# Spectral Mappings for CARLOS BETA

19 September 2011 Filed in: Tuning
Theory

This article is a follow-up to my article Spectral Mappings for CARLOS GAMMA. The reader should refer to it for more details. This article comes a bit late because I have already dismantled the note layout for Carlos Beta on my Chameleon that is getting ready to be played in Carlos Alpha and 15ED2.

I had created a possible list of ratios to calculate the frequency of the first sixteen partials of a “spectrally mapped” sound to be played with Carlos Beta tuning system but never tried to apply it to a real sound. So, it remains only as a reminder of a way to calculate similar spectral mappings.

One step of Carlos Beta is the 11th root of 3:2 = 1.037548235794 or , said another way, 1.037548235794^11 = 1.5

caret (^) is computerese for “raised to the power of”

Using this coefficient (1.037548235794) to approximate ratios of a harmonic sounds results in a similar list:

1.037548235794^0=1

1.037548235794^19=2.01446

1.037548235794^31=3.13514

1.037548235794^38=4.05803

1.037548235794^44=5.0625

1.037548235794^49=6.08705

1.037548235794^53=7.05407

1.037548235794^57=8.17472

1.037548235794^60=9.13057

1.037548235794^63=10.1982

1.037548235794^65=10.9784

1.037548235794^67=11.8183

1.037548235794^70=13.2002

1.037548235794^72=14.2101

1.037548235794^74=15.2973

1.037548235794^75=15.8717

The frequency ratios of the 16 partials were chosen in order to minimize the “

*perceptual change*” of the “

*destination*” spectrum compared to a 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 the 11th root of 3/2 = 1.037548235794