TUNING digest 1387

Reinhard:
>> Schoenberg describes temperament as an
>> indefinitely extended truce (Harmonielehre p.25) and says
>> the chord is the synthesis of the tone. Would either
>> Partch or Erlich disagree? =


Erlich:

>>The first comment is vague. I don't know what the
>>war was that resulted in the truce. If it was
>>the war between temperament and just intonation,
>>then one side clearly won the war; there was no
>>truce.

I assume that from Scho"nberg's point of view the
conflict was within just intonation itself and due,
although he would not have expressed it in exactly
this way, to the incommensurability of the primes.
Temperament was the 'truce' in this conflict,
allowing intervals whose rational expressions were
unequal to be mapped onto identical intervals in a
given temperament.

Of course, whether the solution employed in the
truce was 'more beautiful, more pleasant, or more
practical' (Scho"nberg's own words) is a central
and unresolved matter in our continuing
discussions. While what is more 'beautiful' or
'pleasant' remain a matter of individual taste
(personally, I find just intervals to be usually
more beautiful, but relish the ornamental use of
beating intervals while intensely disliking tonic
'slips' by commas) what is the more 'practical'
seems to be temporally determined. At the moment,
the most available and affordable technology still
seems to give the edge to temperaments (albeit not
necessarily the most useful temperaments - witness
the inconsistant approximation of 3/2s and 9/8 by
the TX81Z) and I look foward to a time when the
technology is not so biased. =

A couple of math-type queries for anyone who can answer them:

1)In my study of E.Dunne's page on Pianos & Contd. Fractions, I have come
to the halt in trying to figure out how one obtains the contd. fraction
expansion for a logaritm,in this case,log[2](3).I've understood how one
obtains Contd. Fractions in general,but I assume for a log one might
require a tool such as that for sq.roots in the denominator (multiplying
the top & bottom of the fraction by (sq.rt.2 + 1).

2)In Dunne's excursion into using the maj 3rd (5/4) for the computation
(on how many notes could be there in an ET octave), where ( log[2](5/4) =
log[2](5) - log[2](4) )He says "since log[2](4) is an integer,the crux of
the approximation is
that of log[2](5) . Why is log [2](4) to be ignored ?


3)What the world desperately needs is a web site dealing with math for
music,sort of a 'Math for Absolute Dummies ' !!! I'm telling you, a lot
of people could be put off by the math involved in microtonality.Good
thing I'm the persistant type !


Thanks in advance.

Cheers,
Drew

>2)In Dunne's excursion into using the maj 3rd (5/4) for the computation
>(on how many notes could be there in an ET octave), where ( log[2](5/4) =
>log[2](5) - log[2](4) )He says "since log[2](4) is an integer,the crux of
>the approximation is
>that of log[2](5) . Why is log [2](4) to be ignored ?

I haven't seen this particular expose', but I suspect that it's not so
much that it can be ignored as that it isn't problematic. The base-2 log
of 5 is irrational, so approximating it with a sum of multiples of
logarithms of integers is much more difficult for the log of 5. The answer
for 4, on the other hand, is trivial and well-known as 2.





>3)I'm telling you, a lot
>of people could be put off by the math involved in microtonality.

That's a well-known and well-founded concern. And it has in fact
already taken quite a few people off the list.