Thread: Linear equation: Mapping 9 known points to a 3x3 grid

Hi!

I need some help to determine if it's possible what I'm trying to do.

I don't want to confuse you, but here's what I'm trying to do:
I have a software synthesizer that has 2 band pass filters. I can map the frequencies of these 2 filters to two modulation parameters ModX and ModY. I want to set the filters in a way to produce vowel sounds (formants) and fade between them.

The formant frequencies of the vowels are given values, see: Formant - Wikipedia, the free encyclopedia
there are 9 of them: u, o, ɑ, a, ø, y, ɛ, e and i
each of these vowel has two formant frequencies. (see the table on Wikipedia)

So what I need to do is set the filter frequencies to match the formants of a vowel.

Based on how the synth works, I can adjust the base frequency of each filter and also add or subtract a value based on the modulation parameters (ModX and ModY).

I need to figure out the base frequencies and the values I have to add/subtract to get my formants. Both modulations can add/subtract from both filters. I will set up the modulation to have 9 distinct pairs of X/Y.

So, here comes the math part:
Filter 1 frequency = f1, Filter 2 frequency = f2,
Filter 1 base frequency = B1, Filter 2 base frequency = B2

Modulation offset for f1, based on 3 states (1, 2, 3) of ModX and ModY parameters:
f1x1, f1x2, f1x3,
f1y1, f1y2, f1y3

Modulation offset for f2, based on 3 states (1, 2, 3) of the ModX and ModY parameters:
f2x1, f2x2, f2x3,
f2y1, f2y2, f2y3

for ModX=1 and ModY=1, the frequencies are calculated like this:
f1 = B1 + f1x1 + f1y1
f2 = B2 + f2x1 + f2y1

for ModX = 1 and ModY = 2:
f1 = B1 + f1x1 + f1y2
f2 = B2 + f2x1 + f2y2

for ModX = 1 and ModY = 3:
f1 = B1 + f1x1 + f1y3
f2 = B2 + f2x1 + f2y3

etc.

now I need to map each of the 9 modulation settings (11, 12, 13, 21, 22, 23, 31, 32, 33) to the 9 formant pairs by finding the base frequencies and offset frequencies for each state.

So, the 9 sets of f1 and f2 are the known values. Variables are B1 and B2 and each of the 12 modulation offsets.
for each of f1 and f2 there are 9 equations adding one of the base frequencies and 2 of the offsets.

I hope that it somehow makes sense to you... is this doable/solvable?

Well, I think it's not possible... not even with a 2x2 modulation and 4 vowels

sorry for confusing you