prime - This is the class to which 3-limit and 5-limit harmonies belong on account of their key functions in just diatonic and just chromatic music. The first pair of 3-limit pitches give rise to the Dominant and Serviant harmonies, with the exact role of each 3-limit pitch being determined by your chosen tonality's direction of construction; the second pair of 3-limit pitches give rise to the Supertonic and Subtonic, with the exact functions of each of these also being determined by your chosen tonality's direction of construction; while the third pair of 3-limit pitches give rise to the second-best versions of the Mediant and Contramediant. The first pair of 5-limit pitches give rise to the best versions of the Mediant and Contramediant, while the first combinations of 3-limit and 5-limit give rise to both the Lead and the Reverse Lead- the exact role of each individual pitch is determined by your chosen tonality's direction of construction.
Paradiatonic Prime - This is the class to which 7-limit, 11-limit and 13-limit harmonies belong. The first pair of 7-limit pitches give rise to the Varicant and the Contravaricant, which, depending on circumstance and chord configuration, can serve either as alternate versions of the Supertonic and Subtonic, or, as alternate versions of the Mediant and Contramediant, while the first combinations of 3-limit and 7-limit give rise to the strongest versions of the Varicodominant and Varicoserviant- pitches which are adjacent to and serve as weaker alternatives to the Dominant and Serviant respectively. The first two 11-limit pitches give rise to the strongest versions of the Semiserviant and Semidominant- the two most important paradiatonic functions. The Semiserviant primarily acts as a secondary bridge between the Mediant and the Dominant, while the Semidominant primarily acts as a secondary bridge between Contramediant and the Serviant, though both have the additional function of enabling modulation to keys that are not in the same series of fifths. Finally, the first two 13-limit pitches give rise to weaker versions of the Mediant and Contramediant.
Quasidiatonic Prime - This is the class to which the 17-limit and 19-limit harmonies belong. What sets these apart from the pseudodiatonic harmonies is the potential of harmonies in this class to be easily mistaken for diatonic intervals- a property that gives them significant potential to act as comma pumps, and thus makes them highly valuable for initiating modulations. In other respects, however, they are usually outperformed by the neighboring 3-limit and 5-limit diatonic intervals they approximate.
Pseudodiatonic Prime - This is the class to which the 23-limit, 29-limit and 31-limit belong, and is the lowest rank among proximal harmonies. Unlike with the quasidiatonic primes, the pseudodiatonic primes are mainly only useful in the later stages of modulation as they are clearly not diatonic harmonies by any stretch. In other respects, while they may find a niche use as substitutes for disconnected intervals springing from the paradiatonic primes, they are otherwise of little value.
The span of the harmonic series between the 32nd and 64th harmonics sees the first true appearance of Antitonic harmony in the form of the 45th harmonic. Similarly, the span of the subharmonic series between the 32nd and 64th subharmonics sees the first true appearance of Antitonic harmonies in the form of the 45th subharmonic. Antitonic harmonies, as per their name, have a way of clashing with the Tonic harmony. Prior to these particular octaves, one cannot really speak of the appearance of notes and harmonies that oppose the tonality of the Tonic, hence why the medial harmonies begin at the 32nd harmonic and 32nd subharmonic. From these particular octaves of the harmonic series and the subharmonic series, the medial harmonies extend all the way out to the 1024th harmonic and the the 1024th subharmonic- all on account of various intervals that are closely associated with some of the Tonic's more intimate harmonies. It is in this region that one also begins to see really good approximations for 3-limit and 5-limit intervals that are otherwise disconnected from the Tonic, such as the 77th harmonic and the 77th subharmonic. However, being so far out from the Tonic means that the degree of connectivity with the Tonic is lessened compared to that of the proximal harmonies.
Out beyond the 1024th harmonic and the 1024th subharmonic lie the distal harmonies- these are seldom worth worrying about because they are so far out from the Tonic.
At least the Distal Harmonies maintain a remote connection to the Tonic, but intervals that belong in this class don't have such a privilege. Even though a number of these might act as harmonic entropy minima, such a status means nothing when it comes to connectivity, and thus, intervals in this class are particularly likely to gravitate towards notes other than the Tonic.
Connectivity Hierarchy Hypothesis
The above connectivity ranks seem to have a certain pecking order when it comes to priority. It also seems to be the case that weaker connectivity is out-prioritized by stronger connectivity, especially when one is dealing in terms of tonality, and that this is part of what makes modulation between different keys possible in the first place. The hierarchy would be constructed as follows:
Pitch Class Primes → Diatonic Primes → Paradiatonic Primes → Quasidiatonic Primes → Pseudodiatonic Primes → Medial Primes → Distal Primes → Disconnected Intervals
If such a hierarchy indeed exists, its existence would be consistent with the observation that when the 45th harmonic, a note with only a Medial Harmonic connection with the Tonic, is placed next to the 48th, the center of tonality seems to shift towards the 3rd harmonic and other members of its pitch class. It would also be consistent with my own observation that 8/5 Minor Third- the Contramediant in both Harmonic Major and Harmonic Minor- seems to have the effect of fostering a stronger sense of tonality than does the 5/3 Major Third- an observation which explains the comments of Andy Milne in regards to the Harmonic Major scale. Furthermore, the the disconnect between the conventional 5/3 Major Third and the Tonic seems to provide the grounds for the believability of George Russell's concept of Lydian as the center of tonal gravity insofar as one uses Ptolemy's Intense Diatonic scale as the standard for Ionian mode. Furthermore, it would explain why the 5-limit Major and Minor thirds are not entirely equivalent in terms of their consonance, and it would additionally imply that the properties of a Minor triad built on the Tonic from the bass-up lend themselves to a weakened sense of tonality in bass-up Minor keys, which in turn might explain the reasons for Major being preferred over Minor in the first place.
In light of the Connectivity Hypothesis and its implications, I propose we classify consonances according to not only their minimum Rényi entropy but also to their connectivity. For example, the 3/2 Perfect 5th is both a harmonic entropy minima and has high connectivity with the Tonic, resulting in the 3/2 Perfect 5th being classified as a "Perfect Consonance". As another example, although the conventional 5/3 Major Sixth may be both close to the Tonic on the harmonic lattice as well as a local harmonic entropy minimum, but because it's combination of disconnectedness with the Tonic and close connection with the Serviant seeming proving to be a liability for those who seek to establish a decent sense of tonality, this consonance is thus best classified as an "Imperfect Consonance". However, dissonance, on account of its crucial function as a propulsive force for harmonic motion, is not to be dismissed- rather, it too should be split into two classes based on how they relate to the Tonic in terms of both Harmonic Entropy and Connectivity. For example the pitch related to the Tonic by an interval of 11/8- an interval which I call a "Paramajor 4th"- displays a high degree of Harmonic entropy relative to the Tonic- although less so that the pitches immediately surrounding it- on the flipside, it demonstrates a high degree of connectivity to the Tonic, lending to this interval being classified as an "Imperfect Dissonance". On the flipside, the 17/12 Tritone not only exhibits high degree of Harmonic Entropy, but is also disconnected from the Tonic, leading to its classification as a "Perfect Dissonance".