User:Aura/Aura's Ideas of Consonance: Difference between revisions

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== Connectivity ==

To me, the exact nature of consonance is determined by the confluence of [[Harmonic_Entropy#a.3D.E2.88.9E:_Harmonic_Min-Entropy|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". 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.

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 [[Harmonic_Entropy#a.3D.E2.88.9E:_Harmonic_Min-Entropy|the the minimum Rényi entropy curve]] corresponding to single octaves of the harmonic series, and overlaying them on top of another and averaging 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, 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.

== Connectivity Ranks ==

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== Application ==

In light of the Connectivity Hypothesis and its implications, I propose we classify consonances according to not only their ~~harmonic~~ entropy, but also 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".

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".

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 == Connectivity ==
-To me, the exact nature of consonance is determined by the confluence of [[Harmonic_Entropy#a.3D.E2.88.9E:_Harmonic_Min-Entropy|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".  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.
+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 [[Harmonic_Entropy#a.3D.E2.88.9E:_Harmonic_Min-Entropy|the the minimum Rényi entropy curve]] corresponding to single octaves of the harmonic series, and overlaying them on top of another and averaging 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, 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.
 == Connectivity Ranks ==
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 == Application ==
-In light of the Connectivity Hypothesis and its implications, I propose we classify consonances according to not only their harmonic entropy, but also 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".
+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".