User:Grady/Harmonic similarity: Difference between revisions

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This page serves to document a personal theory of mine that attempts to serve as a generalization of octave equivalence, as well as the various implications of the theory. I've seen other people express similar ideas, but I'm not sure if the concept in this exact form has been articulated before. If anyone knows if it has, I'd love to know more!

If you want to leave feedback about anything on this page, you can leave it on the Discussion page (see the tabs at the top) or [https://discordapp.com/users/241021202976473098 contact me on Discord]!

If you want to leave feedback about anything on this page, you can leave it on the Discussion page (see the tabs at the top) or [https://discordapp.com/users/241021202976473098 contact me on Discord]! Feel free to let me know whether you think these ideas hold up to scrutiny, or whether your own experience agrees or disagrees with what I've written here.

== Quick aside: Naming ==

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=== Margin for error ===

Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like [[12edo]] to be harmonically similar, even if the ratios are inexact.

== Lattice visualization ==

A very good way to visualize a map of harmonically similar pitches is using a [[3-limit]] [[lattice]], with octaves on one axis and tritaves on the other. In order to demonstrate that octaves are a significantly stronger harmonic similarity relation than tritaves, one might consider spacing the notes much farther apart on the tritave axis than the octave axis. (No diagram yet, but that might be something I can add to this page later!)

This essentially ignores the effects of any basic similarity relations beyond the third harmonic, which I believe is reasonable due to the extremely fast falloff mentioned earlier.

== String analogy ==

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In addition to providing an attempted explanation for the perception of octave equivalence, the theory of harmonic similarity provides explanations for various other noteworthy phenomena.

=== ~~Neutrality~~ of the major second ===

=== Colorlessness of the major second ===

The [[9/8|major second]] is often regarded as a dissonant interval, but it seems rather unique in that it produces auditory roughness, but doesn't necessarily feel emotionally jarring and unstable in the way something like one of the various flavors of [[tritone]] does. This is easily explained by the fact that two notes a major second apart are somewhat harmonically similar, due to being related by two factors of 3 and three factors of 2, which gives the major second a more emotionally ~~neutral~~ flavor than other dissonant intervals.

The [[9/8|major second]] is often regarded as a dissonant interval, but it seems rather unique in that it produces auditory roughness, but doesn't necessarily feel emotionally jarring and unstable in the way something like one of the various flavors of [[tritone]] does. This is easily explained by the fact that two notes a major second apart are somewhat harmonically similar, due to being related by two factors of 3 and three factors of 2, which gives the major second a more emotionally colorless flavor than other dissonant intervals.

=== Usage of the circle of fifths ===

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=== Interval quality ===

Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one pitch class would be fixed in place, while ~~another pitch class~~ would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.

Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one of the pitch classes comprising the interval class would be fixed in place, while the other would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.

As an important note, the concept of [[interval class]] is usually defined to refer to undirected interval class: for example, a major third and minor sixth would be considered the same interval class due to being octave complements. However, this theory requires establishing the concept of directed interval class, in which octave complements are separate classes. In order for this to work, we need to define intervals as being directed in general; in other words, that an upward major third and a downward major third be considered two separate intervals. The downward major third falls into the same directed interval class as the upward minor sixth, but the upward major third does not. If the upward major third can be thought of as the ratio 5/4, then the downward major third can be thought of as 4/5. In the absence of a specifier for upward or downward direction, it should be assumed that an interval points in the upward direction.

The theory of harmonic similarity provides credence to the idea that various intervals with the same interval quality specifier (e.g. major, minor, perfect, neutral, etc.) should often sound similar in emotional character, since those interval classes are usually near each other on an interval-based circle or chain of fifths. In other words, just as the notes C4 and G4 sound related to each other since C4 relates to G5 by a tritave, and G5 relates to G4 by an octave, a [[5/4|major third]] and [[15/8|major seventh]] should be expected to sound similar in character to each other since the major third relates to the [[15/4|major fourteenth]] by a tritave, and the major fourteenth relates to the major seventh by an octave.

Something to note about this is that the interval quality terms in wide use are arbitrary categorical boundaries. For instance, this theory posits that out of all the interval classes labeled as major, the major second is the one that's closest to a "perfect" interval quality, while out of all the ones labeled as minor, the minor seventh is the closest in that regard.

=== Brightness and darkness ===

The definition of brightness in wide use within the xenharmonic community is that a scale is brighter if its scale degrees are farther above the root. Most Western listeners will agree that brighter modes of the [[5L 2s|diatonic scale]] sound brighter and darker modes sound darker; however, in my experience and probably the experience of most other Western listeners as well, "brighter" modes (as per the xenharmonic definition) of the [[2L 3s|pentatonic scale]] actually sound darker, and "darker" modes sound brighter. This seems to indicate that the perception of brightness isn't actually determined by distances of scale degrees above the root. The theory of harmonic similarity provides an explanation for this, asserting that the reason brighter diatonic modes sound bright is because they contain more scale degrees of major quality, while darker modes sound dark because they contain more scale degrees of minor quality, and that there exists an actual reason that various scale degrees of the same interval quality should sound similar in character. This logic is of course inverted with the pentatonic scale, where brighter modes contain more scale degrees of minor quality and darker modes contain more of major quality.

In fact, this correlation in scale degree quality between modes of similar brightness levels is effectively only due to the special case where the period of the [[MOS scale]] in question is an octave and the generator is a tritave, as is the case for the diatonic and pentatonic scales. If the generator is any other interval, the new "bright" scale degrees introduced as you ascend the modal brightness order have little harmonic similarity to each other, and the old "dark" scale degrees that get removed also have little harmonic similarity to each other, and furthermore, the scale degrees within the scale at any given time have little harmonic similarity to each other, which means that traversing the modal brightness order doesn't result in a cohesive shift in emotional character in the same way. This certainly aligns with my experiences comparing different modes of xenharmonic MOS scales, where the brightest and darkest modes don't really feel very distinct in character from each other compared to, say, Lydian and Locrian, but it's unclear how much of this is due to the implications of the harmonic similarity theory versus simply cultural conditioning.

=== Attitudes toward microtonal music ===

Microtonal music is often polarizing for the average Western listener, with many finding microtonal intervals jarring or strange to listen to. Although this is usually dismissed purely as cultural conditioning, there could also be another aspect at play here. I believe harmonic similarity could be the key behind why so many Western listeners have such an immediate reaction to hearing a microtonal interval.

Because 12edo has a perfectly tuned octave and near-perfectly tuned tritave, if any given interval is well approximated by the tuning system, then all the intervals that are harmonically similar to that interval will also be well approximated. Conversely, if an interval falls well outside of the tuning system, for example the [[7/4|subminor seventh]], then all the intervals that are harmonically similar to that interval, such as the [[7/6|subminor third]], will fall well outside of the tuning system as well. Because of this, not only do Western listeners not become acclimated to the subminor seventh itself, but they also don't become acclimated to other intervals that are similar in character, like the subminor third. If Western listeners regularly heard the subminor third but not the subminor seventh (which would be the case if [[9edo]] was the dominant tuning system instead, for example), then they might not find the subminor seventh as jarring, since they'd already be very familiar with a similar interval.

12edo even has a reasonable tuning of the [[5/1|fifth harmonic]], so it's arguably possible that this further contributes to the cohesion in harmonic similarity between the notes and intervals within the tuning system. However, it's debatable whether two frequencies separated by a factor of five have any non-negligible amount of harmonic similarity to each other, or whether 12edo's tuning of the fifth harmonic is accurate enough for this to take effect.

=== The tritave versus the perfect fifth ===

Perhaps a more contentious implication of this theory is that it implies the tritave is a slightly stronger similarity relation than the perfect fifth. In Western music theory, they'd be thought of as equal in similarity, due to the abstraction of octave equivalence. However, this implication seems to generally align with the experiences of xenharmonic musicians, many of whom claim the tritave works better as an [[Interval of equivalence|equave]] than the perfect fifth.

@@ Line 1: / Line 1: @@
 This page serves to document a personal theory of mine that attempts to serve as a generalization of octave equivalence, as well as the various implications of the theory. I've seen other people express similar ideas, but I'm not sure if the concept in this exact form has been articulated before. If anyone knows if it has, I'd love to know more!
-If you want to leave feedback about anything on this page, you can leave it on the Discussion page (see the tabs at the top) or [https://discordapp.com/users/241021202976473098 contact me on Discord]!
+If you want to leave feedback about anything on this page, you can leave it on the Discussion page (see the tabs at the top) or [https://discordapp.com/users/241021202976473098 contact me on Discord]! Feel free to let me know whether you think these ideas hold up to scrutiny, or whether your own experience agrees or disagrees with what I've written here.
 == Quick aside: Naming ==
@@ Line 40: / Line 40: @@
 === Margin for error ===
 Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like [[12edo]] to be harmonically similar, even if the ratios are inexact.
+== Lattice visualization ==
+A very good way to visualize a map of harmonically similar pitches is using a [[3-limit]] [[lattice]], with octaves on one axis and tritaves on the other. In order to demonstrate that octaves are a significantly stronger harmonic similarity relation than tritaves, one might consider spacing the notes much farther apart on the tritave axis than the octave axis. (No diagram yet, but that might be something I can add to this page later!)
+This essentially ignores the effects of any basic similarity relations beyond the third harmonic, which I believe is reasonable due to the extremely fast falloff mentioned earlier.
 == String analogy ==
@@ Line 67: / Line 72: @@
 In addition to providing an attempted explanation for the perception of octave equivalence, the theory of harmonic similarity provides explanations for various other noteworthy phenomena.
-=== Neutrality of the major second ===
+=== Colorlessness of the major second ===
-The [[9/8|major second]] is often regarded as a dissonant interval, but it seems rather unique in that it produces auditory roughness, but doesn't necessarily feel emotionally jarring and unstable in the way something like one of the various flavors of [[tritone]] does. This is easily explained by the fact that two notes a major second apart are somewhat harmonically similar, due to being related by two factors of 3 and three factors of 2, which gives the major second a more emotionally neutral flavor than other dissonant intervals.
+The [[9/8|major second]] is often regarded as a dissonant interval, but it seems rather unique in that it produces auditory roughness, but doesn't necessarily feel emotionally jarring and unstable in the way something like one of the various flavors of [[tritone]] does. This is easily explained by the fact that two notes a major second apart are somewhat harmonically similar, due to being related by two factors of 3 and three factors of 2, which gives the major second a more emotionally colorless flavor than other dissonant intervals.
 === Usage of the circle of fifths ===
@@ Line 74: / Line 79: @@
 === Interval quality ===
-Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one pitch class would be fixed in place, while another pitch class would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.
+Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one of the pitch classes comprising the interval class would be fixed in place, while the other would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.
 As an important note, the concept of [[interval class]] is usually defined to refer to undirected interval class: for example, a major third and minor sixth would be considered the same interval class due to being octave complements. However, this theory requires establishing the concept of directed interval class, in which octave complements are separate classes. In order for this to work, we need to define intervals as being directed in general; in other words, that an upward major third and a downward major third be considered two separate intervals. The downward major third falls into the same directed interval class as the upward minor sixth, but the upward major third does not. If the upward major third can be thought of as the ratio 5/4, then the downward major third can be thought of as 4/5. In the absence of a specifier for upward or downward direction, it should be assumed that an interval points in the upward direction.
+The theory of harmonic similarity provides credence to the idea that various intervals with the same interval quality specifier (e.g. major, minor, perfect, neutral, etc.) should often sound similar in emotional character, since those interval classes are usually near each other on an interval-based circle or chain of fifths. In other words, just as the notes C4 and G4 sound related to each other since C4 relates to G5 by a tritave, and G5 relates to G4 by an octave, a [[5/4|major third]] and [[15/8|major seventh]] should be expected to sound similar in character to each other since the major third relates to the [[15/4|major fourteenth]] by a tritave, and the major fourteenth relates to the major seventh by an octave.
+Something to note about this is that the interval quality terms in wide use are arbitrary categorical boundaries. For instance, this theory posits that out of all the interval classes labeled as major, the major second is the one that's closest to a "perfect" interval quality, while out of all the ones labeled as minor, the minor seventh is the closest in that regard.
+=== Brightness and darkness ===
+The definition of brightness in wide use within the xenharmonic community is that a scale is brighter if its scale degrees are farther above the root. Most Western listeners will agree that brighter modes of the [[5L 2s|diatonic scale]] sound brighter and darker modes sound darker; however, in my experience and probably the experience of most other Western listeners as well, "brighter" modes (as per the xenharmonic definition) of the [[2L 3s|pentatonic scale]] actually sound darker, and "darker" modes sound brighter. This seems to indicate that the perception of brightness isn't actually determined by distances of scale degrees above the root. The theory of harmonic similarity provides an explanation for this, asserting that the reason brighter diatonic modes sound bright is because they contain more scale degrees of major quality, while darker modes sound dark because they contain more scale degrees of minor quality, and that there exists an actual reason that various scale degrees of the same interval quality should sound similar in character. This logic is of course inverted with the pentatonic scale, where brighter modes contain more scale degrees of minor quality and darker modes contain more of major quality.
+In fact, this correlation in scale degree quality between modes of similar brightness levels is effectively only due to the special case where the period of the [[MOS scale]] in question is an octave and the generator is a tritave, as is the case for the diatonic and pentatonic scales. If the generator is any other interval, the new "bright" scale degrees introduced as you ascend the modal brightness order have little harmonic similarity to each other, and the old "dark" scale degrees that get removed also have little harmonic similarity to each other, and furthermore, the scale degrees within the scale at any given time have little harmonic similarity to each other, which means that traversing the modal brightness order doesn't result in a cohesive shift in emotional character in the same way. This certainly aligns with my experiences comparing different modes of xenharmonic MOS scales, where the brightest and darkest modes don't really feel very distinct in character from each other compared to, say, Lydian and Locrian, but it's unclear how much of this is due to the implications of the harmonic similarity theory versus simply cultural conditioning.
+=== Attitudes toward microtonal music ===
+Microtonal music is often polarizing for the average Western listener, with many finding microtonal intervals jarring or strange to listen to. Although this is usually dismissed purely as cultural conditioning, there could also be another aspect at play here. I believe harmonic similarity could be the key behind why so many Western listeners have such an immediate reaction to hearing a microtonal interval.
+Because 12edo has a perfectly tuned octave and near-perfectly tuned tritave, if any given interval is well approximated by the tuning system, then all the intervals that are harmonically similar to that interval will also be well approximated. Conversely, if an interval falls well outside of the tuning system, for example the [[7/4|subminor seventh]], then all the intervals that are harmonically similar to that interval, such as the [[7/6|subminor third]], will fall well outside of the tuning system as well. Because of this, not only do Western listeners not become acclimated to the subminor seventh itself, but they also don't become acclimated to other intervals that are similar in character, like the subminor third. If Western listeners regularly heard the subminor third but not the subminor seventh (which would be the case if [[9edo]] was the dominant tuning system instead, for example), then they might not find the subminor seventh as jarring, since they'd already be very familiar with a similar interval.
+edo even has a reasonable tuning of the [[5/1|fifth harmonic]], so it's arguably possible that this further contributes to the cohesion in harmonic similarity between the notes and intervals within the tuning system. However, it's debatable whether two frequencies separated by a factor of five have any non-negligible amount of harmonic similarity to each other, or whether 12edo's tuning of the fifth harmonic is accurate enough for this to take effect.
+=== The tritave versus the perfect fifth ===
+Perhaps a more contentious implication of this theory is that it implies the tritave is a slightly stronger similarity relation than the perfect fifth. In Western music theory, they'd be thought of as equal in similarity, due to the abstraction of octave equivalence. However, this implication seems to generally align with the experiences of xenharmonic musicians, many of whom claim the tritave works better as an [[Interval of equivalence|equave]] than the perfect fifth.