Enhanced Overtone Pleiades square, triangular, sawtooth waves

If we spectrum analyze a periodic wave we find sinewave components which are intenger number multiples of a frequency called the fundamental. These are usual called harmonics. This is the Fourier theorem by the famous French mathematician. A great achievement of mankind.


And of course it is correct but the outside of it assumptions may not be when we try to apply to music.


Who said that musical instrument sounds are periodic?


Who said that integer multiples in frequency of sinewave said sound harmonic to our brain? Pythagoras was not measuring frequency but length. The natural inharmonisity of real strings can be used constructively for producing overtones that sound corrected to our brain compared to the fundamental. It depends on the material the string is made and how much it is tensioned. And Pythagoral was measuring pitch (his brain perception) not frequency. So we can't derive that the frequency had the same intenger ratios to length used. This is because the speed of the wave depends also slightly on frequency in a material. And this creates the "inharmonisity" that by good design can create Harmony to pitch ie harmony to our brain perception of the height of a note.  What matters is how our brain perceives  a pleasing octave not a textbook which in fact gets away with using the word frequency and not pitch. The same effect of speed of the wave increasing or (different length end effects for the wave that produces the standing wave ie the overtone itself ) must be happening good pipe organ, increasing at the right amount to more than x2 in frequency for example making our hearts happy.


A very easy demonstration with an online sinewave generator will show you that for example 500Hz to 2000Hz is a very shrinked double octave to our brain. Try instead 2020Hz can hear the difference.


It is proposed that Pleiades square, triangular, sawtooth waveforms are synthesized using exactly what sounds correct not to whole numbers multiples but to our brain.


In fact this is how the Pleiades equal temperament was derived.


The DX7ii was used as follows:


A simple patch on the organ (additive synthesis) algorithm was programmed with the 2x and 4x overtones slightly more than x2 and 4x in frequency so that they sounded correct to the mind.


Them the whole instrument was tuned for beatless octaves as a piano tuner might have done.


Then a Korg Triton was tuned for almost zero beats for every note by having both instrument play simultaneously. In fact it was a TX802 connected to a Triton through midi and 2 sets of headphones worn at the same time. In ear headphones for one instrument and the Sennheiser HD580"for the other.


The Pleiades Korg Triton temperament can be found in cent deviation from equal temperament on the euroelectron blogspot. It is open source.


References on stretched tuning, the Pleiades tuning and how it was derived can be found on other euroelectron posts. Also Stretched Tuning - Ernst Terhardt