Duodene

Duodene as a lattice.

Duodene is a 12-note scale in just intonation, representing a natural approach to detempering standard 12edo, when considered as a 5-limit temperament. The scale was named by Alexander Ellis in an 1875 article where he uses it to develop a theory of the chromatic scale in just intonation.

History

While Ellis formalized and named the system, it was first described by French engineer Salomon de Caus in 1615. Marin Mersenne mentions it in his Harmonie universelle (Universal Harmony), and among piano tuners, the system is known as "Mersenne's spinet tuning No. 1." The scale is also found in Euler's Tentamen novae theoriae musicae (Attempt at a New Theory of Music) from 1739.

Musical properties

As a lattice structure, it consists of a chain of three perfect fifths (F – C – G – D) with just major thirds above and below each of these. When arranged on a standard piano keyboard, the white keys of a duodene form a just diatonic scale, specifically Ptolemy's intense diatonic scale.

It can be constructed as a Fokker block with the syntonic comma (81/80) and the enharmonic diesis (128/125) as chromas. It is also an Euler-Fokker genus of [math]\displaystyle{ 675 = 3^3 \times 5^2 }[/math], meaning it comprises all divisors of 675, reduced by octave equivalence.

In Indian musical traditions, it is known as "Gandhar tuning."^{[citation needed]}

As a detempering

Duodene can be tempered to several scales, which it can itself be understood as a detempering of.

Augmented diesis

If the augmented diesis is tempered out (as in 15edo), the multi-period MOS scale 3L 9s is obtained, where the large step represents 27/25 and 135/128, and the small step represents 16/15 and 25/24. This is one possible 12-note chromatic in augmented temperament.

Syntonic comma

If the syntonic comma is tempered out (as in 19edo), the scale becomes reachable through a chain of fifths and becomes the MOS scale 7L 5s, where the large step represents 27/25 and 16/15, and the small step represents 135/128 and 25/24. This is the 12-note chromatic of meantone temperament.

If both are tempered out, the result is 12edo (or an enfactoring, like 24edo).

Schisma

If the schisma is tempered out (as in 53edo), the scale becomes reachable through a chain of fifths and will be contained within the MOS 12L 17s, where the large step represents the gothic comma [dd3] and the small step the pythagorean comma [-d2]. It can instead be viewed as the MODMOS scale 5L 7s; 2|9 #1#3#8#10 or 10|1 b4b6b9b11, where the large step is an augmented unison, the small step a minor second, and the chroma the pythagorean comma.

Thus, ~27/25 is reached by L+3s [-dd2], ~16/15 by L+2s [A1], ~135/138 by L+s [m2], and ~25/24 by L. The sizes of the steps are equidistant, as the augmented diesis is equated to two syntonic commas.

Step pattern

Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step L (27/25), 6 medium steps M (16/15), 2 narrow steps n (135/128), and 3 small steps s (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis.

The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo (s = 2, n = 3, M = 4, L = 5) and 53edo (s = 3, n = 4, M = 5, L = 6). 118edo is optimal (s = 7, n = 9, M = 11, L = 13).

Scala file

! duodene.scl
!
Ellis's Duodene
! Fokblock([81/80, 128/125], [6,5]), genus [33355], Dwarf(⟨12 19 28]), syndie3, Gandhar tuning
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1

Music

• A different 12-tone subset of 34-equal (or thereabouts) on the harpsichord by Cam Taylor (2024)
• Duodene2 by Chris Vaisvil

See also

• Marveldene: the marvel tempered version of this scale.