Extraclassical tonality

The arto and tendo or extraclassical tonality system is a tonality system built from ultramajor (called "tendo") and inframinor (called "arto") triads. Because these chords share many of the same properties as major and minor triads, the extraclassical harmonic system resembles the conventional diatonic system in many ways, however, it also has many distinct characteristics that make it sound nothing like normal major and minor chords.

The name "extraclassical" derives from the mediants' position outside the range of the classical and standard diatonic mediants.

Arto and tendo triads

Arto and tendo triads are the same shape as minor and major triads, but with a more extreme difference between the thirds. This can be characterized in terms of mediants, where the contrastiveness of the arto and tendo mediants is higher than that of minor and major.

An arto triad results by flatting the third of a minor triad by approximately one quarter tone, producing an inframinor third of ~250 ¢.

A tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of ~450 ¢.

The tendo chord is considered to be more consonant than the arto chord, just as the major chord is considered more consonant (if only slightly) than the minor chord. Also, both chords are only slightly more dissonant than the normal major and minor chords.

More generally, mediants of a fifth with a contrastiveness of 25–30% may be considered arto and tendo thirds.

Notation

Assuming the normal major and minor thirds are roughly 12edo or pythagorean, a tendo triad may be notated C–E⁠ ⁠–G or A–C⁠ ⁠–E, and an arto triad may be notated C–E⁠ ⁠–G or A–C⁠ ⁠–E.

For meantone tunings flatter than 1⁄4-comma (e.g. 19edo, which represents 1⁄3-comma meantone), or in just intonation, arto and tendo may be treated as diminished and augmented intervals, such as C–E𝄫–G or C–E♯–G.

An arto chord may be notated with the lowercase letter r, such as Cr or Dr. This is to avoid clashing with the use of A for Augmented.

A tendo chord is notated with the capital letter T.

Cross-tonality

One interesting phenomenon that happens with extraclassical tonality is that it is cross-tonal. Unlike normal major and minor thirds which clash heavily when played together, arto and tendo triads and Intervals are able to co-exist on the same root. For example, in the diatonic scale, usually playing a C major triad with an E♭ would generally sound disruptive and cause dissonance because of the chroma between the E♭ and E, whereas a C arto triad with the third of a C tendo can be played simultaneously without as harsh of a discordance since the interval between them is a more consonant major second. This creates an interesting, subtle ambiguous flavor, but what's even more interesting is that the two triads still sound distinct enough to be considered separate and being played together causes a lack of resolution. This also generates the possibility that both arto and tendo chords can be used in a progression on the same root without going outside of the scale. Likewise, a melody on a tendo chord can play notes of the arto chord of the same root.

Use cases

In 16edo, and especially in armotonic, arto and tendo chords may be used by using the minor 2-mosstep and major 3-mosstep in chords. These are intervals of 225 and 450 ¢, and also serve as simple slendric intervals. Indeed, playing the chord with both arto and tendo intervals results in the same chord as stacking 3 slendric generators.

In diatonic edos sharper than 27edo, the native diatonic major and minor triads are, in fact, arto and tendo; the chromatic scale 5L 7s makes for a good MOS to allow arto/tendo cross-tonality in these systems.

24edo allows both arto/tendo and minor/major chords to be used.

Tunings

The canonical tunings of the arto and tendo thirds are 15/13 and 13/10, which leads to the conclusion in a JI mentality of consonance that the 10:13:15 tendo chord is more consonant than the 26:30:39 arto chord. However, several tuning options are available.

• The thirds could be tuned to the 7-limit intervals 7/6 and 9/7. This slightly compromises cross-tonality with a submajor second in between the intervals instead of a wider major second, however it makes the ratios simpler when played individually.
• Alternatively, they may be tuned to the 7-limit intervals 8/7 and 21/16, at the cost of perhaps not sounding like thirds.
• In 24edo, the thirds can be tuned to 250 ¢ and 450 ¢.
• In slendric systems, especially those with a flat fifth such as 16edo, the arto and tendo thirds may be widened to 1\3edf and 2\3edf.
  • The chord produced by playing the arto and tendo thirds together is actually just the first few slendric generator steps.
• Any pair of JI mediants with a contrastiveness about 25–30%.
• In terms of fractional monzos, the arto and tendo thirds may be tuned as 1\2edf ± 1\2ed(9/8).
• In a fendo system such as 13edo, the JI ratios for the arto and tendo thirds are equated with the suspended mediants of 9/8 and 4/3, meaning that suspended chords and arto/tendo chords