10edo

10edo, or 10 equal divisions of the octave (sometimes called 10-TET or 10-tone equal temperament), is the equal tuning featuring steps of (1200/10) = 120 cents, 10 of which stack to the perfect octave 2/1.

As the double of 5edo, 10edo has its sharp fifth of 720¢, and approximates 7/4 well by 960¢. It divides this fifth in two so that it possesses a neutral third of 360¢ that is extremely close to 16/13. Both 6/5 and 5/4 are mapped to this neutral third, and in this fashion 10edo provides a primitive structural basis for the 7-limit.

10edo occupies a position intermediate between the "albitonic" systems such as 7edo, and "chromatic" systems such as 12edo. While scales such as 3L 4s can be used as a structural backbone when working within 10edo, the 10-form itself can serve as a basic form in tuning systems, more analogous to the 7-form. In particular, 10edo's division of the perfect fifth into six and the perfect fourth into four allows it to represent a trivial tuning of temperaments such as Miracle and Negri.

General theory

JI approximation

10edo contains unambiguous representations of the harmonics within the 2.3.5.7.13.17 subgroup, while primes 11 and 19 fall nearly halfway between its steps. 10edo's representation of this group equalizes an arithmetic division of the perfect fifth into six parts: 12:13:14:15:16:17:18, which contains as subsets 6:7:8:9 (the 2.3.7 subgroup, being its inheritance from 5edo) and 4:5:6.

While 10edo has consistent representations of every interval within the no-11s 17-odd-limit save 10/9 and its complement, it still makes extreme temperings (such as the vanishing of 25/24) that put many of these intervals out of recognition. The primary subgroup in which 10edo is of notable accuracy is 2.7.13.15; in particular, the intervals 16/13 and 15/14 are approximated within 0.6¢.

Approximation of prime harmonics in 10edo

Harmonic		2	3	5	7	11	13	17	19
Error	Absolute (¢)	0.0	+18.0	-26.3	-8.8	+48.7	-0.5	+15.0	-57.5
	Relative (%)	0.0	+15.0	-21.9	-7.4	+40.6	-0.4	+12.5	-47.9
Steps (reduced)		10 (0)	16 (6)	23 (3)	28 (8)	35 (5)	37 (7)	41 (1)	42 (2)

Edostep interpretations

One step of 10edo can be interpreted in the 2.3.5.7.13.17 subgroup as:

• 21/20 (the interval between 8/7 and 6/5)
• 18/17 (the interval between 17/16 and 9/8)
• 17/16 (the octave-reduced 17th harmonic)
• 16/15 (the interval between 5/4 and 4/3)
• 15/14 (the interval between 7/6 and 5/4)
• 14/13 (the interval between 8/7 and 16/13)
• 13/12 (the interval between 16/13 and 4/3)
• 10/9 (the interval between 6/5 and 4/3).

Intervals and notation

As 11edo does not have a chain of fifths (unless one counts the hard 2L 5s scale generated by 6\11) suitable for notation, the task of notating 11edo proves challenging.

One approach is to treat 11edo as a subset of 22edo. The native-fifths notation for 22edo is derived through stacking 22edo's tempered version of 3/2 (13\22) and assigning names accordingly. Only every other position in the chain of fifths is a note of 11edo, so therefore starting from C, the notes C, D, E, Gb, F#, Ab, G#, Bb, and A# are part of 11edo.

In 22edo, as the diatonic MOS has hardness 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C.

This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is smitonic (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.

For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.

"Interval categories" are based on 22edo usage of the ADIN system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and 7-limit qualities.

JI approximations (within the 17-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.

Edostep	Cents	2.9.15.7.11.17 JI approximation	Notation		10-form interval category
			Pentatonic	Mosh
0	0	1/1	C	1	Unison
1	120	16/15, [15/14], 10/9			Second
2	240	9/8, 8/7, 7/6			Unilatus
3	360	6/5, [16/13], 5/4			Third
4	480	9/7, 13/10, 4/3			Fourth
5	600	7/5, [17/12], 10/7			Median
6	720	3/2, 20/13, 14/9			Fifth
7	840	8/5, [13/8], 5/3			Sixth
8	960	12/7, 7/4, 16/9			Antilatus
9	1080	9/5, [28/15], 15/8			Seventh
10	1200	2/1			Octave

Compositional theory

Chords

10edo is arguably the first edo to have three triads consisting of two thirds to make a fifth. However, aside from the neutral triad, the other two use 5edo intervals, which are equivalent to major seconds and perfect fourths. Treating these intervals as thirds relies on the fact that stacking two sharp fifths minus an octave makes an inframinor third, thus fundamentally using oneirotonic logic.

Tendo: 0-4-6-(10)

Neutral: 0-3-6-(9)

Arto: 0-2-6-(8)

See also: Oneirotonic#Chords of oneirotonic

Scales

Example mosh (3L 4s): 0-1-3-4-6-7-9-10

Mosh is the most characteristic scale in 10edo.

Example subaric (2L 6s): 0-1-2-4-5-6-7-9-10

In subaric the "simic pentachord" 0-1-2-4-6 approximates the diatonic minor pentachord in the only way 10edo is able to. A potential temperament for this interpretation of soft subaric is 10 & 2[-7] 2.3.7.17; in other words, Trienstonian plus a 17/12 half-octave.

Detempers

Due to its small size and unique melodic character, it is very easy to detemper 10edo. Example tunings are shown in parenthesis.

Full octave, neutral third generator

• Sharp (13\43), 3L7s
  • Mainly oneirotonic fifth
  • Good approximation of 19:22:25:27:29
  • Similar to Submajor/Interpental temperament
• Flat (11\37), 7L3s
  • Mainly diatonic fifth
  • General 13-limit, especially 2.7.13

Half octave, fifth generator

• Flat (3\32)
  • Mainly diatonic fifth
  • Oceanfront temperament with added 17/14
• Sharp (4\38)
  • Mainly oneirotonic fifth
  • Soft subaric

The 10-form

Main article: 10-form

Intervals in systems approximating 10edo may be conceptualized using the 10-form. This is arguably a more intuitive way of conceptualizing intervals in the 7-limit than the 7-form is.

The 10-form's key features are the presence of the tritone as its own interval category separate from fourths and fifths, and the moving of 9/7 and 7/6 away from the category representing thirds, into the categories corresponding to fourths and major seconds respectively. The interval classes of "second" and "seventh" are split up into correspondents to "minor" and "major". This in effect gives the simplest 7-limit intervals their own pair of categories separate from the simplest 5-limit intervals, much as upgrading from the 5-form to the 7-form gives the simplest 5-limit intervals their own pair of categories.