# The Mathematics of a Synthesizer

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Hello all!

I'd like to do a project involving the mathematics of synthesis. I plan on making a video explaining what's going on behind the scenes in a synth like Razor, including an oscilloscope and such to make changes to a musical function more visual.

I plan on doing a bit of independent research, and I thought here might be a good place to ask (if this doesn't really fit within the forums intended purpose, I understand if an admin wishes to remove it). I have a few questions I am looking for answers to.

Obviously the formula for a sine wave is ƒ(x) = sin(frequency * x),

but what are the, or what is a formula(s) for a:

-Triangle wave?

-Square wave? (also, how does the formula change for varying pulse widths?)

-Saw wave?

-White noise?

A few more questions I have are,

-What is the default amplitude in decibels of most synthesizers, and what is the distance in decibels between an outward vibration of a speaker and inward vibration? (what is the range in decibels of a musical function, usually?)

-Is the default formula for a standard sine wave ƒ(x) = sin(frequency * x)? If so, what is the default frequency used for most digital synthesizers calculations?

-How do you change the period for a different pitch- do you multiply x by the exact frequency value in hertz of the pitch you want? Or do you have to manipulate said hertz value beforehand, and if so, how?

-the default amplitude of the sine function is 1. What does this correspond to in decibels? Let's say I multiplied the enter function by 3. The amplitude is now 3; what does this correspond to in decibels? (I assume that the change in amplitude/volume is not linear?)

Thank you for sharing any knowledge you may have on the topic.

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First, you need to get some audio/digital audio theory 101 under your belt, then an intro to DSP. This is a good place to start for a simple primer:

http://www.rs-met.com/tutorials.html

This book:

An Introduction to Digital Audio

And this one:

Applications of Digital Signal Processing to Audio and Acoustics

And this one:

An Introduction to Parametric Digital Filters and Oscillators

And a nice web resource on filtering:

https://ccrma.stanford.edu/~jos/filters/filters.html

Absorbing the sum of the above should give you a good foundation to write your tutorial.

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Thank you for the quick reply!

I guess knowing more about DSP and the conversion from digital audio to the analogue a speaker needs will help immensely in knowing how formulas are written.

Some additional questions I have though of:

What are the mathematics involved in using:

-low pass filters?

-high pass filters?

-band pass filters?

-band reject filters?

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Some additional questions I have though of:

What are the mathematics involved in using:

-low pass filters?

-high pass filters?

-band pass filters?

-band reject filters?

Virtually all filter types are described in that Stanford link I gave you.

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I guess knowing more about DSP and the conversion from digital audio to the analogue a speaker needs will help immensely in knowing how formulas are written.

Digital to analog conversion is actually the easy part. After that it's straight dB theory because everything's in the analog domain at that point.

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