User:Aura/Aura's Ideas of Consonance

Introduction

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".

Connectivity

To me, the exact nature of consonance is determined by the confluence of the minimum harmonic entropy and another factor which I refer to as "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", and the resulting consonance is thus best classified as an "Imperfect Consonance". This is part of why the 5-limit Major and Minor thirds are not entirely equivalent in terms of their consonance, and 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 doesn't appear in the bass. It should be noted that the utonal equivalent of a virtual fundamental *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. However, when such a minor triad is built on the tonic from the bass upwards, this virtual-fundamental-analog is not of the same pitch class as the Tonic- a fact which lends itself to a weakened sense of tonality in bass-up minor keys, and very likely of the reasons for Major being preferred over Minor in the first place. Furthermore, this same issue of disconnectedness plagues the otherwise pleasant-sounding 5/3 major sixth- as stated on my page on diatonic scales. It may be close to the Tonic on the harmonic lattice, but it's combination of disconnectedness with the Tonic and close connection with the Serviant proves to be a liability for those who seek to establish a decent sense of tonality in their chosen key, for reasons which I shall explain below.

Connectivity Ranks

In order to illustrate why disconnectedness is such a liability, I shall first have rank the various types of harmonies based on their odd-limit as well as their prime-limit- assigning ranks as I go.

Proximal Harmonies

This family of harmonics- 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, 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 Paradominant and Paraserviant- 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 Primes - 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.

Medial Harmonies

The first octave of the medial harmonies is characterized by the first true appearances of Antitonic harmonies as the 45th harmonic and 45th subharmonic- hence the division between proximal harmonies and the medial harmonies at the 32nd harmonic and subharmonic.