Extraclassical tonality

Basics

The arto and tendo tonality system is a tonality system built from 10:13:15 (tendo, or ultramajor) and 26:30:39 (arto, or inframinor) triads (or triads with thirds of a similar size). The idea is that because these chords share many of the same properties as major and minor triads, the arto and tendo harmonic system resembles the diatonic system in many ways, however, it also has many distinct characteristics that make it sound nothing like the diatonic scale.

Arto and tendo triads

Arto and Tendo triads are the same shape as major and minor 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 26:30:39 results by flatting the third of a minor triad by approximately one quarter tone, producing an inframinor third of ~250 cents. The Tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of ~450 cents. The Tendo chord 10:13:15, being of a simpler ratio, is considered to be more consonant than the arto chord which is quite complex. Also, 10:13:15 tendo chord is only slightly more dissonant than the minor chord 10:12:15. Although they appear to be the same, arto and tendo chords are not seven limit, and tuning them to 14:18:21 and 6:7:9 may compromise the cross-tonal nature of the chords; using the septimal second and subfourth may make the intervals too wide/narrow to function as thirds.

More generally, thirds of roughly 250 and 450 cents, or mediants of a fifth with a contrastiveness of 25-30%, may be considered arto and tendo thirds.

Cross-tonality

One interesting phenomenon that happens with arto and tendo is that it is cross-tonal. Unlike the diatonic scale which has two distinct flavors of triads, 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 Eb would generally sound disruptive and cause dissonance because of the chroma between the Eb and E, where as 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 cents, 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.