# Undecimal Primodality

When the 11th harmonic is selected as a numerary nexus to relate other harmonics against, the resultant collection of pitches together forms the undecimal primodality. The undecimal primodality is a gorgeous and flexible family of intervals. It contains possibility of bright diatonic gestures, a rich diversity of neutrals, color opposite qualities, and key super intervals. It's initial genesis step size is 150.6 cents.

## Examining undecimal

Throughout the course of this writing, we will examine the first 2 octaves of the undecimal primodality.

### First octave

11:22 (1p:2p)

0:          1/1               0.000000
1:         12/11            150.637059 undecimal neutral second
2:         13/11            289.209719 undecimal minor third
3:         14/11            417.507964 undecimal major third
4:         15/11            536.950772 undecimal superfourth
5:         16/11            648.682058
6:         17/11            753.637467
7:         18/11            852.592059 undecimal neutral sixth
8:         19/11            946.195074
9:         20/11           1034.995772 undecimal superminor seventh
10:         21/11           1119.462965 undecimal major seventh
11:          2/1            1200.000000 octave

The first octave space contains 11 intervals — 10 completely unique undecimal intervals and 1 octave given by the 1p:2p framework. These first 10 unique intervals are the modal core to undecimal. Understanding the profile of these intervals and their sound is essential to unlocking the identity of Undecimal (/11) as a prime mode tonality.

First, let's examine it's relationship to 12edo major and minor. The 11:14 operates as a 417c major third. This third functions as the "diatonic" major of undecimal which tends to cause Undecimal major qualities to be lifted and brighter. Additionally, 11:21 as a major seventh is bright in parallel to the major third (it's a 2:3 relationship). 11:14:21 thus yields the basic Undecimal analog for maj7. In the realm of minor qualities, the 11:13 at 289c sounds incredibly smooth and operates as the core minor third to the Undecimal family. A parallel minor seventh (something lower like 989) is not found until the second octave of the family.

Secondly, it's "fifth region" is, of course, split in the first octave. No first octave prime families ever contain a 2:3 except /2 itself. As such, Undecimal gives us 753.6 and 648.6 cent "prime fifth" options. These can be challenging to yield as "believable" fifths but by embracing them, one can gain access to new harmonic spaces not typically considered. If a perfect fifth is desired, reaching into the second octave of Undecimal fulfills this while preserving all the previously discussed intervals.

Undecimal has quality super and neutral intervals. The 11:15 at 536.9c is a remarkable superfourth. It's a highly active, interval which has a tendency to bridge almost all the intervals in the family together. The 11:15 superfourth is very tunable by ear and has it's own distinct identity rather than sounding like a flatter version of 8:11. Additionally, it forms a parallel relationship with the 11:20 superminor seventh at 1034.9c. Between the two, a 3:4 relationship exists similar to the familiar perfect fourth relationship between 4 and b7 in 12edo. However, in this family the relationship is shifted much higher to super fourth and superminor seventh. Between this brighter super 4-b7 relationship and the sharper 3-7 relationship, tertian or 12edo-esque structures rendered in Undecimal tend to sound brighter than /13, /17, and /19 color variants. Neutral intervals appear in the first octave as 11:12 and 11:18, 150.6c neutral second and an 852c neutral sixth from 11:18. The 11:19 at 948 cents contributes a unique middle-augmented sixth which tends to sound more like a hypermajor sixth when surrounded by the sharper Undecimal intervals.

### Second octave

22:44 (2p:4p) all previous intervals omitted to reduce clutter

0:          1/1               0.000000
1:         23/22             76.956405
3:         25/22            221.309485 undecimal acute whole tone
5:         27/22            354.547060 undecimal Zalzalian neutral third
7:         29/22            478.259252
9:         31/22            593.717630
11:          3/2             701.955001 perfect fifth
13:         35/22            803.821678 undecimal minor sixth
15:         37/22            900.026096 undecimal major sixth
17:         39/22            991.164720 undecimal minor seventh
19:         41/22           1077.744463 undecimal submajor seventh
21:         43/22           1160.199763
22:          2/1            1200.000000 octave

The second octave space contains 22 intervals — 10 completely unique new intervals, 10 undecimal intervals from the previous octave, 1 octave given by the 1p:2p framework, and 1 perfect fifth given by the 2p:4p framework. Thus, the second octave of a prime contains all the unique material from the first octave, a new set of unique intervals from the second octave, an octave, and a perfect fifth. This makes second octave lineal segments quite ideal for establishing a strong tonicization of the prime.

First, let's examine the second octave's relationship to 12edo major and minor since most ears reading this have their senses rooted in that system. The 37/22 acts identically to 12edos major sixth as it's less than .02 cents away at 900 cents. 35/22 also sits within 4 cents of the 12edo minor sixth. These intervals often come into play when rendering tertian harmony in /11. These can be used to evoke fresh but familiar sensations in average listeners such as demonstrated in Spiritualistica and Timelessness. 31/22 as 593.7 cents offers a lush tritone with it's own attractive quality which can operate as a tritone parallel for Maj#11 or 7#11 sounds.

The 13/11 minor third provided by the first octave finds its fifth in 39/22. With the root added, these form 22:26:39, the Undecimal min7 chord. Since the root has also gained a perfect fifth, we can now create a full, 4 note minor seventh chord 22:26:33:39 representing R-b3-5-b7 with two 2:3s an 11/13th apart.

The second octave grants us an acute major second, 25/22 at 221.3 cents and a lower minor second with 23/22 at 76.9 cents. The 25/22 major second runs sharp along with the 417, 537, and 1119 "diatonic" major intervals. As a result when joined by those intervals, the system begins to sound uniformly sharp but regular (if all are sharp, none are sharp). By contrast, the first octave only gives the neutral second from 12/11 which tends to act more xen relative to the "diatonic" intervals. These colors can be interplayed to form a wide variety of possibilities across an axis of familiarity or novelty relative to your ear.

Undecimal also grants "opposite" color intervals in the form of second octave intervals that go flat relative to key interval areas rather than uniformly sharp like the majority of Undecimal. These "relatively flatter" intervals can be seen in 41/22 at 1077.7 cents as a relatively darker maj7, 29/22 at 478 cents as a subfourth relative to the 417 cent 14/11. These can be combined with the "prime fifths" from the first octave to generate startling combinations which push, pull, and subvert expectation like 22:28:34:41:52:58.

Lastly, The gorgeous Zalzalian neutral third, 27/22 at ~354.5 cents, shows up in the second octave. This can be combined with the rich diversity of neutrals from the first octave to form a strong neutral base.

22:27:30  Undecimal N3S4