Spectral Mappings for CARLOS GAMMA

FFT
Spectral Mappings for CARLOS GAMMA
based on:
Tuning, Timbre, Spectrum, Scale” by William A. Sethares
and
“Tuning: At The Crossroads” by Wendy Carlos
(All calculations done with
LMSO and IntervalCalc by X.J.Scott)



Spectral mapping is a transformation from a “source” spectrum to a “destination” spectrum...it can be used to create inharmonic instruments that retain much of the tonal quality of familiar (harmonic) instruments”.

Why and when such process is desiderable?
For example, when working with nonoctave scales, harmonic instruments usually sound more “
out-of-spectrum” than “out-of-tune” meaning that “partials of the sound interfere when played at certain intervals”.

Carlos Gamma tuning system divides an interval/ratio 3/2 (701.955 cents) into 20 equal steps of 35.09777 cents each and does not repeat at the octave.

Wendy Carlos calls it “
essentially perfect (on the classical just curve)” meaning that it approximates very well many just intervals, except the octave! So, probably, it’s one of the nonoctave tuning systems that works better with harmonic instruments as I have already demonstrated with my previous experiments (Adagio Gamma and Moonlight Gamma Serenade) nonetheless this process can be tried for it too.

If s = 1 step of Carlos Gamma
s = 20th root of 3/2 = 1.02048015365

The following one is one of the many possible spectral mappings for Carlos Gamma.
The frequency ratios of the 16 partials were chosen in order to minimize the “
perceptual change” of the “destination” spectrum compared to an harmonic “source” spectrum (the frequency of the 2nd partial is close to 2 times the frequency of the fundamental, the 3rd to 3 times it et cetera).
All partials are powers of s = 20th root of 3/2 = 1.02048015365

1.02048015365^0=1
1.02048015365^34=1.9923
1.02048015365^54=2.98845
1.02048015365^68=3.96927
1.02048015365^79=4.9609
1.02048015365^88=5.9539
1.02048015365^96=7.00226
1.02048015365^103=8.06993
1.02048015365^108=8.93085
1.02048015365^114=10.086
1.02048015365^118=10.938
1.02048015365^123=12.1049
1.02048015365^127=13.1274
1.02048015365^130=13.9506
1.02048015365^134=15.129
1.02048015365^137=16.0777

caret (^) is computerese for ‘raised to the power of’

Another method for devising “
in-spectrum” partials and increase the consonance of some desirable interval is to start from such interval, for example Carlos Gamma’s repeat ratio 3/2 (1.02048015365^20=1.5) and find functional powers of it to use as “destination” partials of a sound.

Example:

s20

(1.02048015365^20=1.5)
1.5^1=1.5
1.5^2=2.25
1.5^3=3.375
1.5^4=5.0625
1.5^5=7.59375
1.5^6=11.3906
1.5^7=17.0859

Other examples:

s9

(1.02048015365^9=1.20017)
1.20017^4=2.07478
1.20017^6=2.98852
1.20017^12=8.93127

s11

(1.02048015365^11=1.24983)
1.24983^3=1.95233
1.24983^5=3.04968
1.24983^6=3.81159
1.24983^8=5.95398
1.24983^10=9.30057
1.24983^12=14.5282

s14

(1.02048015365^14=1.3282)
1.3282^5=4.13349
1.3282^7=7.29196
1.3282^8=9.68518

s23

(1.02048015365^23=1.59406)
1.59406^3=4.05055
1.59406^5=10.2926

s25

(1.02048015365^25=1.66002)
1.66002^5=12.6057

s29

(1.02048015365^29=1.80025)
1.80025^3=5.83443
1.80025^4=10.5034

s34

(1.02048015365^34=1.9923)
1.9923^1=1.9923
1.9923^2=3.96926
1.9923^3=7.90796

If this article intrigues you but sounds like Greek the best way to shed some light on this matter is to read “
Tuning, Timbre, Spectrum, Scale” by Wiliam A. Sethares, “Tuning: At The Crossroads” by Wendy Carlos and start experimenting with LMSO and IntervalCalc by X.J.Scott.
The sooner, the better!

Michael K. Henderson commented:
Carlo,  
This intrigues me a great deal, and points to a way to build synth patches that I've barely begun to explore. I have wanted to work with Gamma for a long time and I really like what you've been doing with it (from the examples on Jeff's site). Thanks for posting this!