It should be by now well known that a good -to the ear-brain- octave is a bit greater than x2 in frequency. (To do your own experiments you may download any sinewave oscillator and play with the numbers until you hear a satisfying octave. Keep volume down as higher intensities reduce perceived pitch in out brain)
We also know that a good instrument note contains many frequencies, (the harmonic series).
For example an A2 (110Hz) on the piano contains:
the fundamental 110Hz,
2nd Harmonic 220Hz
3rd harmonic 330Hz
etc
Right?
Wrong!
Good piano manufacturers by hundreds of years of evolution can stretch their strings so tightly that the 2nd harmonic would be 220+.somethingHz, the 4th harmonic 441Hz for example etc. This may be an important reason why a Steinway sounds sweet, nice, magical. This is a property of the stifness of steel [1], the fundamental stays 110Hz but the partials sligthly take off!. Do not overdo! This also explains of course why Jazz pianists increased A4 from 440Hz to 444Hz to stretch more the strings. This "inharmonisity" which is not at all inharmonic to the true sense of the word is used to advantage to satisfy God and our senses which is what music should do according to J.S. Bach [2].
These correctly strtetched as the human brain desires partials will then also dictate how the whole piano will be tuned. A good piano tuner, tunes (accords) an octave by listening to beats between say the 2nd harmonic of A2 and the fundamental of A3 which should be 220Hz OOoops 220.4Hz or whtaever. This automaticaly leads to correctly stretched octaves, what the ear-brain ordered!
So the same is done on Yamaha's DX7ii synthesizer.
We create a patch that sounds correct in terms of corrected stretched partials.
Then tune the whole keyboard so that octaves sound correct. Therefore the tuning takes care of itself.
Starting with the (central) temperament octave, the other octaves are high res tuned by comparing the 2nd harmonic of a note to the fundamental of the note one octave higher until beat frequency is minimised (tends to 0) as piano tuners do.
Therefore our ear-brain perception dictates the temperament tuning.
The same procedure can be followed for example on temperament octaves optimised for specific keys, for example modifying the Bminor optimised DX7ii temperement octave instead of the the equal temperement one.
Any temperament octave would need to be sligtly stretched of course and how can again be found by tuning by ear the 1st and last note of the temperament octave (the octave you use to base all the rest of your tuning to) and stretch each semitone slightly to equaly divide frequency distances to all 12 semitones for equal temperament. The res of the DX7ii unfortunately does not allow much accuracy to a temperament octave but it is good enough and you can play with it.
It is important to make your own temperament octave based on the design of a good sounding note (i.e. correct sounding to the ear-brain harmonic partials).
Therefore to make the Pleiades tuning, the following patch was made on the DX7ii.
Algorith 32 was used as it is ideal for specifying each partial seperately. Operator 1 frequency, was set to 1.00, Operator 2 to 4.00 +3 (by ear-brain!) so that the embeded octave feels right to our marvelous senses, and Operator 3 similarly was set to 8.00 +6 (or+5?). Output levels for each partials were reduced from max as you would expect and the following settings were subjectively found to be nice and helpful in listening to beats when time was for tuning octaves.
So to summerise:
Algorith 32
1.00
4.00 +3, 69 Output Level
8.00, +6 (+5), 64
Then the equal tempered octave was slightly corrected limit being the DX7ii resolution.
Then All notes were tuned by octaves by listening to beats, (great fun).
You can find the tuning table derived and enjoyed in playing by several users further down on euroelectron blog.
The Korg Triton tuning was done by tuning each Triton note to each Pleiades tuned DX7ii's (actually Yamaha's TX802) and listening to beats, a bit less fun but certainly very rewarding.
How nice is it that good orchestras do this? by ear-brain-body in real time, and they need to have nice instruments.
This work would not have been possible without the knowledge passed on by the piano tuner Albert Ketenzian.
References:
[1] Musical Acoustics - Donald Hall
[2] Percepts and Principles.., or in other title Instructions to his scholars in music... by J.S. Bach contained in the apendix of Spitta's Biography of Bach
[3] http://www.speech.kth.se/prod/publications/files/qpsr/1971/1971_12_1_051-067.pdf
