The official proposal for e-pi tuning (note: UTF-8 encoding, includes Hebrew characters)

I mentioned this on tuning-math a little bit. I had been experimenting off-and-on with tuning systems involving transcendent numbers. The most famous of these constants are e (~ 2.71828) and pi (~ 3.14159). I tried different equal divisions of both e and pi, but I found something else that interested me: if I produce a Pythagorean-type scale using pi (or e) as the period, and e (or pi/e) as the generator, I get something like 53-EDO, but with the octave stretched by about 7.3 cents.

So here is my tuning explained in brief:

The "octave" is defined as e/1, approximately 1731.23 cents. Pi/1, about 1981.80 cents, can be called a "nonave".

The difference between these two intervals, pi/e (~ 250.561 cents), is called a "tone". Because there are almost seven tones in an "octave" (about 6.90942 tones, precisely), I chose the name for e/1.

But seven tones is sharp of an octave by pi^7/e^8, about 22.6951 cents. Because of its similarity to both the Pythagorean and syntonic commas, as well as the 53rd root of 2 (the "Turkish comma"), I call it the "comma" as well. There are about 11.0403 commas in a tone.

Finally, one special small interval, e^87/pi^76, is about 0.915261 cents. I call that the "schisma". There are about 24.7963 schismas in a comma.

The nominals for the tones could be the letters A through G. But I personally like to use the first seven letters of the common Semitic alphabet to avoid confusion with the conventional understanding of the note names. I'll use Hebrew letters here (but their Arabic or Syriac equivalents could be used):

א aleph ~ 0.000 cents
ב beth ~ 250.561
ג gimel ~ 501.123
ד daleth ~ 751.684
ה he ~ 1002.245
ו waw ~ 1252.807
ז zayin ~ 1503.368

The distance between each of these seven tones is the same, but between ז and א, it's a tone minus a comma, a "small tone" of around 227.866 cents.

For comma differences, I haven't really come up with a good notation, but right now I'll use +/- number of commas. For example, the stretched octave I mentioned earlier is notated ו‎-2, or waw minus two commas. Schisma shifts are notated in a similar manner, but if a tone is raised or lowered by schismas but not commas, it is necessary to add +0 or -0 as a comma modifier.

Finally, about standard tuning. Since the scale is not octave based (not 2/1-octave based anyway), I chose to define "middle aleph" according to Planck's time, a fundamental in quantum physics. (It's the amount of time a photon takes to travel the length of itself - about 5.39121×10^-44 seconds, give or take.) My recommended tuning for middle-aleph is Planck's frequency divided by e^94, or around 278.376 Hz. (That's about a middle C-sharp.)

This tuning I tentatively call "e-pi tuning", unless you want to call it something else. Like the "Geek scale". ;)

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> I mentioned this on tuning-math a little bit. I had been experimenting
> off-and-on with tuning systems involving transcendent numbers.

The only transcendental ratios I know of which have an actual rational
(if you'll pardon the expression) basis are the metallic generators,
the best known of course being the Golden fifth of size
2^((15-sqrt(5))/22). By the Gelfond-Schneider theorem, this is a
transcendental number and the same is true of the whole metallic
family. Aside from that, we have the Lucy-tuned major third of
2^(1/pi), which is almost certainly a transcendental number, though I
know of no proof, and Margo's e-fifth, which is 2^((4+3e)/(7+5e)),
again almost certainly a transcendental number.

Taken as a musical interval, pi is pretty much random. It is flat from
22/7 by less than a cent, which pretty well defines it, I suppose. The
22/7 intervals of 46 or 43 are pretty close, and octaves for these
could be adjusted to make this exact, especially if no one asked why
in the world you would want to. 66 comes much closer yet, but is not a
big name in scale divisions.

e, on the other hand, has an interesting continued fraction, as do
related numbers such as tan(1), (e^2+1)/(e^2-1), (e+1)/(e-1) etc.
Possilbly something could be made of that but I can't imagine what.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Taken as a musical interval, pi is pretty much random. It is flat from
> 22/7 by less than a cent, which pretty well defines it, I suppose. The
> 22/7 intervals of 46 or 43 are pretty close, and octaves for these
> could be adjusted to make this exact, especially if no one asked why
> in the world you would want to. 66 comes much closer yet, but is not a
> big name in scale divisions.

Speaking of pi and 43, 43 is an 11-limit meantone of the "huygens"
variety, which means it has augmented triads of 5/4-5/4-14/11, so that
(11/7)/(5/4)^2 = 176/175 is tempered out. If we insist for some
bizarre reason that the 22/7 of this system be exactly pi, then the
major third
must be sqrt(pi/2), so the fifth has to be (8 pi)^(1/8). This is a
fine pi in the eye fifth for anyone bored with Lucy tuning; it's close
to 1/5 comma (12/61 comma if you want to get fussy about it) and even
closer to 43-et, over which it holds no advantage at all, unless you
think the circle of fifths not closing and the fifth being a
transcendental number are advantages.