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:
Tuning-unit-sizes for some small intervals, with cents-values given for comparison:
interval tuning unitscents 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 grad 60 ~1.955000865 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
--- In tuning@yahoogroups.com, "Gene Ward Smith"
> You might want to include the connection with 12276 equal and atomic
> temperament on this page.
Some other information: the poptimal range for the atomic temperament schisma in TU terms is 60.00389 to 60.00416; hence if we took for instance the schisma to be exactly 60.004 TUs, we would get a pythagorean comma of 720.048 TUs and a Didymus comma of 660.044 TUs.
The whole point of this sort of thing, of course, is that you plan to round off to the nearest integer, but if you do not this would be a good set of tuning values for the TU. It would give a 5/4 as 4*3069-7*60.004 = 11855.972 TUs, 3/2 as 7*3069+60.004 = 21543.004 TUs, and of course the octave is exacty 12*3069=36828 TUs. You can see how closely these lie to the rounded values.
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