Tina (music)

The tina is a logarithmic unit of measure used for MusicAL intervals. Typically tinas are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one tina is much too small to be heard between successive notes.

A tina is approximately 1/7 cent, and its chief advantage over the cent is that tinas can represent a very wide variety of just-intonation intervals accurately using only integer values, obviating the need for decimal places and thus eliminating the possibility of rounding error; the tina thus provides a resolution more than 7 times greater than the cent, with the need to use only 4 digits. The chief disadvantage of the tina is that 8539 is a prime number, thus it cannot be divided exactly into any smaller equal temperament.

8539 tinas are equal to one octave — a frequency ratio of 2:1 — and an equally tempered semitone (the interval between two adjacent piano keys) is equal to ~712 tinas. One tina is precisely equal to 2^1/8539, the 8539th root of 2, which is approximately 1.000081178.

If you know the frequencies a and b of two notes, the number of tinas measuring the interval between them may be calculated by the following formula:

$$n = 8539 \log_2 \left( \frac{a}{b} \right) \approx 28366 \log_{10} \left( \frac{a}{b} \right)$$

Likewise, if you know a note b and the number n of tinas in the interval, then the other note a may be calculated by:

$$a = b \times 2 ^ \frac{n}{8539}$$

To compare different tuning systems, convert the various interval sizes into tinas. For example, in just intonation the major third is represented by the frequency ratio 5:4. Applying the formula at the top shows this to be about 2749 tinas. The equivalent interval on the equal-tempered piano would be 2846 tinas. The difference, 97 tinas, is about a seventh of a half step, easily audible. The just noticeable difference for this unit is about 42 tinas.

The tina was suggested privately in 2004 as an interval measurement by George Secor, then publicly in April 2007 by Gene Ward Smith.

See also

• Interval (music)
• Musical tuning