Collection of chords
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 unless otherwise specified; the provided interval rulers are in 58edo except for the essentially tempered chords.
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 tertian triads, composed of two stacked thirds.
Fifth-bounded triads
These triads are bounded by a perfect fifth.
Suspended triad
├───────────────────────┴─────────┴───────────────────────┐ [24 34]├─────────┴───────────────────────┴───────────────────────┐ [10 34]
Intervals: Unison - Perfect Fourth - Perfect Fifth, Unison - Major Second - Perfect Fifth
Just tunings: Usually 6:8:9; 8:9:12
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 and arto triads
Tendo triad
├─────────────────────┴───────────┴───────────────────────┐ [22 34]
Intervals: Unison - Ultramajor Third - Perfect Fifth
Just tunings: 16:21:24, 10:13:15
Arto triad
├───────────┴─────────────────────┴───────────────────────┐ [12 34]
Intervals: Unison - Inframinor Third - Perfect Fifth
Just tunings: 14:16:21, 26:30:39, 20:23:30
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.
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.
The tendo and arto triads may be used in place of standard major and minor triads to form a tonality system called extraclassical tonality, which can be expressed in a number of types of tuning systems.
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. This phenomenon is called cross-tonality and it makes 5-, 8-, 12- or 9-form scales wherein both thirds coexist over the same root useful to play in. A simple interpretation of a cross-tonal chord is 20:23:26:30, which suggests the tempering out of 300/299.
Cross-tonal chords are functionally unresolved and ambiguous similar to suspended chords, and cross-tonality also allows notes from arto and tendo scales to be played simultaneously without clashing.
Arto/tendo cross-tonality is a major contributor to the "omniconsonant" sound of 5edo.
- In the 34edo system, tendo differs from diatonic major by a sextula; 34edo is 2.3.5.13.17 so arto and tendo harmony becomes one of the primary kinds of xenharmony that retains the fifth-based structure.
- In hemipythagorean in general, arto and tendo thirds have a canonical tuning value, being half of the perfect fourth and major sixth respectively.
- Assuming the diatonic major and minor thirds are close to pythagorean (such as in 12edo or 41edo), tendo is equivalent to semiaugmented.
- Assuming meantone thirds close to golden meantone, tendo and arto are at the augmented and diminished thirds.
- In sharp archy tunings (sharper than 27edo), the supermajor and subminor thirds approach tendo and arto.
- In slendric, the equal divisions of the fifth may be used as particularly wide arto and tendo tunings.
Supermajor and subminor triads
Supermajor triad
├────────────────────┴────────────┴───────────────────────┐ [21 34]
Intervals: Unison - Supermajor Third - Perfect Fifth
Just tunings: 14:18:21
Subminor triad
├────────────┴────────────────────┴───────────────────────┐ [13 34]
Intervals: Unison - Subminor Third - Perfect Fifth
Just tunings: 6:7:9
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.
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.
Major and minor triads
Major triad
├──────────────────┴──────────────┴───────────────────────┐ [19 34]
Intervals: Unison - Major Third - Perfect Fifth
Just tunings: 4:5:6, 64:81:96, 22:28:33
Minor triad
├──────────────┴──────────────────┴───────────────────────┐ [15 34]
Intervals: Unison - Minor Third - Perfect Fifth
Just tunings: 10:12:15, 16:19:24, 54:64:81, 22:26:33
"Major" and "minor" as chord types tend to encompass both nearmajor/minor and farmajor/minor, partially due to the prevalence of meantone tuning and 12edo itself causing the functional conflation of 5/4 with farmajor thirds. The 58edo intervals shown in the diagram sit between the two categories.
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.
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.
Neutral triad
├────────────────┴────────────────┴───────────────────────┐ [17 34]
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.
Other triads
Augmented and diminished triads
Augmented triad
├──────────────────┴──────────────────┴───────────────────┐ [19 38]
Intervals: Unison - Major Third - Augmented Fifth
Diminished triad
├──────────────┴──────────────┴───────────────────────────┐ [15 30]
Intervals: Unison - Minor Third - Diminished Fifth
Latal triads
Main article: Chthonic harmony
Major latal triad
├────────────┴──────────┴─────────────────────────────────┐ [13 24]
Intervals: Unison - Subminor Third - Perfect Fourth
Minor latal triad
├──────────┴────────────┴─────────────────────────────────┐ [11 24]
Intervals: Unison - Supermajor Second - Perfect Fourth
Tetrads
Arto-tendo tetrad
├───────────┴─────────┴───────────┴───────────────────────┐ [12 22 34]
Intervals: Unison - Inframinor Third - Ultramajor Third - Perfect Fifth
Just tunings: [1/1 15/13 13/10 3/2], 20:23:26:30
The combination of the arto and tendo triads. See #Arto triad and #Tendo triad above.
Major seventh tetrad
├──────────────────┴──────────────┴──────────────────┴────┐ [19 34 53]
Intervals: Unison - Major Third - Perfect Fifth - Major Seventh
Just tunings: 8:10:12:15
Minor seventh tetrad
├──────────────┴──────────────────┴──────────────┴────────┐ [15 34 49]
Intervals: Unison - Minor Third - Perfect Fifth - Minor Seventh
Just tunings: 10:12:15:18
Dominant tetrad
├──────────────────┴──────────────┴─────────────┴─────────┐ [19 34 48]
Intervals: Unison - Major Third - Perfect Fifth - Minor Seventh
Just tunings: [1/1 5/4 3/2 16/9]
Contains a tritone from the third to the seventh, and is thus generally a tense chord. Found on the V of major diatonic scales, where it resolves to the tonic.
Harmonic tetrad
├──────────────────┴──────────────┴────────────┴──────────┐ [19 34 47]
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
├──────────────┴──────────────────┴──────────┴────────────┐ [15 34 45]
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. Below are the major and minor harmonic tetrads in 22edo.
├──────┴─────┴────┴───┐ [7 13 18]├─────┴──────┴───┴────┐ [6 13 17]
Trines
A trine is a chord with three notes where the two outer notes are separated by an octave. It might not be considered a triad, but rather a specific voicing of a dyad; trines are not rotation-agnostic.
Conventionally, a trine's middle note lies between an ultramajor third (~450c) and an inframinor sixth (~750c), similar to the qualities of third in a standard fifth-bounded triad generally not exceeding <250c or >450c. The 3-limit trines 3:4:6 and 2:3:4 are very important for 3-limit harmony as concordant conventional triads cannot be constructed.
Trines are also useful in Armodue theory.
Ultramajor third trine
├─────────────────────┴───────────────────────────────────┤ [22 58]
Intervals: Unison - Ultramajor Third - Octave
Just tunings: 10:13:20
Utonal trine
├───────────────────────┴─────────────────────────────────┤ [24 58]
Intervals: Unison - Perfect Fourth - Octave
Just tunings: 3:4:6
Semiaugmented fourth trine
├──────────────────────────┴──────────────────────────────┤ [27 58]
Intervals: Unison - Semiaugmented Fourth - Octave
Just tunings: 8:11:16
Neutral trine
├────────────────────────────┴────────────────────────────┤ [29 58]
Intervals: Unison - Tritone - Octave
Just tunings: 5:7:10, 7:10:14
Semidiminished fifth trine
├──────────────────────────────┴──────────────────────────┤ [31 58]
Intervals: Unison - Semidiminished Fifth - Octave
Just tunings: 8:11:16
Otonal trine
├─────────────────────────────────┴───────────────────────┤ [34 58]
Intervals: Unison - Perfect Fifth - Octave
Just tunings: 2:3:4
Inframinor sixth trine
├───────────────────────────────────┴─────────────────────┤ [36 58]
Intervals: Unison - Inframinor Sixth - Octave
Just tunings: 13:20:26
Essentially tempered chords
An essentially tempered chord, a.k.a. dyadically essentially tempered chord (det chord), is a chord whose structure is dependent on a temperament.
Slendric tetrad
├───────┴───────┴───────┴────────────────┐ [8 16 24] (41edo)
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
├───────────┴───────────┴───────────┤ [12 24 36] (36edo)
Intervals: Unison - Major Third - Minor Sixth
Temperament: Augmented
Distances: 5/4, 5/4
Just tuning: [1/1 5/4 8/5]
Symmetric diminished tetrad
├────────┴────────┴────────┴────────┤ [9 18 27 36] (36edo)
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]
