40/39

Interval information

Ratio	40/39
Factorization	23 × 3-1 × 5 × 13-1
Monzo	[3 -1 1 0 0 -1⟩
Size in cents	43.83105¢
Names	tridecimal 1/5-tone, tridecimal minor diesis, unintendo comma
Color name	3uy1, thuyo unison
FJS name	[math]\text{A1}^{5}_{13}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	10.6073
Weil height (log2 max(n, d))	10.6439
Wilson height (sopfr(nd))	27
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.50925 bits
Open this interval in xen-calc

In 13-limit just intonation, 40/39 is the difference between the third octave of the third 5/4 (5 × 2³) and the fifth of the thirteenth partial of the same root (13 × 3). Within an octave, it is the difference between 39/32 and 5/4 and thus between 13/8 and 5/3. It is also the difference between the perfect fourth (4/3) and the tridecimal naiadic (13/10), and between the Pythagorean whole tone (9/8) and the tridecimal semifourth (15/13).

Temperaments

If treated as a comma to be tempered out in the 2.3.5.13 subgroup, it leads to unintendo temperament.

Unintendo

This temperament was accidentally named when Vector described fendo as tempering out 40/39 in the 2.3.5.13 subgroup. The mistake has since been corrected. Its generators are a sharp perfect fifth and a flat major third. It can be described as 7 & 15 & 10. It equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see 65/64), but the other way around.

Subgroup: 2.3.5.13

Comma list: 40/39

Sval mapping: [⟨1 0 0 3], ⟨0 1 0 -1], ⟨0 0 1 1]]

sval mapping generators: ~2, ~3, ~5

Optimal tunings:

• CWE: ~2 = 1200 ¢, ~3/2 = 710.153 ¢, ~5/4 = 383.023 ¢

Badness (Dirichlet): 0.227

Notation

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal ⁠ ⁠ and is called the 13/5 small diesis, or 13/5S for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F⁠ ⁠⁠ ⁠. The primary role of ⁠ ⁠ is 6400/6561 (25S). The upward version is called 5/13S or 13/5S up and is represented (in a secondary role) by ⁠ ⁠.