Re: Nyquist tuning

[edgar-rft]

Question to the music scientists on this list

It's approx. 30 years ago that I learned music composition theory, but
maybe anybody on this list knows wether there already exists a notation
system for such tuning scales?

The usual "C4, CS4, DF4, D4, DS4, EF4 ..." does not provide enough values
for tuning scales with a high number of semitones per octave.

With Nyquist you can create tuning systems with as many semitones per
octave as can be represented by XLisp FLONUMS. (Okay, wether a notation
for such a tuning system other than FLONUMS would make sense is a
different question).

But the reason why I ask is that I find it rather tedious to compute (in
my head) everything back and forth from a440 to write down the correct
TET parameters.

If anybody has an idea how this cold be solved (in a meaningful way of
course) I would like to integrate the formula into the TET transformation
functions.

Thanks,

- edgar

-----
a copy of Roger's mail follows:
-----

> I just got my digest with lots of traffic on tuning.
> One thing I can add is that Nyquist is pretty agnostic about tuning. Of
> course you have to have a measurement system. Tuning people often use
> cents. Nyquist uses Hz and Steps. But Nyquist doesn't tell you what
> *values* to pass as parameters. E.g. you can write (OSC 64) in C-major
> and get an equal-tempered third (E4) based on A440, but you can also
> write (OSC 63.86) to get an E that's 14 cents flatter, approximately a
> just third. Or, you can write
>    (OSC (HZ-TO-STEP (* (/ 5.0 4.0) (STEP-TO-HZ C4))))
> to get a 5/4 frequency ratio relative to C4 -- a more exact way to
> specify a just third.
>
> You can also redefine all the global step values (..., C4, CS4, DF4, D4,
> DS4, EF4, ...) to values of your choosing. E.g. you could write a
> function JUST-TUNING-IN-THE-KEY-OF that takes some parameter indicating
> the key and redefines all the globals. (Not recommended if you are going
> to change keys in the middle of some audio computation.) Nyquist already
> has a function to allow you to retune these globals to equal temperament
> using some base other than A440.
>
> If you use the score functions to create and manipulate scores before
> synthesizing them, there are mapping functions that would allow you to
> map equal-tempered pitch specifications to some other tuning.
>
> One weakness in all this is that Nyquist represents pitch/frequency with
> numbers, whereas traditional music notation makes a distinction between
> sharps and flats.
>
>
> -Roger

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[paul beach-2]

Explanations of Cents;
the formula is:

c = 1200 * log (f1/f2) / log (2)

We will see how Ptolemy's scale compares to Equal Temperment.
Ptolemy's scale
C  C#  D Eb E F F#  G Ab A Bb B C
1/1  16/15  9/8  6/5  5/4  4/3  45/32  3/2  8/5 5/3  9/5  15/8  2/1

Put 16/15 into f1/f2
c(C#) = 111.7312853 cents which is sharper than equal temperment which
has 100 cents per semitone. In other words ET sounds about 10% ugly.

Western Music Theory

1. It is mainly about resolution of the tritone. A full course spends
hundreds of page dealing with resolution of the dominant seventh. If you
are not sure what to do: the seventh falls and don't double the leading
tone.  
2. Generally don't double the third of a major chord, though Bach does
so (sometimes) when he resovles a dominant seventh. His major third is
not as (ugly) sharp as Equal Temperment.

Is 31 scale equal temperment rubbish? No, I tried a major chord.

        Ptolemy     12-ET  31-ET
root    440           440          440
Third   550           554.36       550.2
Fifth   660           659.25     658.0

For 12-ET, the third attracts attention to itself; while 31-ET, the
fifth sticks out. In general the 31-ET is more consonant with Ptolemy's
scale. There are of course more notes to create special effects. Small
birds have two vocal chords. It sounds like a 1/4 tone trill (between
the vocal chords?) that glide up or down; or by a more expresive manner.

Note that tension on a string or vocal chords is NOT expressed by an
integer scale. In linear theory, it would be square root of mass plus
spring constant.

Regarding software. There was a utility where I could do some crude work
with forumula's. It was called Waves and Bitmaps by a Russian-Finnish
programmer--otherwise there is nothing.

This is just grunt work, will be happy to put up a reference page--you
programmers are making these simple things complicated.






On Wed, 11 Jun 2008 17:12:10 +0200, edgar-rft@... said:

> Question to the music scientists on this list
>
> It's approx. 30 years ago that I learned music composition theory, but
> maybe anybody on this list knows wether there already exists a notation
> system for such tuning scales?
>
> The usual "C4, CS4, DF4, D4, DS4, EF4 ..." does not provide enough values
> for tuning scales with a high number of semitones per octave.
>
> With Nyquist you can create tuning systems with as many semitones per
> octave as can be represented by XLisp FLONUMS. (Okay, wether a notation
> for such a tuning system other than FLONUMS would make sense is a
> different question).
>
> But the reason why I ask is that I find it rather tedious to compute (in
> my head) everything back and forth from a440 to write down the correct
> TET parameters.
>
> If anybody has an idea how this cold be solved (in a meaningful way of
> course) I would like to integrate the formula into the TET transformation
> functions.
>
> Thanks,
>
> - edgar
>
> -----
> a copy of Roger's mail follows:
> -----
>
> > I just got my digest with lots of traffic on tuning.
> > One thing I can add is that Nyquist is pretty agnostic about tuning. Of
> > course you have to have a measurement system. Tuning people often use
> > cents. Nyquist uses Hz and Steps. But Nyquist doesn't tell you what
> > *values* to pass as parameters. E.g. you can write (OSC 64) in C-major
> > and get an equal-tempered third (E4) based on A440, but you can also
> > write (OSC 63.86) to get an E that's 14 cents flatter, approximately a
> > just third. Or, you can write
> >    (OSC (HZ-TO-STEP (* (/ 5.0 4.0) (STEP-TO-HZ C4))))
> > to get a 5/4 frequency ratio relative to C4 -- a more exact way to
> > specify a just third.
> >
> > You can also redefine all the global step values (..., C4, CS4, DF4, D4,
> > DS4, EF4, ...) to values of your choosing. E.g. you could write a
> > function JUST-TUNING-IN-THE-KEY-OF that takes some parameter indicating
> > the key and redefines all the globals. (Not recommended if you are going
> > to change keys in the middle of some audio computation.) Nyquist already
> > has a function to allow you to retune these globals to equal temperament
> > using some base other than A440.
> >
> > If you use the score functions to create and manipulate scores before
> > synthesizing them, there are mapping functions that would allow you to
> > map equal-tempered pitch specifications to some other tuning.
> >
> > One weakness in all this is that Nyquist represents pitch/frequency with
> > numbers, whereas traditional music notation makes a distinction between
> > sharps and flats.
> >
> >
> > -Roger
>
>
> --
> The author of this email does not necessarily endorse the
> following advertisements, which are the sole responsibility
> of the advertiser:
>
> _____________________________________________________________________
> Der WEB.DE SmartSurfer hilft bis zu 70% Ihrer Onlinekosten zu sparen!
> http://smartsurfer.web.de/?mc=100071&distributionid=000000000066
>
>
> -------------------------------------------------------------------------
> Check out the new SourceForge.net Marketplace.
> It's the best place to buy or sell services for
> just about anything Open Source.
> http://sourceforge.net/services/buy/index.php
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--
  paul beach
  sniffyraven@...


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[paul beach-2]

Hello,
Edgar's  TET 31 code works fine. Is there a loop instruction for (seq
(whatever)) ?

This did not seem to work in the Nyquist prompt.
;;One step further, a 31-TET sine oscillator would look like this:

(defun 31-tet-osc (k)
  (osc (hz-to-step (31-tet k))))

This works OK in the Nyquist prompt
;;(setf new-hertz-value (exp (* (log 2.0) (/ 1.0 31.0))))
(defun 31-tet (k)
  (* (exp (* (log 2.0) (/ k 31.0))) 440.0))
(seq
(osc (hz-to-step (31-tet 1)))
(osc (hz-to-step (31-tet 2)))
)
--
  paul beach
  sniffyraven@...


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