Various internet trolls will argue, quite violently, that A=432 Hz is a dirty, stinking lie – and that there is no historical record of it ever being… bla, bla, bla.
While we agree that A is not the foundation of resonance (B-flat is), the “A” note, nevertheless, at 432 Hz nonetheless is a harmonic of our B-flat reference pitch – and there is at least one documentation of this which survives to this day.
- In the second column, Zarlino’s frequencies in the first column have been divided by 2 a few times to bring them into familiar octaves
- The third column adds up the digits of each frequency according to “gematria” and reveals that most of them factor to 9
- The fourth column indicates (in red) if these frequencies exactly match the harmonic frequency for that note as we determined it, from our “magic” base frequencies of 5.4 Hz and 7.2 Hz – and indicates in black if they are close matches
- The last column highlights in red where Zarlino’s frequencies match our frequency for that note when a different sequence of harmonics is used than the one we chose
Let’s examine the frequencies from Zarlino’s table, to see how he come up with the notes of his scale.
It does seem that he used A = 432 Hz as his “generator” frequency for the the rest of his just-intonation scale:
- A as 432 Hz / 3 x 2 (3rd harmonic below) = 288 Hz: Zarlino D
- D as 288 Hz / 9 x 8 (9th below) = 256 Hz: Zarlino C
- C as 256 Hz * 3/2 (3rd harmonic) = 384 Hz: Zarlino G
- A as 432 Hz x 3/4 (3rd harmonic above) = 324 Hz: Zarlino E
- A as 432 Hz x 9/8 (9th harmonic) = 486 Hz: Zarlino B
- B as 486 Hz x 9/8 (9th harmonic) = 546.75 Hz: Zarlino C#
- B as 486 Hz x 3/4 (3rd harmonic) = 364.5 Hz: Zarlino F#
In sequence, this 8-note collection of Zarlino harmonic notes is: A B C C# D E F# G
Zarlino was the first advocate in Western culture, that I know about – unless you count Pythagoras, for Just-Intonation. He used whole-numbers to derive a scale based on harmonics of a root note.
So, it seems that Zarlino based his harmonic, just intonation scale on A = 432 Hz. Where he got the idea that A should be 432 Hz, I don’t know. I got my idea of A = 432 Hz as a harmonic by-product of the 3 frequencies I “found” on a frequency generator (A is the 5th harmonic of my F note (345.6 Hz * 5/4 = 432 Hz)). Perhaps if I can read his book in Italian, I’ll be able to find out.
So this is a piece of corroborating evidence from history that we are on the right track with these frequencies. And also, an interesting insight that 432 Hz has a known historical provenance in Western music history going back to at least the 16th Century. So, it’s older than Verdi – and actually, as we’ve discussed elsewhere, it’s there because as a naturally occurring phenomenon, different cultures have found different ways of discovering it, throughout history.
Here’s a little of my not-very-good playing of a “harpsichord” in Apple Logic tuned to Zarlino’s frequencies, and playing only his notes. It’s quite a nice-sounding tuning, enjoy!