Collection of chords: Difference between revisions
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This page covers chords with fewer than 5 pitch classes. At 5 pitch classes or above, the "chord" becomes more treatable as a scale, and may be found at [[Collection of scales]]. | This page covers chords with fewer than 5 pitch classes. At 5 pitch classes or above, the "chord" becomes more treatable as a scale, and may be found at [[Collection of scales]]. | ||
Chords here are [[rotation]]-agnostic. | |||
== Triads == | == Triads == | ||
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These triads are bounded by a perfect fifth. | These triads are bounded by a perfect fifth. | ||
==== | ==== Suspended triad ==== | ||
Intervals: '''Unison - Perfect Fourth - Perfect Fifth''' | Intervals: '''Unison - Perfect Fourth - Perfect Fifth''' | ||
Just tunings: Usually 6:8:9 | Just tunings: Usually 6:8:9 | ||
The sus4 triad is usually tuned with the perfect fourth and fifth octave complements, so it is | The suspended triad refers to two triads, sus4 and sus2, which are often each other's inversions. The sus4 triad is usually tuned with the perfect fourth and fifth octave complements, so it is the inversion of a corresponding sus2 triad. Sus triads are the simplest kind of triad in Pythagorean tuning, however they are considerably less concordant than other triads, especially with flatter tunings of the fifth (and thus sharper tunings of the fourth, and flatter tunings of the second). | ||
==== Tendo triad ==== | ==== Tendo triad ==== | ||
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Just tunings: 16:21:24, 10:13:15 | Just tunings: 16:21:24, 10:13:15 | ||
The tendo triad is generally similarly consonant to the sus4 triad. The intervals themselves are more complex, but the smaller interval is also wider, making the triad feel more open. Notably, the tendo triad's third is separated from that of the arto triad by over 200 cents, meaning both can be played at the same time and avoid clashes, similarly to a | The tendo triad is generally similarly consonant to the sus4 triad. The intervals themselves are more complex, but the smaller interval is also wider, making the triad feel more open. Notably, the tendo triad's third is separated from that of the arto triad by over 200 cents, meaning both can be played at the same time and avoid clashes, similarly to a suspended chord. | ||
==== Supermajor triad ==== | ==== Supermajor triad ==== | ||
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The arto triad is the "minor" counterpart of the tendo triad, and serves a similar role in harmony to a standard minor chord, allowing arto and tendo triads to, in a sense, fully replace minor and major ones and form their own system of harmony. As mentioned previously, the two chords can be played both at once, avoiding clashes. | The arto triad is the "minor" counterpart of the tendo triad, and serves a similar role in harmony to a standard minor chord, allowing arto and tendo triads to, in a sense, fully replace minor and major ones and form their own system of harmony. As mentioned previously, the two chords can be played both at once, avoiding clashes. | ||
=== Other triads === | === Other triads === | ||
Revision as of 08:00, 11 December 2025
This page covers chords with fewer than 5 pitch classes. At 5 pitch classes or above, the "chord" becomes more treatable as a scale, and may be found at Collection of scales.
Chords here are rotation-agnostic.
Triads
A triad is a chord containing 3 pitch classes. That is, C-E-G is a triad, but C-E-C is not a triad because even though the Cs may be separated by an octave, they are the same pitch class. Some theorists restrict the word 'triad' to exclusively refer to stacks of two thirds.
Fifth-bounded triads
These triads are bounded by a perfect fifth.
Suspended triad
Intervals: Unison - Perfect Fourth - Perfect Fifth
Just tunings: Usually 6:8:9
The suspended triad refers to two triads, sus4 and sus2, which are often each other's inversions. The sus4 triad is usually tuned with the perfect fourth and fifth octave complements, so it is the inversion of a corresponding sus2 triad. Sus triads are the simplest kind of triad in Pythagorean tuning, however they are considerably less concordant than other triads, especially with flatter tunings of the fifth (and thus sharper tunings of the fourth, and flatter tunings of the second).
Tendo triad
Intervals: Unison - Ultramajor Third - Perfect Fifth
Just tunings: 16:21:24, 10:13:15
The tendo triad is generally similarly consonant to the sus4 triad. The intervals themselves are more complex, but the smaller interval is also wider, making the triad feel more open. Notably, the tendo triad's third is separated from that of the arto triad by over 200 cents, meaning both can be played at the same time and avoid clashes, similarly to a suspended chord.
Supermajor triad
Intervals: Unison - Supermajor Third - Perfect Fifth
Just tunings: 14:18:21
The supermajor triad contains a supermajor third, and is usually seen as a 7-limit triad containing the interval 9/7. It is interestingly the *less* stable of the two basic 2.3.7 triads, despite being major, giving 2.3.7 harmony a unique sound.
Major triad
Intervals: Unison - Major Third - Perfect Fifth
Just tunings: 4:5:6, 64:81:96
The major triad is one of the two standard varieties of triad. It has a stable, rooted sound coming from its simple enumeration and low placement in the harmonic series. It is represented in 12edo and in the standard tuning of the diatonic scale.
Neutral triad
Intervals: Unison - Neutral Third - Perfect Fifth
Just tunings: 18:22:27, 26:32:39, 2:sqrt6:3
The neutral triad is associated with the 11-limit and 13-limit. It exists between major and minor triads, and often wants to resolve to either one, creating a tense, unstable sound similar to that of a diminished triad, or in other cases an "ambiguous" quality. While 2:sqrt6:3 isn't strictly just intonation, it is the tuning that utilizes the equal division of the perfect fifth into two identical neutral thirds, so it is in a sense as fundamental as the JI tunings for this particular case.
Minor triad
Intervals: Unison - Minor Third - Perfect Fifth
Just tunings: 10:12:15, 16:19:24, 54:64:81
The minor triad is the other of the two standard triads, appearing in and giving its name to the minor scale. While it is conventionally less stable than the major triad, the influence of 16:19:24 in 12edo, according to some, gives it a sense of stability. It is represented in 12edo and in the standard tuning of the diatonic scale.
Subminor triad
Intervals: Unison - Subminor Third - Perfect Fifth
Just tunings: 6:7:9
This type of triad is usually thought of in the 7-limit, where it is significantly more stable than its major counterpart, owing to its much simpler position in the harmonic series. This essentially makes the 2.3.7 subgroup work somewhat backwards melodically compared to the 5-limit - melodically similar chords are stable in 2.3.7 and unstable in 2.3.5.
Arto triad
Intervals: Unison - Inframinor Third - Perfect Fifth
Just tunings: 14:16:21, 26:30:39
The arto triad is the "minor" counterpart of the tendo triad, and serves a similar role in harmony to a standard minor chord, allowing arto and tendo triads to, in a sense, fully replace minor and major ones and form their own system of harmony. As mentioned previously, the two chords can be played both at once, avoiding clashes.
Other triads
Augmented triad
Intervals: Unison - Major Third - Augmented Fifth
Diminished triad
Intervals: Unison - Minor Third - Diminished Fifth
Major latal triad
Intervals: Unison - Subminor Third - Perfect Fourth
Minor latal triad
Intervals: Unison - Supermajor Second - Perfect Fourth
Tetrads
Arto-tendo tetrad
Intervals: Unison - Inframinor Third - Ultramajor Third - Perfect Fifth
Just tunings: [1/1 15/13 13/10 3/2]
The combination of the arto and tendo triads. See #Arto triad and #Tendo triad above.
Harmonic tetrad
Intervals: Unison - Major Third - Perfect Fifth - Subminor Seventh
Just tunings: 4:5:6:7
This tetrad approximates the harmonic series and can be seen as a tetradic analogue of 4:5:6 that extends it to the 7-limit.
Minor harmonic tetrad
Intervals: Unison - Minor Third - Perfect Fifth - Supermajor Sixth
Just tunings: 70:84:105:120
Exists in JI, but in jubilic temperaments it becomes a clear "minor" counterpart to the harmonic tetrad.
Essentially tempered chords
An essentially tempered chord is a chord whose structure is dependent on a temperament.
Slendric tetrad
Intervals: Unison - Supermajor Second - Subfourth - Perfect Fifth
Temperament: Gamelic
Distances: 8/7, 8/7, 8/7
Just tuning: [1/1 8/7 21/16 3/2]
A special case of an arto-tendo chord where all three distances between the notes are equal and equated to 8/7, while the chord's span is a perfect fifth 3/2. This can only happen in slendric temperament.
Symmetric augmented triad
Intervals: Unison - Major Third - Minor Sixth
Temperament: Augmented
Distances: 5/4, 5/4
Just tuning: [1/1 5/4 8/5]
Symmetric diminished tetrad
Intervals: Unison - Minor Third - Tritone - Major Sixth
Temperament: Diminished
Distances: 6/5, 6/5, 6/5
Just tuning: [1/1 6/5 36/25 5/3]
