Saddle chord: Difference between revisions

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A '''saddle chord''' is a chord that represents a ''saddle point'' in the [[~~Harmonic_Entropy|~~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.

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>

In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many [[~~quarter_tone|~~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.

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} in [[24edo]] 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.

[[Category:~~chords~~]]

[[Category:Chord]]

[[Category:Consonance and dissonance]]

~~[[Category:dissonance]]~~

~~[[Category:pentad]]~~

~~[[Category:sonance]]~~

~~[[Category:tetrad]]~~

~~[[Category:triad]]~~

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-A '''saddle chord''' is a chord that represents a ''saddle point'' in the [[Harmonic_Entropy|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.
+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>
-In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many [[quarter_tone|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.
+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} in [[24edo]] 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.
-[[Category:chords]]
+{{todo|add examples|inline=1}}
+[[Category:Chord]]
 [[Category:Consonance and dissonance]]
-[[Category:dissonance]]
-[[Category:pentad]]
-[[Category:sonance]]
-[[Category:tetrad]]
-[[Category:triad]]