HT files in the Library

[Richard Moody]

----- Original Message -----
From: Robert Scott <rscott@wwnet.net>
To: <pianotech@ptg.org>
Sent: Tuesday, August 15, 2000 1:47 PM
Subject: HT files in the Library


>
> The suggestion from Michel Lachance about a FILE SHARING LIBRARY got
> me thinking about something I would like to see in such a depository -
> Historical Temperament files.
>
> In Jorgensen's book on Tuning there are 57 HTs described in terms
> of cents offset from equal temperament.  ........

>  So if anyone has suggestions on how to generate
> ETD offsets from aural desriptions I would like to hear them.
>
> -Robert Scott
>  Ypsilanti, MI

Most of the HT's have been given in "offset from ET"or "cents difference
from ET.  I am assuming this is what you mean by "ETD offets from aural
descriptions..."
In regards to inharmonicity it is my understanding that a program such as
"FAC" in the SAT first computes the inharmonicity of the instrument being
tuned and adjusts the machine according to that.  So that when the
theoritical "offests"  (from ET) are entered,  the machine adjusts these to
its computed inharmonicity of the particular instrument.
    The "settings" for HT's or the cents differences from ET (Equal
Temperament) are of course theoritical. However with two decimal places,
(1/100 cent) the calcualtions can agree perfectly with the "aural
descriptions".  Thus when a temperament prescribes a 5th that is narrowed by
one quarter of a comma we need to know what comma they are using and the
cents value of that. In the case of Meantone it is the comma of Didymus or
21.506.  This is gotten from the ratio of a 3rd formed by four sucessive
5ths or 81/64.  This ratio can be converted to cents using the formula
cents = Log(F/f)/Log(2)*1200.  This is for systems using logs of 10.  (such
as calcuators and PC's)  Since music octaves (and any  note inbetween) can
be expressed in Log 2 the original formula reads "log sub 2 of the frequency
ratio times 1200.  If you had a slide rule with a Log 2 scale you should be
able to get cents from that.  So you need to know the numerical ratio of the
intervals you want the cents of.
    From aural instructions giving only beats per second, the ratio of the
interval must be figured from that and that ratio converted to cents using
the above formula.
    A spread sheet does this very conveniently, details available upon
request.
    What is the aural description you wish to convert to cents?  Using a
spread sheet I should have the cents difference from ET in less than 20
minutes. The reference pitch should be known, as it makes a difference if a
temp is being described starting with A435 or A440.   Most HT's are given
from middle C.  This is if only beat rates are given.
    Since most HT's have a theoretical basis it is better to compute cents
from that. Such as quarter tone Meantone, that would indicate a series of
5ths narrowed by one fourth of 21.5 (the Comma) or 5.3x  cents.  Now this is
cents
from PERFECT.  To get cents from ET you must know the difference of the ET
interval from the Perfect interval. It is the numerical value of the
differences from pure. You need to be careful if it is + or minus.
    Using a spread sheet, what I do is compute a scale of frequencies from
middle C (from A440) according to ET.  Then I compute another  scale of
frequencies of the temperament I am studying.  This gives me the benefit of
computing beat rates if I desire.  But if I want the difference from ET then
the ratio of the same two notes are computed and converted to cents.  In
Meantone for instance it turns out that  A4 is close to 435.   If it were
then using 435/440 I can compute difference in
ents---obtaining  -19.78574734628 cents on the W95 calculator. Thus in
Meantone A4 is -19.78x cents from A4 at ET.     You are at
the first step of "cents from ET", but suppose the musicians want to tune to
A440 and play in Meantone?.  Simply add 19.786 cents to each note?  Since
cents is an additive quantity it seems it should be that simple.  I would
test it on a spread sheet though.  Calcualte the freq of C4 19.786 cents up,
and make a new scale of the HT from that freq and see if A4 comes out to
A440.   ---ric