The Idea

In my projects, I use pitched tempo to find the fundamental frequency of my just intonation system.

For example, 120 BPM equals 2 Hz (120 ÷ 60). Octaved up: 4, 8, 16, 32 Hz, etc. These frequencies are assigned to the 2^n/1 ratio of the system.

From there, I build the other intervals in just intonation:

  • Fifth: 3/2 × 32 Hz → 48 Hz
  • Minor third: 6/5 × 32 Hz → 38.4 Hz
  • Major third: 5/4 × 32 Hz → 40 Hz
  • etc.

Some Tempos and their Fundamentals

| BPM | Base frequency | Practical octaves | |-----|----------------|-------------------| | 60 | 1 Hz | 32, 64, 128 Hz | | 90 | 1.5 Hz | 48, 96, 192 Hz | | 120 | 2 Hz | 32, 64, 128 Hz | | 150 | 2.5 Hz | 40, 80, 160 Hz | | 180 | 3 Hz | 48, 96, 192 Hz |

Mixing Application

In theory, mixing a track at 120 BPM with another at 125 BPM should keep the keys in tune, since the pitch shift applied to match tempos corresponds to the frequency shift of the system.

(In practice I never do this, but it should work.)

Fast Rolls

Something more interesting: fast rolls become tuned!

For example, at 120 BPM:

  • Tempo at 2 Hz
  • Sixteenth note at 8 Hz
  • Thirty-second note (1/32) at 16 Hz
  • 1/64 at 32 Hz → starts to become audible

Honestly I don't know if it has much effect. I don't think many people will think "oh hey the roll is in tune", but it's nice in principle.

Further Reading

Some articles that cover this concept in more depth: