<html>
<head>
<meta content="text/html; charset=ISO-8859-1"
http-equiv="Content-Type">
</head>
<body bgcolor="#FFFFFF" text="#000000">
<div class="moz-cite-prefix">Dear <span>Ildar,<br>
</span><br>
You wrote:<br>
</div>
<blockquote
cite="mid:1386087111.529.YahooMailNeo@web125301.mail.ne1.yahoo.com"
type="cite">
<div style="color:#000; background-color:#fff;
font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial,
Lucida Grande, sans-serif;font-size:12pt"><br>
<div><span>As for the mathematical toys, such as a stack of
fifth--since Pythagoras some inquisitive scientists are
trying to harness music with the rules of high school math.
If you believe that diatonic is the product of the stack of
fifth, take a tuning fork and tune your piano by perfect
fifth. Gook luck! In Greece there has been a controversy
(canoncs vs harmonics), but practical musicians (including
Aristoxenus and his father) did not adopt Pythagorean views.
They were too abstract and impractical. </span><br>
</div>
</div>
</blockquote>
<br>
<br>
I can assure you that the chain of fifths (and octaves) is not a
mere "mathematical toy".<br>
The math may be easy, primary school level will do, but the
consequences profound and fundamental.<br>
Our notation system is based on the chain of fifths, and most of our
music theory along with it.<br>
The chain of fifths is what makes a diminished fourth a different
interval from the major third etc, and at the same time gives no "2
different major thirds"<br>
And the reason this all works so perfectly is that the chain of
fifths corresponds to how the human brain quantizes the pitch space
/ how we automatically categorize intervals.<br>
Tune the fifths and octaves perfect and you get Pythagorean tuning.<br>
I will not get into a further debate on tuning as it is probably to
unrelated to the topic of this thread, but I will say that it is not
merely an old controversy in Greece but that Pythagorean is also
still practiced today naturally by violin, singers, trombone etc
when playing solo (both taught and automatically done so). And was
the default tuning system for polyphonic music in the west until
replaced by 1/4 comma meantone and other meantone tunings. And the
12-tone equal tempered tuning system that we use today for fixed
pitch instruments is very close to Pythagorean tuning, it is hard to
hear the difference with Pythagorean as long as one follows the
enharmonic notation correctly (don't play a diminished sixth as if
it is a perfect fifth, or augmented second when there should be a
minor third etc)<br>
<br>
It is very easy to tune the diatonic system based on perfect fifths
on a piano. Tuning fork is not even needed as it can be done by ear
with ease.<br>
Lets start with F tune perfect fifths above it and reduce them by
perfect octaves when needed and one gets:<br>
F-C-G-D-A-E-B, or C D E F G A B C when going up in steps from C.
Containing all the strict diatonic scales. A chain of 6 perfect
fifths linking 7 tones like described.<br>
Tune one fifth extra and you get a chromatic interval, F-F#. And one
can continue adding fifths till one gets D#-F a diminished third.<br>
Makes good sense to call these the chromatic intervals, from 7 to 11
fifths. So the chromatic scale covers 12 tones linked by perfect
fifths.<br>
If we add one more fifth we get E# - F, which are enharmonically
related. So a chain of 12 fifths and upwards are the enharmonic
intervals.<br>
To me this is the best description there is.<br>
It also allows the following clarification which corresponds to how
we perceive music.<br>
Something like the harmonic minor scale is by this definition a
chromatic scale.<br>
However, one can use it melodically in such a way that the melody
only moves by diatonic intervals within this scale.<br>
So the augmented second is not used as a melodic step. This is the
most common use of this scale in western music so one can still
speak of diatonically based music in a way.<br>
<br>
Kind regards,<br>
<br>
Marcel de Velde<br>
Zwolle, Netherlands<br>
<a class="moz-txt-link-abbreviated" href="mailto:marcel@justintonation.com">marcel@justintonation.com</a><br>
<br>
<br>
</body>
</html>