You may quote me: "I have gravitated towards a digital keyboard because I find it easier to push buttons to switch temperaments than use a saw and hammer on my other instruments".
How many 432ers know that before 1834 there was no way to measure cycles per second? The whole concept of A = 432 Hz as some sort of musical magic (a right-brained fantasy) is based on a supposed connection between pitch, Vedic numbers, Pythagoras's Music of the Spheres, and the Western system of measuring pitch in Hertz values which are cycles per second. Hertz or Hz references were completely unknown to the Grand Masters of Classical music who predated 1834, often by centuries - Vivaldi, Handel, Bach, Hayden, Mozart, Beethoven, Schubert, Schumann, Mendelsohn, Chopin. None of them used the same pitch for tuning or if they did, it was by sheer accident. 'A' could have been anywhere between 425 and 450 and they wouldn't have even known its reference value. Claims that Vivaldi used 432 Hz are just nonsense!
At first I, too, thought that the "second" was arbitrary, nothing but yet another human construct. It turns out that it may not be after all. But it still doesn't make a clear-cut difference without exceptions.
Upon inquiry I was surprised to find a correlating time unit in... wait for it... Vedic cosmology, also connected with 108, 432, etc. Without going too far into the philosophy, the Vedas are held to be composed - not by humans - but by God or empowered persons (specifically Vyasadeva aka Vedavyasa - I guess for those of Abrahamic faiths, somewhat like Moses and the Ten Commandments). So, in Hindu, Jain and Buddhists' etc minds, the time unit used wouldn't be considered arbitrary at all, but on the contrary, "God-given".
There are several Kala (Vedic time) systems depending on what time periods you are measuring - the blink of an eye, a breath, a day; a month, a year, the manifestation of the material world. Their system of diurnal timekeeping is different yet similar enough to end on a proportionate unit. To start with they divide the day not into 24:60:60 hours:minutes:seconds but with 30 and 2 (to begin with) to get a muhurta then a ghati or danda of 24 minutes. From here it varies from system to system. One I used divides a muhurta of 48 minutes into 30 laghu of 96 secs, then by 15, 5, and 3 down to the respective units of a kashtha of 6.4 seconds, a kshana of 1.28 sec and a nimesha of 426 milliseconds.
Straight away I'm thinking it may not take much juggling to convert 1.28 into harmonic equivalents of the frequencies we are dealing with (read: "want"). Instead of cycles per second we now have cycles per kshana. Surely if we are going to use the "sacred geometry of the Vedas" as a basis for our magical thinking then our pitch will have to be based on the orbital frequency of the Earth and a pure tuning system like Just Intonation instead of Equal Temperament, using cycles per kshana instead of cycles per second (cps or Hz)!
Turns out not much of it makes sense. Drat!
432Hz = 552.96 cycles per kshana; and 256Hz = 327.68 c.p.k.
Not bad, close to a C# and an E respectively. Sort of.
So, using 432 cycles per kshana what do we get? (Comparisons are to note values from 432Hz ET)
432 cpk = 337.5 Hz (halfway b/n a G and an F#); and
256 cpk = 400 Hz (somewhere b/n G and G#).
Forget it! However, if we use another system (I have no idea what they're called) based on 30:2:30:2:6:10 we get a better result. Incidentally that ratio is 1/3600th, showing promise for whole integers, common demoninators, etc that 432ers fancy. The previous system is 1/3375ths, probably why it went nowhere as far as the results we are looking for... 426 milliseconds ain't gonna cut it.
For want of a better name, the muhurta-ghati-kala-pala-prana-vipala (30:2:30:2:6:10) method gives either a prana of 4 seconds or a vipala of 0.4 sec. It should be obvious that by using the larger prana unit of 4 seconds, all we are in fact doing is changing the octave. This, and only this, is where the connection with the Western second to the Vedic prana kicks in - 432 cycles per prana becomes A6 @ 1728Hz, the A two octaves higher.
It also becomes apparent that to use the smaller vipala for calculations as it means you are converting by 2/5ths which will lead nowhere, eg:
432Hz x 0.4 sec (1 vipala) = 172.8 cpv (cycles per vipala), so for the same pitch referenced in Vedic terms instead of the Western 'arbitrary' second:
432Hz = 172.8 cpv (F? 171.4)
512Hz = 204.8 cpv (G#? 203.9)
Again referenced to A=432Hz ET. But if we reference to Just Intonation with C = 256:
432Hz = 172.8 cpv (F? 170.6)
512Hz = 204.8 cpv (G# exactly - finally, a direct hit!)
Close, but... nah, nothing like the results we get converting the frequency of the Earth's orbit around the Sun into cycles per seconds: 1 ÷ 31556925.250732 = 0.0000000317; which raised to the 32nd octave = 136.1Hz, EXACTLY C# in the A=432Hz Equal Temperament scale - the so-called Om note (ॐ)!
By the way, do you know how traditionally they used to tune up (Indian musicians) before a concert? They set pitch to the singer – what ever it is, it then becomes the shadja. Fair dinkum. Now, THAT is truly ARBITRARY! Legend also has it that the shadja (the root note, usually around C#) is found by the "orchestra guru" meditating on the Heart chakra which supposedly vibrates to the Earth's orbital frequency. He then hums it and the ensemble tunes to that. Hard to believe I know but in days of yore they lit fire sacrifices with mantras, not matches.
Conclusion:
108 cycles per prana = 432Hz; but it doesn't make a lot of sense with the Indian scale which is based on the Shadja!
This is hard to put into words succinctly. When we use the prana unit, a conflict arises when trying to establish a correlation between natural harmony and the vibrational frequency of 432Hz tuning using Vedic units of time measurement and 'Sacred Geometry' i.e. the 'magical properties' of whole integers such as 256 and 432. These represent C and A in the Pythagorean Temperament but are derived from the arbitrary Western second. Trying to use Vedic units of vipala, kshana, etc. however, it completely breaks down.
Yet another conflict arises where although Pythagorean (and derivatives thereof - the ONLY temperament with A & C @432 & 256) uses whole integer ratios, it actually sounds pretty terrible imho (unlike JI which also uses whole integer ratios but has pure thirds and is very harmonious despite its wide 4ths and narrow 5ths). Equal Temperament based on A=432Hz sounds a lot better than PythT but then every note in the scale is technically out of tune with the A - vibrating on a frequency that does NOT harmonically concur, causing beating.
Even if we do use the prana unit, which corresponds with A=432Hz frequencies (being a few octaves higher) and the traditional Indian method of tuning (of which Just Intonation and Ptolemaic Intense Diatonic Scale are mere simplifications) it breaks down again -depending on which 'magic number' 256 or 432 we chose as C or A - because a C of 256 gives an A of 426.6; and an A of 432 creates C = 256.9 (close, but again, no cigar! Remember we're talking about harmonics converging via whole integer ratios etc).
It is at about this point that I leave the desk and wander over to my keyboard and play some soothing chords in meantone. Ahhh, I just love those pure thirds!
