Simon Wascher wrote:
>
> hello,
>
> I wrote:
> > >consecutive fifths wich are about just intonation and divide the
> > >divergence between 12 just intonated fifths and the octave between the
> > >other fifths. As I remember, this specific system also includes a
> > >correction for the thirds.
> > Phil Taylor wrote:
> > I stand corrected. However, if the system used involved distributing
> > the accumulated error from twelve perfect fifths among all the notes,
> > the result will surely be an equally-tempered scale, even though it's
> > mathematical basis is different?
>
> I think I was not quite clear in my writing. what I wanted to describe
> is a intonation system based on 12 fifths which are from two (or more)
> different sizes. All 12 together have the same size as in an equal
> temperament but inside there are bigger and smaller fifths what makes it
> possible to have perfect fifths (and thirds) in favorised keys and such
> that are not so good at the far end of the circle of fifths. In case of
> Werckmeister the circle of the fifths is "closed" not only because
> thelve fifths equal the octave but also because all fifths are of
> musically usable size. For example there are no fifths which are to big,
> what would sound very bad (there are tonal systems wich include such
> fifths, meaning that one cannot play in all keys, but in the range of
> four or five keys these tuning systems sound very brilliant - these
> systems are very usefull for music from the sixtenth and sevententh
> century europe). So the kind of tunings people like Werckmeister worked
> out made it possible to play music like the "welltempered piano" because
> all the scales starting from all 12 (piano) keys are well sounding and
> even better, each has an individual coulor because of their different
> distance to the more brilliant center of tonalities arround these four
> or five perfect fifths.
>
> for music which does not use a 12 key (piano) keyboard there is no real
> need to use those intonation compromises. The color (intonation) of
> every interval, step or harmony can be choosen more freely and the A
> bevore the modulation must not be the same as after . Typical example:
> The A wich is the sixth of C is lower in many just intonation systems
> than the A as secund of G if C and G are common to and a perfect fifth
> in both keys. For computer generated music it should be possible to give
> up the equal tempered intonation and to calculate the intonation
> outgoing from the tonic center, maybe even when using a twelve key piano
> keyboard for input.
>
> Back to abc and traditional music this would for example mean that the
> K: signature should influence the intonation.
>
> Simon Wascher - Vienna, Austria
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Just intonation:
Pathagorian commas are too hard for me to keep track of, so I'm
going to forget them. Just remember that (3/2)^m is never equal
to 2^n. This is what complicates non equal temperament scales.
The basic just intonation scale, as I see it, is 21 notes (as for ABC
player ABZWEB12.EXE on my website, where you can display the scale). All
7 diatonics and each also sharped and flatted. Extending this to ## and
bb one has 35 notes. This is for any one keynote/mode combination. Some
key/mode combinations have A= 445.5 rather than A=440, unless you want
to renormalize the whole scale. [This is the start of an ABC player
program using the 35 note just intonation scale. That's now in Limbo
until I finish a paying project.]
With just intonation for chords one has:
D1FACEGBD
3 or more successive notes is a chord, major if starting note is
an even numbered one, and minor if odd. D1 = 293 1/3 starting a
minor triad, but D = 297 for ending a major one.
True Basic source code (in ASCII) for the 35 tone one is given as
SCALE35C.TRU in the Tunes, modes, etc. section on my website.
You'll have to modify a few statements to run it in Quickbasic.
Basic scale C/ionian is displayed at the start, then you select
key, then scoring mode for the scale of your choice. For the 21
note just intonation one just eliminate the first and last
frequencies of all 7 letter notes.
Bruce Olson
Old English, Irish and, Scots: popular songs, tunes, broadside
ballads at my website (no advs-spam, etc)- www.erols.com/olsonw
or click below <A href="http://www.erols.com/olsonw"> Click </a>
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