_______________ c. the unbinding of heaven [fixing the piano] PDF Print E-mail
Written by D. Bond   
Saturday, 31 October 2009 10:58

c. the unbinding of heaven [fixing the piano]


            No, no. I told you, ‘You always have the same pitch’,

            but I want very different pitches, if you’d please.

            pea, pay, tah, boi, tock!


                        Karlheinz Stockhausen admonishes the piano player.[1]



Time to get serious. Recall the scale:


            1/1, 16/15, 9/8, 6/5, 5/4, 4/3, [ ? ] 3/2, 8/5, 5/3, 7/4,15/8, 2/1


We could construct any number of alternates for this, but let’s look at that troubling fraction in the middle, the tritone, affectionately named,  diablos en musica


What ratio should we use hear? The fifth to b2? Fourth of b3? M3 above 9/8? Here are two possibilities given by wikipedia: 64/45-45/32]. Can you see their relationship to the scale?


But what is it we’re precisely trying to find?


Basically we are looking for a ratio that divides the octave evenly such that if we plugged it into the ratio iteration patch the series it produced would be:


            1, r, 1, r, …


The tone between 4/3 and 3/2 would be (8/6+9/6)/2=17/12. When plugged into ratio_cycler results in a slow ascent in pitch. We can use the continued fraction representation series to find better ratio approximation of this quantity given by:


            1 + 1/(2+1/(2+1/…


The first few terms give us:


            3/2, 7/5, 17/12, 41/29 …


41/29 is very close, descending slightly. Successive terms alternate between rising and falling, each a tiny bit closer, narrowing ever closer to the ratio but never exactly finding it. A precise ratio does not exist.[2]


When we do the math, we find the ratio to approach:




and that


            1/sqrt(2) = sqrt(2)/2


Plugging this value into an iteration cycle and we find that, for all intents and purposes, the value does indeed oscillate steadily without any rise or fall.


This is the difference between the geometric mean [sqrt(2)] and the harmonic mean [3/2, the average between 1/1, 3/2, 2/1].


When we map a keyboard to play specific ratios in each octave, and then use a designated portion of the keyboard [or foot pedals, a second keyboard, etc.] to designate transposition ratios, and using a bit of math to transpose the pitch center and keyboard center, we can create an electronic instrument capable of modulating to new pitch centers while retaining just relationships.[3]


We can manage the transpositions of this adapted keyboard in a variety of ways, using ET or the iteration of just ratios. The way we approach this would depend on the role the instrument is to play at any particular time. With a push of a button it can all be remapped and reset.


Neatly, we find that while traditional acoustic instruments are incapable of modulating with respect to just relationships without retuning the entire instrument [or having many versions available], this does not hold true for electronic instruments. Furthermore, the tendency of a musician to adjust a strange interval in favour of a more familiar one can be avoided using electronics.


The problem of keyboards and their fixed keyboard layout remains though, in that we are limited to a selection of relationships less than or equal to the number of physical keys we have, at most 88. But when we allow ourselves a few modulation ratios, suddenly 12 can become 144, or via iteration, limitless.


Here is a simple patch that demonstrates a keyboard that retains its relationships while shifting tonal centers. The top three octaves play the chosen ratios, and the  lowest octave points to the transposition level.


            max patch: the keyboard unglued


With a little planning and a good map, there is no trouble at all in playing a keyboard justly in any tonal center we care to choose, rational or irrational.


next 6.d. in the garden

[2] refer to the proof that sqrt(2) is an irrational number.

[3] I recently found out Wendy Carlos [wikipedia] used a similar method in her album Secrets of Synthesis, employing a second keyboard. I am unsure if she designated transposition by Just or ET ratios.

Last Updated on Tuesday, 25 May 2010 22:20