  # tuning-unit, temperament-unit (TU)

[Joe Monzo, with data from Brad Lehman]

A term used by John Brombaugh to designate the tiny musical interval describing 1/720th part of the pythagorean comma.

The temperament unit is calculated as the 720th root of the ratio 531441:524288, or [ 2-19 * 312 ](1/720), thus with a ratio itself of approximately 1:1.000018821. It is an irrational number. The width of this tuning unit interval is ~0.032583348 (pretty close to 1/31) cent, and exactly 1/60 grad.

This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into ~36828.6282 equal parts. Thus the tuning unit represents one degree in ~36828.6282-edo "non-octave" tuning, or its audibly identical "octave"-based relative 36829-edo.

There are just over 3069 tuning units (a more exact figure is ~3069.05235, about 30691/19) in a Semitone.

The formula for calculating the temperament-unit-value of any ratio r is: tuning units = log10(r) / log10[ 2(-19/720) * 3(1/60) ].

A tuning-unit is:

• ~0.817380431 (~9/11) jot,
• ~1.334613924 (~1 1/3) dodekamus,

Tuning-unit-sizes for some small intervals, with cents-values given for comparison:

```		      interval           tuning units            cents

Pythagorean comma   720                      ~23.46001038
syntonic comma  ~660.0392862              ~21.5062896
kleisma  ~248.8166324 (~248 4/5)    ~8.107278862
heptameride  ~122.3542465 (~122 1/3)    ~3.986710963
savart  ~122.762094 (122 3/4)       4
skhisma   ~59.96071375 (~59 49/51)  ~1.953720788
millioctave   ~36.8286282  (~36 5/6)     1.2
Türk sent   ~34.74398887 (~34 3/4)    ~1.132075472
cent   ~30.6905235  (~30 2/3)     1
jot    ~1.223420529 (~1 2/9)    ~0.039863137
cawapu    ~0.749280359 (~3/4)      ~0.024414062
midipu    ~0.18732009  (~1/5)      ~0.006103516
```
. . . . . . . . .
[Gene Ward Smith, Yahoo tuning group, message 55893 (Thu Aug 26, 2004 10:07 pm)]

--- In tuning@yahoogroups.com, "Gene Ward Smith" wrote:

> You might want to include the connection with 12276 equal and atomic 