Just Intonation: Making music come to life

           Just Intonation: Making music come to life
                         By Gasman / RA

   "...The way I used to type in the notes was directly
    into the assembler editor as pairs of numbers, without
    labels, so I used to be able to tweak the numbers up
    and down to make the chords 'ring' properly."
   - Music god Tim Follin, speaking on comp.sys.sinclair

With all  the sound  trackers available  on the  Spectrum today,
musicians can create  their compositions without  worrying about
the way their notes are translated into sound waves. However, if
you have some knowledge of the science involved, you can achieve
a sound quality higher than anything produced by a tracker.

Before the age of computer  music, this science belonged to  the
exciting world of piano tuning. No, really.

You see, there's  more than one  'correct' way to  tune a piano.
Most people would  find it hard  to tell the  difference between
them, but each  method has precise  details which conflict  with
the others. The only thing they all agree on is the octave. Take
any note you  like, double the  frequency, and you  get the same
note  an octave  higher. Double  it again  and you  get another
octave.

We're now faced with the problem of dividing the octave up  into
12 notes. (Yes, it is 12, and not 8: C, C#, D, Eb, E, F, F#,  G,
Ab, A,  Bb, B.)  The obvious  thing to  do is  have them equally
spaced: just multiply  the frequency by  'wibble' to get  to the
next note, and once you've multiplied by 'wibble' 12 times,  you
reach twice the frequency of  the original note - precisely  one
octave  above. A  quick bit  of calculator  prodding shows  that
'wibble' is 2^(1/12), which is approximately 1.0594631.

The big advantage here is that  you don't have to start from  C.
If you start  with B and  then do the  multiplying, you hit  the
same sequence  of notes.  This is  good news  for piano  tuners,
because  it means  that you  can get  a Bach  fugue in  D minor
sounding just as good as a Mozart sonata in G major. This method
is called 'equal temperament', and it's used in everything  from
saxophones to synthesisers.

Unfortunately, this  scheme results  in poor  quality sound.  To
explain why, let me tell you about Pythagoras. After a hard  day
of proving stuff about triangles, Mr P would chill out with  his
Fender Stratocaster (hmm, you have to use your imagination a bit
here).  He  discovered that  the  coolest sounds  came  when the
frequencies of the notes had  a nice simple ratio between  them.
For instance, if two notes are an octave apart, they have a  2:1
ratio, which  is about  as simple  as it  can get.  On the other
hand, 1.0594631:1 is not a nice  ratio at all, and as a  result,
equal temperament doesn't sound as nice as it should.

2500 years of music later, it's been decided that a 'nice' ratio
is one where the numbers are  all multiples of 2, 3 and  5. This
is known as 'Five-limit Just Intonation', because you're limited
to the  prime numbers  up to  5. You  can get  other systems  by
choosing a different limit, but they're a bit pointless and they
just confuse the issue. Trust me on this, okay...

So, sticking to five-limit just intonation, these are the ratios
to go for:

C  -  1: 1
C# - 25:24
D  -  9: 8
Eb -  6: 5
E  -  5: 4
F  -  4: 3
F# - 45:32
G  -  3: 2
Ab -  8: 5
A  -  5: 3
Bb -  9: 5
B  - 15: 8
C  -  2: 1

Of course, if you want more than one octave, you can move up  or
down one  octave by  doubling or  halving the  ratio. A  word of
warning though - THESE RATIOS ONLY  WORK IF YOUR TUNE IS IN  THE
KEY OF C! The ratios aren't actually connected to the note names
themselves; I've just chosen C as a starting point.

To illustrate this, take the sequence of notes C F Bb. We'll say
that the frequency  of C is  10OOHz; it doesn't  matter what you
choose.  Working  from  the  ratios  above,  we  get:  C=10OOHz,
F=13ЗЗHz, Bb=18OOHz.
Now, let's do the same thing in the key of F: the notes turn out
to be F Bb Eb. We now know that F should be 13ЗЗHz. However,  we
can't just read the values from the table now, because F is  the
starting point,  not C.  And this  time, Bb  really means  'five
semitones above the starting point',  and the ratio for that  is
4:3. So, it  should be calculated  as 1333 x  (4/3), which works
out as 1778Hz. Likewise, Eb  is ten semitones above F,  and this
turns  out  to  be  24OOHz.  So,  we  get:  F=13ЗЗHz, Bb=1778Hz,
E=24OOHz.

Hang on a  moment... Bb was  18OOHz a minute  ago, and now  it's
1778Hz! No,  it's not  a mistake.  Both values  can be  correct,
depending on the key your tune is written in. Obviously this  is
impossible on a piano, because the  Bb key can only be tuned  to
one note - so  the piano tuner has  to compromise and use  equal
temperament instead.

On the Spectrum, there's no such problem, because we can produce
any frequency we want (well, near enough). The first step is  to
transpose your tune to the key of C (either C major or C minor).
Next, decide which frequency to use for C. Anything will do,  so
choose one that doesn't make your tune too high or low. Finally,
use the handy table of ratios to calculate the frequency of each
note.

It's as easy as that!  Well, that's not the full  story, because
you now have to write your  own routine to play the music  - but
that's the easy  part, because you've  already done most  of the
work. Anyway, if it's good enough for Tim Follin, why would  you
want to disagree...

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