A Couple of Useful Formulas in Microtonality

If you have read my last two blogs, you noticed how I provided interval size measurements using units of cents. But, how did I calculate them and how can you? In this blog, I will discuss the two major formulas that have helped me as a student in converting ratios to cents and vice versa, along with a procedure to approximate decimal figure ratio into a fraction-form called “continued fractions”.

Note, it is very efficient to review fractions, decimals, exponents and logarithms, also some algebra, from Khan Academy.

While I had people teaching me microtonality, I spent time reviewing high-school pre-algebra content that also helped me understand the formulas verbally and mathematically.

Before we proceed, what are cents? It is "a unit of interval measurement invented by Alexander Ellis and appearing in his appendix to his translation of Helmhotz's On the Sensations of Tone [1875] (p. 41, and Appendix XX Section C, p 446-451, in the 1954 Dover edition)" This information was quoted from Monzo, 2018.

The formulas:

CENTS = 1200*log_2(frequency ratio) where log_2 means “logarithm base 2”.

In this formula 1200 appears because an octave is defined as 1200 cents, and 2 appears because an octave is the 2:1 frequency ratio.

For example, the 2nd harmonic (check the previous blog) is an octave above the fundamental, forming a 2:1 frequency ratio with it.

log_2(2) = 1

1200*1 = 1200 cents and so, the 2:1 ratio is 1 octave, as expected.

Many calculators only provide logarithms base 10. Using the logarithmic properties, we can express logs base 2 using only logs base 10 (this is an application of the “change of base formula” (Khan Academy, n.d.)):

log_2(ratio) = log_10(ratio)/log_10(2) (Monzo, 2018)

For another example, let’s look at the interval from the fundamental to the 43rd harmonic, which is 43:1

1200*log_2(43) = 6511.51771 cents.

1 octave = 1200 cents which means we can easily determine how many octaves fit within the interval we got — 6511.51771 cents. To wit, 1200 + 1200 + 1200 + 1200 + 1200 = 6000 cents, which is 5 octaves, so the interval is between 5 and 6 octaves.

6511.51771 - 6000 = 511.51771 cents. Conventionally speaking, just 11.5 cents above 12-edo perfect fourth which is 500 cents. So, in the harmonic series (see previous blog), the 43rd harmonic is 5 octaves plus 511.5 cents above the fundamental.

Why use log_2? Simple, because 2:1 or 2/1 comes out as 1 meaning 1 octave, and the 32nd harmonic or 32/1 comes out as 5 octaves. I will post a screenshot in that regard:

(Monzo, 2011)

The second formula is RATIO = 2^(cents/1200) (Monzo, 2018)

Now, this is again because 2 and 1200 are related, meaning that 2 is the frequency ratio that gives us 1200 cents.

Using logarithmic properties, we can express this in an infinite number of different ways.

For example, RATIO = 3^(cents/1901.955), since 3:1 frequency ratio = 1901.955 cents, 4^(cents/2400), and so on.

Going back now to the first formula,

CENTS = 1200*log_2(ratio), we can write that as SEMITONES = 12*log_2(ratio), so for example the interval from the fundamental to the 3rd harmonic is 12*log_2(3) = 19.01955 semitones (or 19.01955 degrees of 12edo).

And of course, 19.01955 semitones * 100 = 1901.955 cents.

In 12-edo tritone is half of an octave, so we can split say C1-C2 into two equal intervals. F#1 or Gb1 is 6 semitones above C1 and 6 semitones below C2. The frequency ratio F#1 or Gb1 forms with C1 or C2 is therefore 2^(600/1200). 600/1200 = 0.5 and so 2^0.5 = 1.41421356 (which is the square root of 2) is the frequency ratio for the 12-edo tritone.

We use logarithmic interval measures like cents and semitones because the logarithm converts multiplication of frequency ratios (as when stacking them) into addition, and division of frequency ratios into subtraction, reflecting much more directly and easily how musicians hear and reckon with intervals (Paul Erlich, personal communication, June 25th, 2019).

So, in addition we say 1+1 = 2 -- this doesn’t work intuitively with frequency ratios – it would appear to imply that a unison plus a unison is an octave, which is wrong. However, 1.41421356*1.41421356 = 2 which means stacking of 2 tritones such as C1-F#1 and F#1-C2 has an outer interval (C1-C2) equal to an octave.

Let’s look at the two formulas’ algebraic relationship – they are equivalent ways of saying the same thing. If 1200*log_2(ratio) = cents, dividing both sizes of the equation turns it into log_2(ratio) = cents/1200. Raising 2 to the power of equal exponents gives equal results, so we can write that as 2^(log_2(ratio)) = 2^(cents/1200). Now by the definition of log base 2, the log and exponentiation on the left cancel one another out, giving ratio = 2^(cents/1200) (Weisstein, n.d.).

We have seen that using a large number for the denominator and a small number for the numerator gives us a positive integer i.e. 3/2 just means 3rd harmonic over the 2nd harmonic an approximated answer 701.955 cents. However, having the inverse 2/3 would give us a negative 701.955 cents. An intuitive way of looking at this is having the large harmonic “higher” in pitch above the fraction symbol (—) and the lower harmonic “lower” in pitch below the fraction symbol (—). Since the 3rd harmonic is G4 (approximately) and 2nd harmonic is C4 if our fundamental remains C3. We can conclude that G4 is higher in pitch than C4 having the higher harmonic above and the lower harmonic below.

While I was learning how to find cents and ratios, something bothered me in regards on how I always had decimal figures instead of fractions or whole numbers. I was then told about a method of converting decimals into fractions using a mathematical concept called “continued fractions”. This turned out to be useful and it gave many decent approximations of decimal figures. For example, 600 cents does not exist in the harmonic series but can be approximated. We convert 600 into ratio form, 2^(600/1200) = 1.41421356, we then use this as our input in doing continued fractions.

I learned how to do it using this simple video: Infinite fractions and the most irrational number (Polster, 2016).

After having to do enough of them I started using a calculator called "A continued fraction calculator" to save myself some time (Dr. Knott, 2016).

REFERENCES:

Dr. Knott, R. (2016). A continued fraction calculator. Retrieved from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html

Khan Academy. (n.d.). Exponentials & logarithms. Retrieved from https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions

Monzo, J. (2011). 12-Tone Equal-Temperament. Retrieved from http://tonalsoft.com/enc/number/12edo.aspx

Monzo, J. (2018). Cent, ¢, 1200-ed2. Retrieved from http://tonalsoft.com/enc/c/cent.aspx

Polster, B. [Mathologer]. (2016, July 30th). Infinite fractions and the most irrational number [Video File]. Retrieved from https://www.youtube.com/watch?v=CaasbfdJdJg

Weisstein, E. W. (n.d.). Logarithm. Retrieved from http://mathworld.wolfram.com/Logarithm.html

APPENDIX:

Frequency and Pitch. (n.d.). Retrieved from http://www.animations.physics.unsw.edu.au/jw/frequency-pitch-sound.htm

John Napier (1550-1617) had discovered logarithms which has been a powerful and useful tool till today.

Loy, G. (2006). Musimathics: The Mathematical Foundations of Music, Volume 1. Cambridge, Massachusetts: The MIT Press

Scales: Just vs Equal Temperament. (n.d.). Retrieved from https://pages.mtu.edu/~suits/scales.html

Here below is my one of my mentors, friend, and a modern microtonalist composer who uses cents to define a pitch of a note for his pieces, Johnny Reinhard.