arithmetic divisions for 2/7-comma meantone? (Zarlino 1558)

hold the presses!!

in his private email to me, accompanying his English
translation of cap. 42 and 43 of part 2 of Zarlino's
_le institutione harmoniche_ (1558), Leonardo Perretti
wrote:

>> This chapter [43] describes how to make the temperament,
>> but does not specify the actual, practical procedures,
>> that are described in the Part 2, cap. 18 - 24 for
>> making the scale (JI, I think). I have given a look
>> there, and it seems that Zarlino uses a sort of
>> monochord to establish the length of the strings;
>> he is pretty practical there, and never speaks of
>> beatings or something that could be referred to them.
>> Also, if you look at the method described in this
>> chapter, it suggests a procedure based essentially
>> on measurements (...to narrow a fifth, place a comma
>> at its high end, divide it by seven, and take away
>> the two higher parts...).

i have just realized that if Zarlino actually
intended to *measure* 2/7-comma meantone on a monochord,
he most likely was not using (81/80)^-(2/7) as his
division, which is what we've been assuming all along,
but rather, he probably divided the monochord
*string-length* of the syntonic comma into 7
*arithmetically* equal parts!

in cap. 43 of part 2,

http://sonic-arts.org/monzo/zarlino/1558/cap42-43.txt

Zarlino describes these divisions as "equal", and
insists that they therefore cannot be "rational",
as are the regular JI intervals he had presented a
few chapters before this. this would lead the
reader to believe that he is indeed discussing
logarithmic divisions.

but he then continues to describe how to determine
the tempered intervals by *dividing the string-length
of the comma into 7 equal parts* and measuring off
various numbers of those parts.

this procedure would result in the following
divisions for the "5th":

ratio string ~cents
length
3:2 "perfect 5th" 560 701.9550009
3:2 less 1/7-comma 561 698.8662685
3:2 less 2/7-comma 562 695.7830369
3:2 less 3/7-comma 563 692.7052867
3:2 less 4/7-comma 564 689.6329983
3:2 less 5/7-comma 565 686.5661524
3:2 less 6/7-comma 566 683.5047298
3:2 less 1 comma 567 680.4487113
1:1 840 0

if this is how Zarlino calculated his temperament,
then his 2/7-comma meantone "5th" is ~695.7830369
cents, and not the ~695.8103467 cents which results
from the logarithmic division of the comma, and it
is indeed a rational number: 840/562.

a tempered approximation of this "5th" is 2^(40/69).

-monz

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 09, 2003 7:10 AM
> Subject: [tuning] arithmetic divisions for 2/7-comma meantone? (Zarlino
1558)
>

> hold the presses!!
>
> <snip>
>
> i have just realized that if Zarlino actually
> intended to *measure* 2/7-comma meantone on a monochord,
> he most likely was not using (81/80)^-(2/7) as his
> division, which is what we've been assuming all along,
> but rather, he probably divided the monochord
> *string-length* of the syntonic comma into 7
> *arithmetically* equal parts!
>
> <snip>
>
> if this is how Zarlino calculated his temperament,
> then his 2/7-comma meantone "5th" is ~695.7830369
> cents, and not the ~695.8103467 cents which results
> from the logarithmic division of the comma, and it
> is indeed a rational number: 840/562.
>
> a tempered approximation of this "5th" is 2^(40/69).

~cents
695.7830369 = 840/562
- 695.6521739 = 2^(40/69)
---------------
0.130863021

i realize that's a difference of only ~1/8-cent.
but i think all previous descriptions of Zarlino's
"2/7-comma" meantone have been mathematically
inaccurate.

assuming that i am correct in believing that
Zarlino intended to use arithmetic string-length
divisions, i would have to go thru his entire
monochord division to give an accurate accounting
of the ratios for this tuning. hopefully soon ...

-monz

oops.

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 09, 2003 7:18 AM
> Subject: Re: [tuning] arithmetic divisions for 2/7-comma meantone?
(Zarlino 1558)

> > From: "monz" <monz@attglobal.net>
> > To: <tuning@yahoogroups.com>
> > Sent: Sunday, March 09, 2003 7:10 AM
> > Subject: [tuning] arithmetic divisions for 2/7-comma meantone? (Zarlino
> 1558)
> >
>
> > hold the presses!!
> >
> > <snip>
> >
> > i have just realized that if Zarlino actually
> > intended to *measure* 2/7-comma meantone on a monochord,
> > he most likely was not using (81/80)^-(2/7) as his
> > division, which is what we've been assuming all along,
> > but rather, he probably divided the monochord
> > *string-length* of the syntonic comma into 7
> > *arithmetically* equal parts!

no. my bad.

in cap. 43 of part 2 of _le institutione harmoniche_
Zarlino refers to cap. 25 for his description of how
to divide an interval logarithmically equally by
means of the mesolabium, a device which geometrically
measures divisions of a string-length.

we'll know more after an upcoming translation ...
so forget my fanciful speculation.

;-)

-monz