What tuning is this?

It's a twelve tone scale and it's not equal temperament, pythagorean tuning or just intonation.
I just divided the frequency of the unison by twelve and multiplied the unison by the product for example: frequency 261.6255hz/12 = 21.802125 x 261.6255hz = 283.427625hz x 21.802125 = 305.22975
It continues that way until the octave is reached.

--- In tuning-math@yahoogroups.com, "andrewstevenmolina" <andrewstevenmolina@...> wrote:
>
> It's a twelve tone scale and it's not equal temperament, pythagorean tuning or just intonation.
> I just divided the frequency of the unison by twelve and multiplied the unison by the product for example: frequency 261.6255hz/12 = 21.802125 x 261.6255hz = 283.427625hz x 21.802125 = 305.22975
> It continues that way until the octave is reached.
>

It appears that you are not multiplying the unison, but *adding* to it. So your scale is in fact the harmonic series starting with 12. The first octave is therefore 12:13:14:15:16:17:18:19:20:21:22:23:24.

This means that there will be 12 notes in the first octave, 24 notes in the second octave, 48 notes in the third octave, and so on. In other words, your scale becomes more dense in the upper registers.

Ryan Avella

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "andrewstevenmolina" <andrewstevenmolina@> wrote:
> >
> > It's a twelve tone scale and it's not equal temperament, pythagorean tuning or just intonation.
> > I just divided the frequency of the unison by twelve and multiplied the unison by the product for example: frequency 261.6255hz/12 = 21.802125 x 261.6255hz = 283.427625hz x 21.802125 = 305.22975
> > It continues that way until the octave is reached.
> >
>
>
> It appears that you are not multiplying the unison, but *adding* to it. So your scale is in fact the harmonic series starting with 12. The first octave is therefore 12:13:14:15:16:17:18:19:20:21:22:23:24.
>
> This means that there will be 12 notes in the first octave, 24 notes in the second octave, 48 notes in the third octave, and so on. In other words, your scale becomes more dense in the upper registers.
>
>
> Ryan Avella
>
I'm sorry your right I meant adding, so it would be instead 261.6255hz/12 = 21.802125 + 261.6255hz = 283.427625 + 21.802125 = 305.22975

>
> This means that there will be 12 notes in the first octave,
> 24 notes in the second octave, 48 notes in the third octave,
> and so on. In other words, your scale becomes more dense
> in the upper registers.
>
>
> Ryan Avella

Or he could octave-repeat that first series of 12 harmonics, as Johnny
Reinhardt does with his harmonic-series scales. So, the size (measured
in Hertz) of the scale steps would double with each octave.

Just another way to approach it, if a 12-step scale is what he's after.

Jason Conklin

Hello Group,

I'm curious if anybody has tried or found any scales in which all the ratios are transcendental. Since just intonation is with rational numbers and equal temperament is with algebraic numbers, it seems like something new to try. For example, we could use numbers like 1/pi, 1/pi^2, sqrt(e), etc...

Chris Arthur

On Wednesday 03 July 2013 0:51:11 christopher arthur wrote:
> Hello Group,
>
> I'm curious if anybody has tried or found any scales in
> which all the ratios are transcendental. Since just
> intonation is with rational numbers and equal
> temperament is with algebraic numbers, it seems like
> something new to try. For example, we could use numbers
> like 1/pi, 1/pi^2, sqrt(e), etc...

LucyTuning is based on the whole tone being 1/pi octaves:

http://lucytune.com/

You can take any temperament and choose a trancendental
number for the generator. The formulas for optimal tunings
tend to lead to algebraic numbers. (LucyTuning is very
close to metameantone, where major thirds and fifths beat
equally. But 1/pi is easier to remember.) Trancendental-
ness isn't something you can hear.

Graham

also quadratic imaginary roots, like sqrt(-163). There might be different ways to interpret complex numbers in music, like taking complex twelfth roots of unity, and/or of -1 ...in this case, just pretend the octave might be -2, instead of -1, for example, there are lots of options, so sqrt(163) would just be like sqrt(41*4 - 1) etcPGH

To: tuning-math@yahoogroups.com
From: gbreed@gmail.com
Date: Sat, 13 Jul 2013 18:24:19 +0100
Subject: Re: [tuning-math] Transcendental numbers for tuning

On Wednesday 03 July 2013 0:51:11 christopher arthur wrote:

> Hello Group,

>

> I'm curious if anybody has tried or found any scales in

> which all the ratios are transcendental. Since just

> intonation is with rational numbers and equal

> temperament is with algebraic numbers, it seems like

> something new to try. For example, we could use numbers

> like 1/pi, 1/pi^2, sqrt(e), etc...

LucyTuning is based on the whole tone being 1/pi octaves:

http://lucytune.com/

You can take any temperament and choose a trancendental

number for the generator. The formulas for optimal tunings

tend to lead to algebraic numbers. (LucyTuning is very

close to metameantone, where major thirds and fifths beat

equally. But 1/pi is easier to remember.) Trancendental-

ness isn't something you can hear.

Graham

--- In tuning-math@yahoogroups.com, christopher arthur <chris.arthur1@...> wrote:
>
> I'm curious if anybody has tried or found any scales in which all the
> ratios are transcendental. Since just intonation is with rational
> numbers and equal temperament is with algebraic numbers, it seems
> like something new to try.

This may be off-topic, but have you considered algebraic (and hence non-transcendental) numbers like phi = (1+sqrt(5))/2, which are neither just ratios, nor based on an equal division of a just ratio?

For example, ratios based on simple "noble numbers" (= certain numbers derived from phi) are pretty good at avoiding simple just ratios, which may be useful if you're interested in dissonance.

At the same time, there's Heinz Bohlen's 833 cents scale which has ratios based on phi, and which has some interesting properties related to combination tones, and may be relevant in the case of a distorted audio signal:

http://www.huygens-fokker.org/bpsite/833cent.html

And then there are linear tunings where the generator divides the interval of equivalence by a noble number. A prominant example is golden meantone, where the generator fourth is (1+2phi)\(2+5phi), or (phi+3)\11 b  503.786 cents. Those tunings have some interesting properties regarding their MOS's, and also have the fractal property that intervals scaled by a multiple of phi are still in the same tuning.

There are still lots of other intervals based on algebraic numbers to explore.

- Gedankenwelt

P.S.: My last two messages on this list were delayed by 1 and 2 weeks, resp.; I hope this one arrives a little earlier