Hi all . I am a long time fan of audacity (presently 2.1.3). I want to use it to demonstrate features in musical sound to my pupils. I find there are many basics that are simply glossed over (mathematically speaking)

I wanted to start with simple pure sine waves and show additive synthesis but the generate/tone feature gives a tone with way too many sub harmonics eg starting with 500 Hz I get 250, 125 giving a very impure result for demonstration purposes. (strange cos harmonics are normally higher multiples of the fundamental? - cant get my head round that? Is it something to do with the way Aud approximates a sine wave - from a wave table maybe?

Can anyone suggest another way to get a pure note from Aud?

I can only think of making my own WAV and using that (if I can make one pure enough)

That’s a pretty old version. The current version is 2.3.2 and is available via the Audacity website: Audacity ® | Downloads

I suspect this is due to some kind of “user error”, because “Generate > Tone”, when “Sine” is selected, produces an extremely pure sine wave.
(If “Sawtooth” or “Square” are generated, there will be sub harmonics due to aliasing).

Assuming that you have not accidentally generated the wrong shape waveform, how are you measuring the waveform that indicates sub-harmonics?

thanx steve I have duly updated my audacity to 2.3.1. I tend to go on the priniciple, if it aint broke dont fix it, so no doubt my old version was 10 years old - tribute to Audacity (but then I still keep a working machine with XP sr3)

Indeed I think I am misunderstanding the screen traces. I am trying to get a handle on Analysis/plot spectrum - more work needed

One thing that has been bothering me about additive synthesis is

sin(A)+ sin(B) = 2 (sin((A+B)/2) x cos((A−B)/2))

This is a well known trig formula

but it doesnt shown the beat frequency you get from mod| B-A|

Im not sure how to obtain the harmonics from the product of two sines (some kind of fourier expansion technique?)

Hope you can see what I am trying to explore here
As is well known to musicians, when you add two tones you end up with a timbre and a beat

I want to be able to explain how eg you add a tonic say middle A 440 plus a fifth above 660. You will get a diad chord and a sub tonic at 220hz. I presume in this case that the subtonic will also interact with the tonic plus the fifth above etc

I have never found this aspect dealt with anywhere

I thought Aud would be a good experiment vehicle if I can master the interpretation and limitations of the FFT

No, it won’t do, because the FFT resolves the waveform into A and B.

You need to approach it the other way round:
Generate a “Square, no alias” waveform at 100 Hz. It looks like this when you zoom in:

If I zoom out a bit and select 0.1 seconds, then as expected, you can see 10 positive going peaks within the 0.1 second selection:

From this we can deduce that the fundamental frequency is 100 Hz (10 cycles in 0.1 seconds = 100 cycles per second).
We know that a square wave can be created from the sum of harmonics (and in fact, that’s how the “Square, no alias” is generated), but can we “see” these harmonics? Yes we can, by looking at the Track Spectrogram view (Spectrogram View - Audacity Manual). I’ve stretched the track vertically so that we can see it better:

and also in Plot Spectrum:

Audacity has a scripting language built in called “Nyquist”. It is very good for this sort of thing.

Here’s an example script that sums the first 8 harmonics of a “square” wave:

;version 4

(setf harmonics 8)  ;number of harmonics
(setf hz 100)       ;fundamental frequency

(setf wave (s-rest 1))
(setf amp 0.5)

(dotimes (hnum harmonics wave)
  (setf f (* hz (+ 1 (* hnum 2))))
  (setf amplitude (* amp (/ 1.0 (+ 1 (* hnum 2)))))
  (setf wave (sum wave (mult (hzosc f) amplitude))))

You can run this in the Nyquist Prompt effect, by copying and pasting.

Some brief explanation:

A semicolon indicates the start of a comment. Comments are ignored by Nyquist.

The first line (;version 4) is a special comment that tells Audacity that we are using version 4 syntax. It is not strictly necessary in this case, but it’s good practice to include the syntax version number.

“Nyquist” is based on the LISP language. Functions are always written as:

(function-name arguments)

For example, where we would normally write:
2 + 3
In LISP / Nyquist, we write:

(+ 2 3)

“setf” is an assignment function. It sets the value of the first argument, to the value of the second argument.

(setf A 4)  ;sets the value of 'A' to 4
(print A)   ;prints 4

When a Nyquist script runs in Audacity, the final result is returned to Audacity. In this case, the returned value is a sound which I have called ‘wave’.

Here’s an overly commented version of the script:

;version 4

(setf harmonics 8)  ;number of harmonics
(setf hz 100)       ;fundamental frequency

;; Begin with "duration" of silence
(setf wave (s-rest 1))
;; and an initial amplitude of 0.5
(setf amp 0.5)

;; Add the sine tones
;; 'hnum' is the harmonic counter, and we increment it,
;; starting from zero, on each loop, up to the value of 'harmonics',
;; and return 'wave'.
(dotimes (hnum harmonics wave)
  ; Calculate required frequency:
  (setf f (* hz (+ 1 (* hnum 2))))
  (print f)  ;prints the frequency to the debug output
  ; Calculate the required amplitude:
  (setf amplitude (* amp (/ 1.0 (+ 1 (* hnum 2)))))
  ;; Generate the new sine wave and add it to 'wave'
  (setf wave (sum wave (mult (hzosc f) amplitude))))