EDphi6 tuning with adjusted spectrum
An example of a piece I made with this is here: droned
As a starting point, I think phi
, in the context of harmonic spectra, is mainly interesting to generate maximum
dissonance. Isn't it true that phi can be interpreted as one of the hardest
numbers to approach by a ratio of small integers?
I next started thinking about adjusting the harmonic spectrum to be in a better
concordance with phi. I came up with the following construction:
Take the regular harmonic spectrum, designated as:
1 2 3 4 5 6 ...
then lower all partials by the relative amount 2-phi or 0.38197 to get the series:
0.61803 1.61803 2.61803 ....
For ease of notation of this series here, let's define phiN (N = 0 1 2 ...) to be phi + N - 1 (so phi1 == phi):
phi0 phi1 phi2 phi3 ...
The first two intervals generated by this series are:
phi1 / phi0 == phi2 == phi^2
phi2 / phi0 == phi^3 (which makes phi2 / phi1 exactly phi)
These nice 'coincidences' are of course completely in the line of phi properties. So if we start with phi as basic interval of a scale (which I gather has been adopted as 'phitave' by some), these intervals are trivially accounted for.
The next two are:
phi3 / phi0 = 5.8541 which, as an interval is 3.67228 phitave
phi4 / phi0 = 7.47214 --- 4.17941 phitave
To accommodate these intervals in an ET, we have to find an optimal division
of the phitave to approximate these fractions (.67228 and .17941). Obviously,
6 is a surprisingly simple and good candidate.
If we express the series as steps in an equal division of the phitave by 6,
we get:
0 12 18 22.034 25.076
Of course, instead of lowering the partials by 2-phi, we can also increase
by (phi-1), giving the same series, but without the first, phi0
1.61803 2.61803 3.61803 ....
With the same basic properties.
Please take note of the fact that we have a musically correlated sound quality with a related scale construction, without any reference to regular JI intervals, especially the octave.
Now about ways to obtain that spectrum. FM would be a solution, if you avoid reflected sidebands.
Another nice way is shifting the partials of any generated harmonic sound by completing the analytic signal it stands for. We have to fabricate the imaginary part of the signal, which we can do with a Hilbert transform. We can then shift the complete spectrum by multiplying with a fixed frequency complex signal. This is easily done in Csound.
; suppose p4 is desired fundamental
frequency
idfr = - 0.3819660113 * p4
ifr = p4 - idfr
;
; generate any harmonic sound with fundamental frequency ifr into a1
;
areal, aimag hilbert a1
asin1 oscili 1, idfr, 1
acos1 oscili 1, idfr, 1, 0.25
ash1 = areal * acos1 - aimag * asin1
; ash1 is the shifted spectrum with p4 as shifted fundamental
<CsScore>
f 1 0 16384 10 1
----------------------
Now that I have integrated sound synthesis in IL
I’m doing it like this:
shiftdown =
"
sndfun = %1; frq = %2;
idfr = (phi - 2) * frq;
ifr = frq - idfr;
snd = sndfun( ifr );
csnd = hilbert( snd );
ph = phasor( idfr );
shsnd = re( ph * csnd );
";
shiftup =
"
sndfun = %1; frq = %2;
idfr = (phi-1) * frq;
ifr = frq - idfr;
snd = sndfun( ifr );
csnd = hilbert( snd );
ph = phasor( idfr );
shsnd = re( ph * csnd );
";