- In the second column, the 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, numerologically 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 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 the frequency of that note if it is determined via a different harmonic approach than the one we chose
- Being as Zarlino was an intelligent fellow, we’ll keep track of these frequencies in case they are useful when it comes to the practicalities of building a Just Intonation musical instrument
To summarize the findings – of the eight notes that Zarlino lists, A, B, C, C#, D, E, F#, G:
- We exactly match A, B, D, E
- And we exactly match C#, F#, G if we calculate these harmonics still starting from B-flat, but using an alternative harmonic of harmonics
- The only one that’s a miss is C – probably because he perhaps wanted to take the simplistic approach that 1 x 2 = 2, 2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16, 16 x 2 = 32,
32 x 2 = 64, 64 x 2 = 128, and 128 x 2 = 256. In other words, for this one note, he seems to have determined this note mathematically – (and perhaps this is the correct frequency. We’ll keep that one in our hat, also)
I’m not going to argue with Zarlino on the correct way to determine the harmonic frequencies – what is significant here is that we came up with a bunch of musical frequencies based on a chance encounter with vibration in a Johannesburg hotel room – and those frequencies, one way or another, also match what Zarlino recorded in his book published in 1558.
How is this possible? Well, first of all, Zarlino was the first advocate in Western culture for Just Intonation. Secondly, he must have started with the same frequencies as I did in order to calculate his just, harmonic series; either that, or the frequencies were passed down to him from some earlier culture. I will have to read his book to find out.
Just another confirmation from history that we are on the right track with these frequencies – and good to point out to the internet trolls (professional or otherwise) that you “cannot prove a negative” – as we used to say when I was 8. In other words, you can’t prove that something doesn’t exist. All you can prove is that you haven’t found that evidence of a documentation of A = 432 Hz – but we have! – and it matches and corroborates our findings.