Arto and tendo theory

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.

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.

Tunings

The canonical tunings of the arto and tendo thirds are 15/13 and 13/10. 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 250c and 450c.
• 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.
• Any pair of JI mediants with a contrastiveness about 25-30%.
• In terms of fmonzos, the arto and tendo thirds may be tuned as 1\2edf ± 1\2ed(9/8).