Music and mathematics: ratios, cycles and useful models

Intervals compare proportions

Suppose one tone has frequency 200 Hz and another 400 Hz. Their ratio is 2:1. The pair 300 Hz and 600 Hz has the same ratio even though its numerical difference is larger. In the usual octave-based description, both pairs span an octave. Adding a fixed number of hertz therefore does not preserve a musical interval across registers; multiplying by a fixed ratio does.

This is useful when transposing a mathematical example. If a melody model uses frequencies 200, 250 and 300 Hz, multiplying all three by 1.2 gives 240, 300 and 360 Hz. The ratios between corresponding notes remain unchanged. This calculation concerns a simplified frequency model, not a guarantee that an instrument or singer will retain identical timbre, articulation or perceived effort in the new register.

References: [1]

State the tuning system before calculating

Twelve-tone equal temperament divides an octave into twelve equal ratios. If r is the ratio of one step, then r multiplied by itself twelve times must equal 2. Therefore r = 2^(1/12), approximately 1.05946. From a reference frequency f, a note n steps higher has frequency f × 2^(n/12). This is a definition of a particular tuning model, not a claim that every musical culture uses it.

Cents offer a logarithmic unit for comparing intervals: 1200 × log2(f2/f1) gives the size in cents between positive frequencies f1 and f2. An octave has 1200 cents, and an equal-tempered semitone has 100. A decimal measurement can make comparison precise while leaving the musical judgement open. A small difference from an equal-tempered reference is not automatically an error in a performance that uses other intonational practices.

References: [2]

Count positions, then hear the pattern

For a constant pulse of b beats per minute, one beat lasts 60/b seconds. At 75 beats per minute, a beat lasts 0.8 seconds and an eight-beat cycle lasts 6.4 seconds. If four evenly spaced events fit into each beat, their spacing is 0.2 seconds. Always say which level your number describes: beat rate, subdivision rate and phrase length are not interchangeable.

A common-grid calculation can clarify a 3-against-2 relationship. Divide a shared span into six equal positions. One layer attacks at positions 0, 2 and 4; another attacks at 0 and 3. Both begin together again at position 6. The arithmetic locates attacks, but it does not specify accent, timbre, expressive timing or how a listener will group the result. Those remain musical questions to explore by listening.

Do not confuse a calculation with a complete explanation

In acoustics, the difference between two nearby frequencies can produce an audible beating rate. Two steady tones at 300 and 303 Hz provide a simplified example of a three-per-second amplitude fluctuation. These acoustic beats are a different use of the word from a musical pulse. Keeping the terms separate prevents an attractive equation from answering the wrong question.

The same care applies to pattern counting. A system may generate hundreds of pitch sequences, but that does not establish hundreds of meaningful ragas, compositions or culturally recognised forms. A useful research habit is to write three things beside every formula: what it represents, which assumptions make it valid, and which aspects of the music it leaves out. The resulting model is easier to test, explain and improve.

References: [3]

What to remember

• Frequency ratios preserve interval relationships across registers.
• Tuning calculations require explicit assumptions and units.
• Arithmetic describes selected structure; musical meaning requires further evidence.