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 interval latitude with respect to the perfect fifth, wherein arto and tendo have latitudes more extreme than major and minor.

An arto triad results by flatting the third of a minor triad by approximately one quarter tone, producing an inframinor third of about 235-255 ¢.

A tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of about 445-465 ¢.

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, medials of a fifth with a latitude of ~25-30 degrees may be considered arto and tendo thirds, where the outer bound (and one of the most common tunings) is 3edf, corresponding to slendric.

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. One interpretation of such a chord is 20:23:26:30, which is notably isoharmonic on its opening two elements, especially if 300/299 is tempered out so that 23/20 and 13/10 are made fifth-complements.

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.

This property of arto and tendo chords is a major contributing factor to the "omniconsonant" property of 5edo.

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, and even in, 22edo, the native diatonic major and minor triads begin to border on 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.

Arto and tendo scales are difficult because the tendo third has to be on the same note as a fourth, and those are two very similar notes. As such, arto and tendo tonality is best used in scales with sharper fourths (and thus, flat fifths). 21edo provides reasonable tunings of these chords that can be used to build a scale, so does 16edo.

Alternatively, one can temper the arto and tendo thirds together with the major second and fourth respectively. A JI interpretation of this, assuming the canonical JI tunings of the thirds, is fendo temperament. 15edo does this.

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.
• 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
• In blackwood systems, the suspended and slendric chords (and hence a possible tuning of the extraclassical chords) are the same thing.
• Any pair of JI medials with a latitude around ±25 degrees.
• In terms of fractional monzos, the arto and tendo thirds may be tuned as 1\2edf ± 1\2ed(9/8).

The following table lists EDFs up to 31 (and corresponding EDOs up to 53) that have arto and tendo thirds (which are defined broadly as a latitude of 22.5 to 30 degrees, which is moderately extended from the "canonical" arto and tendo latitude range of 24 to 28 degrees).

Fifth	Edos	Arto third	Tendo third	Notes
3	5, 10, 15, 16, 20, 21, 25, 26, 30, 31, 36	5: 240c	5: 480c	Widest range; corresponds to slendric. Diatonic thirds in multiples of 5edo. Same intervals are shared with multiples of 3edf up to 21, so they are grouped here. 5edo fendo. (8edo's thirds aren't here cuz they're technically neogothic!)
8	13, 14, 27, 28	14: 257c 13: 277c	14: 428c 13: 462c	Narrowest range, 13edo fendo. Same intervals are shared with 16edf, so it is grouped here. Diatonic thirds found in 27edo.
11	18, 19, 37, 38	19: 253c	19: 442c	Same intervals are shared with 22edf, so it is grouped here. 19edo saj and sin thirds. 18edo fendo. 37edo diatonic.
14	24, 47*, 48	24: 250c	24: 450c	Same intervals are shared with 28edf, so it is grouped here.
17	29	29: 248c	29: 455c	29edo saj and sin thirds.
19	32, 33	32: 263c 33: 254c	32: 450c 33: 436c	32edo diatonic.
20	34, 35	34: 247c	34: 459c
23	39, 40	39: 246c	39: 462c
24	41	41: 234c, 263c	41: 439c, 468c	Intersection of the 3 and 8 categories; first tuning with 2 pairs. Slendric in 41edo. 41edo saj and sin chords.
25	42, 43	43: 251c	43: 447c
26	44, 45	44: 245c 45: 240c	44: 464c 45: 453c
27	46, 47	46: 235c, 261c 47: 230c, 255c	46: 444c, 470c 47: 434c, 460c	Slendric in 46 and 47edo, 47edo sharp fifth generates both thirds in arto/tendo range
29	49, 50	50: 240c	50: 456c
30	51, 52	51: 235c, 259c	51: 447c, 470c	Slendric in 51 and 52edo.
31	53	53: 249c	53: 453c	Very close to 13-limit intervals

Arto and tendo as interval qualities

By utilizing the logic of 24edo, "arto" and "tendo" can be generalized to interval qualities that follow interval arithmetic, corresponding to semi-diminished and semi-augmented, which are closer to their definition as interval regions in diatonic scales close to Pythagorean; in meantone tunings, these tend to correspond to supermajor and subminor.

For a diatonic edo to have arto and tendo intervals by this definition, its chroma must be an even number of edosteps.

Interval	24edo	31edo	17edo	41edo
Tendo unison	50c	39c	71c	59c
Arto second	50c	77c	0c	29c
Tendo second	250c	232c	282c	263c
Arto third	250c	271c	212c	234c
Tendo third	450c	426c	494c	468c
Arto fourth	450c	465c	424c	439c
Tendo fourth	550c	541c	565c	556c
Arto fifth	650c	659c	635c	654c
Tendo fifth	750c	735c	776c	761c
Arto sixth	750c	774c	706c	732c
Tendo sixth	950c	929c	988c	966c
Arto seventh	950c	968c	918c	937c
Tendo seventh	1150c	1123c	1200c	1171c
Arto octave	1150c	1161c	1129c	1141c

In just intonation, the "arto" and "tendo" labels may be used for the diesis 416/405, which relates the Pythagorean and arto/tendo thirds in just intonation.