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#X graph graph1 0 0 40 1 151 551 551 301;
#X array array-ampdb 41 float 1;
#A 0 0.01 0.0112202 0.0125893 0.0141254 0.0158489 0.0177828 0.0199526
0.0223872 0.0251189 0.0281838 0.0316228 0.0354813 0.0398107 0.0446684
0.0501187 0.0562341 0.0630957 0.0707946 0.0794328 0.0891251 0.1 0.112202
0.125893 0.141254 0.158489 0.177828 0.199526 0.223872 0.251189 0.281838
0.316228 0.354813 0.398107 0.446684 0.501187 0.562341 0.630957 0.707946
0.794328 0.891251 1;
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#A 0 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3
0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 0.525 0.55 0.575 0.6 0.625
0.65 0.675 0.7 0.725 0.75 0.775 0.8 0.825 0.85 0.875 0.9 0.925 0.95
0.975 1;
#X array array-4thpow 41 float 1;
#A 0 0 3.90624e-07 6.25001e-06 3.16406e-05 1e-04 0.000244141 0.00050625
0.000937891 0.0016 0.00256289 0.00390625 0.00571914 0.0081 0.0111566
0.0150063 0.0197754 0.0256 0.0326254 0.0410062 0.0509067 0.0625 0.0759691
0.0915063 0.109313 0.1296 0.152588 0.178506 0.207594 0.2401 0.276282
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0.903688 1;
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#X text 53 27 (here's how I computed the three transfer functions...)
;
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#X restore 53 608 pd otherstuff;
#X text 292 403 linear;
#X text 279 509 decibels;
#X text 387 518 quartic;
#X text 45 5 QUARTIC CURVES AS THE IDEAL AMPLITUDE AND FREQUENCY SCALERS
;
#X text 346 611 updated for Pd version 0.34;
#X text 246 578 units-->;
#X text 45 447 amplitude;
#X text 79 429 |;
#X text 79 420 |;
#X text 79 410 |;
#X text 79 402 |;
#X text 78 398 ^;
#X text 38 149 The graph below shows that a simple quartic curve \,
x-to-the-fourth-power \, twists like decibels but--unlike decibels--actually
hits zero at left. You get the best of both worlds. Moreover \, raising
something to the fourth power is very cheap: just two multiplications--whereas
\, if you're computing envelopes in dB \, eventually you'll have to
exponentiate \, sample by sample \, to get to linear units.;
#X text 36 34 It's an old saw that we perceive amplitude and frequency
logarithmically. But using decibels as a unit for controlling amplitude
and frequency gets ugly for two reasons. First \, it's expensive to
do the conversion. Second and more profoundly \, decibels grow by shifting
\, and things should grow by scaling \, so that \, for example \, zero
really means "nothing.";
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