Tunings

None of the instruments i make are in tune.  They all feature direct access to the raw mapping of the underlying machinery of the oscillator or resonance.  Contrast to midi instruments that are always mapped exactly to 12 tone equal temperament, the scaling of Bach's "Well Tempered Clavier".  Not dissing Bach's work, but have you noticed that when a Klingon wants to fire energy torpedoes at a peaceful federation vessel, he says "BACH!".

Moving on, I would like to list for elucidations sake, a few of the raw mappings I have promulgated over the years, in chronological order:

• Ancient Greek Kithara, with seven or twelve strings tuned to the tetrachordal scales catalogued by Aristoxenus.  Plato's interpretations of the scales: lydian being whiney, mixolydian for bachanales, phrygian for war, etc, feel contrived in a modern sense, but Aristoxenus gave us a fine matrix of microtones.  However, they are probably wrong or at least "transmitted wrong" for we actually know nothing of the ancients.
• CSound virtual instrument using the, again wrong, catalogue of Ancient Greek scales by Kathleen Schlesinger.
• Electric and classical guitars, as well as a banjo, with the frets ripped off and replaced with the following systems: 22 tone equal temperament, a mainstay; 31 tone equal temperament, crazy; also several thrift store guitars with completely arbitrary fret placement based on no system whatsoever other than whimsy, these created the most wonderful "north african dusk" pitch relationships.  Later, helped my friend also to build a nineteen tone equal temperament guitar, and seventeen tone, which is currently in 2013 my favorite.
• Using 8038 sine/triangle/square oscillator chip with linear current input to tune a drum/drone machine by ear.  Linear input means that when the voltage, converted to current by a resistor, doubles, an octave is made: 1volt, 2volt, 4volts roughly for the octaves.
• Learning to use transconductance amplifiers with linear input for the same purpose, then the Arp exponential converter is encountered.  This comes closer to the idea of "1 volt per octave", i.e. a logarithmic but continuous tuning scale.  Note that there are little humps and bumps, subtleties of resistor configurations, that give variety across its uses.  We are still discussing a transconductance triangle wave oscillator, which takes a current input to manifest "slope speed": higher current is faster waveform.  But the width is set.
• In Fourses experimental instrument, taking this triangle oscillator and, in addition to controlling its slope speed, also gating different signals, static or dynamic, to the edge boundaries of the triangle wave.  A smaller/closer boundary means a higher, quieter waveform, and wider means low pitch and loud.  Finally, using other waveforms for the boundaries, a simulation of four "bouncing balls in a box" that bounce off of each other and thus have chaotic/undefined bounds.
• In resonant filter using transconductance state variable circuit, an Arp exponential converter is used to make the knob logarithmic, and its output which is naturally current because it is the collector of a transistor, is fed into a Serge style differential pair which also has a "pseudo logarithmic" feel.  It actually is in the shape of an s curve: steep and linearlike in the middle, and then sloping out on both ends.  It has two inputs: verso and inverso.  It can also feed two separate outputs, although I often use just one.  Thus it can lower one current, and at the same time raise another, on the other side.  Both sides, as I said, have a pseudo logarithmic taper on the bottom, making it somewhat like "1volt per octave", but also it tapers on the top, so at a certain note it really becomes a fine tuner for that one note.  Thus the Arp at the base can feed a base pitch and the Serge at the top can be a tuner.  This configuration is what i use for most of the analog circuits now.  The Arp part gives it that nice wide range feeling, and the Serge adds its own "s slope" on top.
• Not to forget all the other, "homey, drifty, and rustic" styles of pitch control, such as a raw singular transistor biased ever so gently, that it is highly sensitive to heat, you wave your hand above it and the pitch drifts; using a sort of emitter follower to generate current at the collector that is bound within a certain region, good for noisemakers; and the other things like using cadmium sulfide which has its wonderful "warm up period" and each piece has its own range, here is a simple transistor calamine resonator and multivibrator, from the "TIMARA CURRICULUM"
  
• And now, I would like to speak of some of the digital tunings I developed for Shbobo, for use on the Shnth device.  It all flowed pretty naturally from my analog experience, especially with the triangle oscillator that has two controls: one for the slope/speed, and the other for its bounds.  The former control will be called the "nume", for numerator; the latter control is the "deno" for denominator.  You see, they two actually come together to make an "alternative" form of expressing pitch than logarithmic 12tone equal temperament, or what might be used in virtual synthesizers, floats which are converted exponentially.  The nume/deno system is Just Intonation, and you can of course read Harry Partch's book for more info on that.  Also you can see his wonderful misinterpretation/California embellishment of the Ancient Greek kithara.  The first program I wrote in this system was called Just Ints, and like the name it only uses the int type, but it can express pitch in actually very fine increments, using only 8 bits for nume and 8 for deno.  You can multiply that out and not discounting repeat tones, there are about 64k pitches there.  Of course some of them are in very high prime limits, like 197, 199, 211, or 223, so I also developed a way of expressing/modulating the prime limit as you scroll through pitches.  There are two ways developed to scroll through pitches: kingal, tuning the nume and deno separately; and queenal, which uses a dictionary of pitches to allow you to scroll through them "chromatically", although with a high prime limit it almost sounds glissando/analog scroll.
• Speaking of analog scroll, backing up to the triangle wave of my mainstay analog line; for years the Sidrazzi organ had this arbitrary tuning system called the "analog scroll".  You could push a button for each bar, and it would go "woowoowoo", moving at a brisk pace up and down.  When you released the button, it would hold whatever pitch it was on.  The philosophy behind this is quite whimsical: in the age of arbitrary invasions by an arbitrary president GWB, i wanted to create an organ so that such energy could be diverted into musical arbitrariness, not physical harm.  That is how the analog scroll came about.  It was originally quite fast, like a sub bass oscillator, so you couldn't even pick the pitch: it would give you a random one each time.   You could go, "i don't like that one, there's no oil there, let's invade another pitch," and press it again.  The button eventually slowed down so you could pick the pitch, and now it is replaced by a slider pot, so you actually pick pitches in a conventional way.
• Although when I say conventional, I must point out there is a whole tuning system of slopes on the tetrax organ, that has its own intricacies that leads to a wonderful variety of timbres.  On that organ, there are two separate master pitch controls: one for the "up slope" and one for the "down slope".  If they are set roughly the same, then you get a triangle wave.  Anything else is some mutation of triangle or saw wave.  And then the individual tuning pots also work in this system, scrolling the waveform from energy on the up slope to energy on the down slope.  So you see, it is really tuning both pitch and timbre at the same time, and it leads to some interesting modulations that are solely due to the fact that it is not 1 volt per octave or any old fashion like that.
• Also, when you make a State Variable Filter as a resonance, there are actually two current inputs, because you use a dual transconductance amplifier.  To use it in a normal, VCF type fashion, you feed both sides the same current, so it can get higher and lower in unison, for wet analog sounds.  But also there is another hypothesized way, where you take a serge differential current source, and use both sides to modulate the two separate integrators of the filter.  I think one is bandpass and one is lowpass but it really doesn't work like this, actually to understand the State Variable Filter, you must understand the wave equation/harmonic oscillator of physics; it's all there in that equation.  It's second order differential which means it has x, dx, and d2x.  It is the x and dx which have an omega miniscule input, which means frequency control.  It is the omega miniscule which is usually varied in unison, but here we are talking about varying it inverse proportionally.  If you think about it, it is bringing the lowpass and the bandpass in and out of each others phase, so sometimes the resonance is highly damped and sometimes it is very ringy: we are thus able, like in the tetrax organ, to modulate the filter's pitch and q at the same time.  BTW, this solves an age old 20th century problem, which was how to make a VCF with only one transconductance amplifier?  Buchla, I think, would pop in a third stage, a CM3060, just to modulate the Q.  But Q modulation is a tricky thing and not many synthesists actually use it.  However, on the Plumbutter, you can flip the switch of "Gongue" into "Gonz" mode, and you will be using this type of modulation that affects both pitch and Q.  It makes for some wonderful woody, ringy type effects, and it is a great variation from always just "modulatin the cutoff" up and down like an ambulance.  The ambulance effect, BTW, is the worst.  I should make a CD someday of songs made exclusively from the ambulance effect.
• By the way this is how the human mouth filters our various vowel sounds: there are two or three completely separate formants that move around independently to put various information onto the signal.  Some vowels are very resonant because all the formants act in unison, whilst other vowels are more "woody" because the formants are more spaced.  Note that there's at least two tunings in human voice: the rich wave from our vocal chords and then the resonant filterings created by mouthforms.
• Making my way back to the digital discussion.  We already covered the Just Intonation triangle oscillator with nume and deno.  That is the "Yang: sunlight", and its moon should be the wet State Variable Filter, which is also programmable digitally.  An oscillator and a filter: a song for Becke.  The State Variable Filter, or wave equation, is commonly programmed for all sorts of simulations/stimulations.  But you will note that there is usually another block of code before it which is to "linearize" the pitch.  Yes, the pitch control is actually in a sort of "s curve" kind of like the one for the serge differential amplifier: tapered and logarithmic at the bass and also dwelling in microtones at the top, whilst linear in the middle.  Needless to say, I eliminated the compensating block of code at the beginning.  When you use the "Wave" as it is called in Shlisp, you give it a "rate" parameter, and it is this rate that exhibits its own tuning system, that is like an s curve.  So if you use the same control for the rate of a wave and the nume of a triangle, you will get a slightly different mapping.  It is this mapping difference, this "dissonance of curves" that brings more variety to any composition, i feel. 
• Then there is also the digital version of the Tetrax wave, which is a triangle, mutating through different varieties of saw: controlling both pitch and timbre.   
• Another one pops into my mind, and this is from farther back, in the "homey, whimsical, and rustic" category.  It was a paper circuit called the "Dogvoice", that had two sections, like the Yang and Yin triangle and resonance, but much dirtier and doggier.  The first, an oscillator, was made out of two ultrasound harshers, using a strange comparator to heterodyne them.  Then the product of this heterodyne was sent into a Moog type ladder filter.   Now there are two tuning systems at play here, like as discussed earlier with digital Yang and Yin.  The first: a heterodyne signal, has a lightly linear feel, but also due to aliasing it wraps around and around.  So there is a repeating, ever changing fractal tuning system.  The second system, in the ladder filter, is quite dependent on the ladder itself: a stack of four or five or more differential transistor configurations, each one contributing a hump to the mapping of control voltage to pitch and also, q or timbre of the filter.  The ladder filter is known for its sweet and salsafied humpy nature, looking like terraces on a mountainous rice paddy, it also creates whirls of phase change around each steppe.  These "homey, whimsical, and rustic" categories of tuning definitely have a great attraction for being programmed and emulated in digital, but unfortunately it is quite difficult.  They are great as paper circuits though: accessible, circuit bendable, and sublime/enigmatic in their workings.
• The most newest tuning I have been listening to a lot lately, is the unintentional sounds of my CNC wood working machine as it prepares parts for analog instruments.  It uses sine waves, cosine waves, but I usually express these as a complex Euler that is taken apart into real and imaginary.  It uses triangle waves to create standoffs for the parts, and much of it is box shapes too.  The bit is a chaotic little squealer, for which I am always wearing ear protection, but I can still hear it vibrating at a plethora of frequencies at once, the resonance of the wood comes into play too.  It is quite complicated, and 100% controlled by my signature ruby gcode scripts.

Discussion

It should thus be apparent that there are many ways of tuning electronic instruments, string instruments, and digital instruments, not to mention woodwinds and brass, and drums.  A platonic tuning system would only control the pitch, and only according to strict, universal rules that are for the betterment of the metropolis.  However, tuning systems are not always quite this way, and I have found it worthwhile to explore the differences amongst tuning systems.  The control of pitch and timbre at the same time are quite common in electronics, but you can also have like a buzzy fret on a guitar, or a squeaky note on a bassoon. I have always had a mantra: "eliminate compensation", which has always brought users closer to the machinery, making it more visceral, rather than farther away, more Platonic, more classical, and "BACH".  I want to emphasize that there are infinite kinds of tuning systems, and most stem from some underlying machinery, like the triangle wave with nume and deno, or the tetrax wave with timbre/pitch duality, or the digital state variable filter with its scurved mapping.  They are like what you get on playing a microtonal guitar, some arbitrary system within which to work.

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