For your Listening Pleasure - tuning thoughts

[Richard Brekne]

Hi Bernhard

Well... this was a mouthful :)   I hadnt actually thought along the 
lines of redesigning the keyboard and redefining music per se...  This 
stuff will understandably  take a bit of digesting to get to its merits 
or not.  But do you really think its practical to think in terms of 
trying to redesign the keyboard layout to accommodate a repeating 12th 
pattern ? 

I would think pure social inertia would preclude any possibility of 
realization.  Otherwise... grin.. I would very willingly voice for your 
sanity !

Be all that as it may... dividing the present instruments tuning into 
12ths instead of octaves yields a very fine tuning that is quite 
compatible with octave based tuning curves, and yet provides a higher 
degree of simplicity in figuring.

Simply tune a perfect 12th and take the 19th root of 2 to find the 
semitones in between... or pull the 5th (taken up from the lowest note 
in the 12th) down a bit ( from D3 to A3 about 0.4 bps for the 3:2 5th) 
and run a quadratic interpolation to get a curve.  Then tune perfect 
12ths all the way up from those original notes.

My own uses D3(3), A3(2), and A4(1) at 440.4, 440, and 440,4 
respectively and after ascertaining the resulting 3rd partial 
frequencies for A3 and A4 utilizes  Rober Scotts quadratic interpolation 
calculation to create the curve using these three as anchors

Works like a charm

Cheers
RicB



Bernhard Stopper wrote:

>Hi Ric,
>my thoughts on the pure 12th tuning is  not only a tuning idea, its relates
>to a complete new musical system, as described in my publication.
>
>I know that there were several persons that had this tuning idea probably
>independant from mine around the world at about the same time. Also some old
>concert tuners used the match of M 6ths with double octave + M 3rd, whithout
>exactly knowing, that this has nothing to do with the normal 5ths circle
>theory. (as result the 12th was pure, what was not recognized by them)
>In my publication i described the 12th tuning as a basis of a complete new
>musical system that is described by a 12 - 12ths circle that fits with 19
>octaves, with consequences that goes much far than "just another tuning"
>idea.
>
>The classic circle of 12 5ths and 7 octaves can be represented as (pc =
>pythagorean comma)
>
>(3/12)^12 = 2^7 * pc
>
>the dodecachord system this transforms the 12th circle (that fits with 19
>octaves) to:
>
>3^(12) = 2^(19) * pc
>
>The "dodecachord" system (was realized on a harpsichord built by Michael
>Scheer, shown on the Frankfurt Musikmesse 1990) has a keyboard with a key
>pattern that repeats after a 12th instead of an octave.
>The dodecachord together with the pure 12th tuning as its base, is a new
>music system that shows interesting symmetric musical structures that are
>hidden when looking on the scales only over the span of an octave.
>
>The center key pattern is (1 for tone, x for half tone):
>
>1 x 1 1 1 x 1 1 1 x 1
>
>instead of the octave pattern on the "normal" keyboard that has a pattern of
>
>1 x 1 1 1 x 1
>
>the major and minor scales over a 12th are simply inverse symmetric, when
>viewed from the 4. step upward and downward from the center:
>
>major scale (4. step upwards):
>
> > 1 1 x 1 1 1 x 1 1 x 1
>
>minor scale (4. step downwards):
>
>1 x 1  1 x 1 1 1 x 1 1 <
>
>other symmetry pairs are: dorian/mixolydian, phrygian/lydian.
>
>Another major importance in the dodecachord system is the possibility to
>represent every keyboard interval as a ratio combination of 3^y, with y as
>x/19 and x as any integer, what is a main key for representing musical
>structures in 3 dimensional structures that are similar to structure like
>crystal zeolithes (1) or on neural networks (and what makes it possible for
>us to "recognize" musical structures). The reason, why a pure 12th tuning
>sounds that great, since is that it "fits" easier in the brains
>landscape....
>
>(1) Spektrum der Wissenschaft, 9/89, p. 94
>
>Ric, still believe i am not crazy? <g>
>
>best regards,
>
>Bernhard
>
>BTW the Mensurx 5.0 is out now, and i have included the MiniMens string
>simulator.
>
>  
>