Saddle chord: Difference between revisions
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A '''saddle chord''' is a chord that represents a ''saddle point'' in the [[ | A '''saddle chord''' is a chord that represents a ''saddle point'' in the [[harmonic entropy]] surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher. | ||
Chords <span style="line-height: 1.5;">at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.</span> | Chords <span style="line-height: 1.5;">at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.</span> | ||
In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many [[ | In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many [[24edo|quarter tone]]-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} with each note a quarter tone apart. | ||
For triads and higher, though, there are other possibilities, corresponding to [https://en.wikipedia.org/wiki/Monkey_saddle monkey saddle] and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds. | For triads and higher, though, there are other possibilities, corresponding to [https://en.wikipedia.org/wiki/Monkey_saddle monkey saddle] and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds. | ||
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[[Category:Consonance and dissonance]] | [[Category:Consonance and dissonance]] | ||
Revision as of 21:58, 22 December 2024
A saddle chord is a chord that represents a saddle point in the harmonic entropy surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher.
Chords at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.
In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many quarter tone-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} with each note a quarter tone apart.
For triads and higher, though, there are other possibilities, corresponding to monkey saddle and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds.
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Todo: add examples |