General formulas for the equal-tempered interval

It is copied from wikipedia's article "Equal temperament", now in "View history" section.

Background theory

Basic concepts:
From logarithms: The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. For example: (98)² = 8164. The logarithm of 8164 to base 98 is 2, or

${\displaystyle \log _{\frac {9}{8}}{\frac {81}{64}}=2.}$

To compare logarithms, we use some standard base, e.g. 10 or e. Usually, in mathematical notation "log" means "logarithm with base 10". For example: log 98 = log 1.125 = 0.0511525224473813, and, log (98)² = log 8164 = log 1.265625 = 0.1023050448947626. The latter logarithm divided by the former yields 2, the exponent as above.
From physics: For a given velocity of the wave on the string, length and frequency of the sound produced from a string when it is plucked, are inverse proportional. For the same tension in the string, and the same string’s linear density, the velocity of the wave on the string is the same.
From acoustics: Pitch is the distinctive quality of a sound, dependent primarily on the [fundamental] frequency of the sound waves produced by its source.^[45]
Psychoacoustical law: It was known to Pythagoras^[46][47] and perhaps to ancient Chinese philosophers, that our subjective sensation of the difference (interval) in pitch (tone) is inversely proportional to the exponent of the ratio of string (chord) lengths. In modern terms, it is proportional to the logarithm of the ratio of fundamental frequencies of sounds produced by two different lengths of the string.

Example

Consider two pairs of a string's lengths: 90 cm, 80 cm, and, 81 cm, 64 cm. Our subjective sensation of pitch difference (interval) between the sounds produced by string's lengths 81 cm and 64 cm, the ratio 8164 = (98)², is double that between 90 cm and 80 cm, the ratio 9080 = 98 (see above, From logarithms and Psychoacoustical law). However the pitch of sound from the 90 cm is lower than that from the 80 cm, and from the 81 cm is lower than that from the 64 cm. Now consider two pairs of fundamental frequencies (from other lengths): 90 Hz/80 Hz and 81 Hz/64 Hz. Our subjective sensation of the ratio 8164 = (98)² is also twice that of 9080 = 98, but the pitch of 90 Hz is higher than that of 80 Hz, and that of 81 Hz higher than that of 64 Hz. The greater the string's length, the lesser the fundamental frequency, and the lower the pitch.
Let A and B be two periodic sounds with stable fundamental frequencies (stimuli) ffreq_A and ffreq_B. Then, the ratio ffreq_Affreq_B is called the ratio interval. Our subjective sensation of the interval between these sounds is proportional to log ffreq_Affreq_B.
Let C and D be two other periodic sounds with stable fundamental frequencies: ffreq_C and ffreq_D. Their ratio interval is ffreq_Cffreq_D. Let

${\displaystyle m={\frac {\log {\frac {\mathrm {ffreq_{A}} }{\mathrm {ffreq_{B}} }}}{\log {\frac {\mathrm {ffreq_{C}} }{\mathrm {ffreq_{D}} }}}}}$

(see above, logarithms, second example, and below, main formula for d = 1). Then

${\displaystyle log{\frac {\mathrm {ffreq_{A}} }{\mathrm {ffreq_{B}} }}=m\cdot \log {\frac {\mathrm {ffreq_{C}} }{\mathrm {ffreq_{D}} }}.}$

Our subjective sensation of the sound interval A/B (represented by log ffreq_Affreq_B), is proportional, by proportion m, to our subjective sensation (represented by log ffreq_Cffreq_D) of the sound interval C/D. The proportion m is magnitude, and log ffreq_Cffreq_D the unit for the tempered interval A/B.
Indeed, if, in the last equation, we multiply m by d and divide log ffreq_Cffreq_D by d, that is

md ${\displaystyle {\frac {\log {\frac {\mathrm {ffreq_{C}} }{\mathrm {ffreq_{D}} }}}{d}}}$

then md is the magnitude and

${\displaystyle {\frac {\log {\frac {\mathrm {ffreq_{C}} }{\mathrm {ffreq_{D}} }}}{d}}}$

is the unit, as with every quantity, e.g. length L = 5 m. If the magnitude 5 is multiplied by 100, and m (certain length, serving as unit) divided by 100 (that is, cm), then L = 500 cm. When ffreq_Cffreq_D = 2 and d = 1200, unit is cent (one 1200th of octave)."
Let t(r) be the tempered interval of ratio interval r. Then we have:

${\displaystyle t(r)=\log r=m\cdot {\frac {\log b}{d}},}$

where m is magnitude and log bd is the unit of tempered interval. The main formula below is obtained from this one.
In the example 2 of the main formula, it is

${\displaystyle {\frac {\log {\frac {3}{2}}}{\log {\frac {9}{8}}}}=3.442...,}$

which means our subjective sensation of the ratio interval 32 (a perfect fifth) is 3.442 times our subjective sensation of the ratio interval 98 (a major second). The number 3.442 is the magnitude, and the tempered interval t(98) = log 98 is the unit of tempered interval t(32) = log 32. Multiplying by 100 yields 344.2 hundredths of the tempered interval t(98).
Note: Pitch depends to a lesser degree on the sound pressure level (loudness, volume) of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases. For instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder. See Pitch and frequency (last paragraph, "Pitch depends to a lesser degree...").

Main formula

${\displaystyle m={\frac {\log r}{\log b}}\cdot d={\frac {\ln r}{\ln b}}\cdot d}$

where m is the magnitude of tempered interval, r the ratio interval, b the base interval (2 for octave) and d the divisor of base interval (1200 for cents).
Here are some examples of the formula:

1. Ratio interval r = 32, and unit cent (b = 2 and d = 1200), then m = 701.955... and t(32) = 701.955 cents.
2. Ratio interval r = 32, and unit hundredth of the whole tone 98 (b = 98 and d = 100), then m = 344.247... and t(32) = 344.247 hundredths of the whole tone.
3. Ratio interval r = 2 (octave) and unit hundredth of the whole tone 98 (b = 98 and d = 100), then m = 588.4949... and t(2) = 588.4949 hundredths of the whole tone.
4. Ratio interval r = 2 (octave), and unit hundredth of Pythagorean limma (b = 256243 and d = 100), then m = 1330 and t(2) = 1330 hundredths of Pythagorean limma.
5. Ratio interval r = 800729 (idiosyncratic of Byzantine music), and unit tenth of Pythagorean comma (b = 531441524288 and d = 10), then m = 68.58... and t(800729) = 68.58 tenths of Pythagorean comma.
6. Ratio interval r = 256243 (Pythagorean limma), and unit hundredth of the fourth (b = 43 and d = 100), then m = 18.1158... and t(r) = 18.1158 hundredths of the fourth.
7. Ratio interval r = 98 (whole tone), and unit hundredth of the fourth (b = 43 and d = 100), then m = 40.942... and t(98) = 40.942 hundredths of the fourth.
8. Ratio interval r = 1.359 (1 plus random decimal 359), and unit cent (b = 2 and d = 1200), then m = 531.05... and t(1.359) = 531.05 cents.
9. Ratio interval r = 0.052 (random decimal part), and unit cent (b = 2 and d = 1200), then m = -5118 and t(0.052) = −5118 cents.
10. Ratio interval r = 1, and unit cent (b = 2 and d = 1200), then m = 0 and t(1) = 0 cents.
11. Ratio interval r = 2 (octave), and unit thousandth of tempered 10 (b = 10 and d = 1000), then m = 301.03 and t(2) = 301.03 1000ths of t(10).
12. With natural logarithm (ln): Ratio interval r = 2 (octave) and unit thousandth of tempered e (b = e = 2.71828... and d = 1000), then m = 693 1000ths of t(e) with natural logarithm.

Remarks:
1. For 11th example: log(10) = 1, so log(b) (the unit) becomes 1 and is ignored. So we could call it "absolute" or "abstract" unit with log.
2. For 12th example: ln(e) = 1, so ln(b) (the unit) becomes 1 and is ignored. So we could call it "absolute" or "abstract" unit with ln.

Derived formula

${\displaystyle r=b^{\frac {m}{d}}}$

where r, b, d, m are as in the main formula, above. This can be obtained by taking the exponent of each side in the main formula.

1. When b = 2 (octave), d = 1200, m = 100 (tempered semitone), then r = 1.0594630943592953, ratio interval of tempered semitone.
2. When b = 9 8 (whole tone), d = 100, m = 344.247..., then r = 1.5 = 32 (perfect fifth). Compare example 2 of main formula.
3. When b = 256243, d = 100, m = 226, then r = 1.125 = 98 (whole tone).
4. When b = 256243, d = 100, m = 1330, then r = 2 (octave). Compare example 4 of main formula.
5. When b = 256243, d = 100, m = 655 (random number), then r = 1.406859...
6. When t(r) = −351 (random) cents (b = 2, d = 1200), then r = 0.816485969595...
7. Let note NH = 250 Hz, and we need some note's frequency NH plus 98, plus 800729, minus (descending diesis) 13 of t(256243) (tempered Pythagorean lemma). Calculation is following:

Desired frequency = 250 Hz × 98 × 800729 × (256243)⁻¹³ = 250 Hz × 1.125 × 1.097393681 × 0.982778 = 303.3 Hz.

8. When b = 10, d = 1000, m = 301.03, then r = 2 (octave).

Notes

• The numbers r, m, b, d could be random under following conditions:
  1. The magnitude (m) can be any real number.
  2. The ratio interval (r) must be a positive number: r > 0.
  3. The base interval (b) must be greater than unity: b > 1.
  4. The divisor (d) must be a natural number: d = 1, 2, 3, ...
• The relations between r and m are:
  • When 0 < r < 1, then m < 0.
  • When r = 1, then m = 0.
  • When r > 1, then m > 0.

as with logarithms in general.