Salinas and 19edo (was:: Dictionary updates: 2/7-comma ...)

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 30, 2003 10:09 PM
> Subject: Re: [tuning] Re: Dictionary updates: 2/7-comma and 1/3-comma
meantones
>
>
> hi paul,
>
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> > To: <tuning@yahoogroups.com>
> > Sent: Sunday, March 30, 2003 2:41 PM
> > Subject: [tuning] Re: Dictionary updates: 2/7-comma and 1/3-comma
> meantones
> >
> >

> > > http://sonic-arts.org/dict/19edo.htm
> >
> > i just scrolled down to the end to see what i would find:
> >
> > > It was noted by Salinas in 1577 that 19-EDO was audibly
> > > indistinguishable from 1/3-comma meantone.2(11/19) =
> > > 696.7741935 cents.
> >
> > i though[t] it *wasn't* noted by salinas. where did he note it?
>
>
> i've been away from Salinas for several months and will
> have to look into it again, but my memory tells me that
> he did note that 19edo was nearly the same as 1/3-comma MT.

i've been doing some more research on Salinas.
i still haven't grappled with translating his latin
myself, but have relied on Daniels 1965, "Microtonality
and Mean-Tone Temperament in the Harmonic System of
Francisco Salinas", _Journal of Music Theory_ vol. 9.

i've come to this conclusion
(which now ends my 19edo page):

"
Salinas in 1577 (De Musica, book 3, chapter 16) described
1/3-comma meantone with mathematical exactitude for the
first time.

First he constructed a 24-tone JI system, which had
duplicate pairs of some pitches a syntonic comma apart,
the higher of which was labeled "superius" and the
lower "inferius".

Then he explained the amount of tempering for each
of the meantone pitches. By tempering out the full
comma which exists between the 5 pairs of
"superius/inferius" pitches, he reduced the number
of pitches from 24 to 19.

Below is a lattice which places Salinas's 1/3-comma
meantone in prime-space and shows its relationship
to his JI system, as he describes it; the slanted
arrows represent the syntonic-comma:

[lattice diagram]

It can be seen that, with the exception of A# superius,
all pitches of the meantone are either exactly those
of Salinas's JI system, or are 1/3, 2/3, or a full comma
higher or lower than those in his JI system.

He goes on to explain how to temper the 24-note JI system
into 19 notes of 2/7-comma meantone, and then also into
19 notes of 1/4-comma meantone, the latter of which he
finally declares to be the best of the three temperaments.

Salinas did not explicitly mention the equal nature
of the 8ve-division in his 1/3-comma meantone, but he
would have known about it himself and it can be inferred
from the measurements he described. The 19-tone systems
of the 2/7-comma and 1/4-comma meantone are less
equally-spaced, being rather closer to subsets of 50edo
and 31edo, respectively.
"

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> Salinas did not explicitly mention the equal nature
> of the 8ve-division in his 1/3-comma meantone, but he
> would have known about it himself

how do you draw this inference?

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, March 31, 2003 4:21 PM
> Subject: [tuning] Re: Salinas and 19edo
> (was:: Dictionary updates: 2/7-comma ...)
>

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > Salinas did not explicitly mention the equal nature
> > of the 8ve-division in his 1/3-comma meantone, but he
> > would have known about it himself
>
> how do you draw this inference?

i'd have to actually slog thru the numbers
Salinas gives in order to make it explicit,
but hopefully it will suffice to say this for now:

(paul, you know me ... if i can find the time
to devote to this, i'll eventually add it to
the webpage)

Salinas certainly knew the measurement of
the "5th" (generator) of 1/3-comma meantone,
and after calculating a 19-tone chain, which
he manifestly did (and explained), it would have
been obvious that one more step in the chain
would result in a pitch very close to the one
he started with.

i hope to actually translate the latin text
so that i can see exactly what he said about
his 19-tone 1/3-comma meantone tuning. who
actually was the first author to note the
equivalence of 19edo with 1/3-comma MT?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> Salinas certainly knew the measurement of
> the "5th" (generator) of 1/3-comma meantone,
> and after calculating a 19-tone chain, which
> he manifestly did (and explained), it would have
> been obvious that one more step in the chain
> would result in a pitch very close to the one
> he started with.

i, too, would like to think it's obvious. but he did start with a ji
scale and reduced it, rather than going through the process of
generating the scale one fifth at a time. so maybe -- just maybe --
the thought never occured to him.

> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > Salinas certainly knew the measurement of
> > the "5th" (generator) of 1/3-comma meantone,
> > and after calculating a 19-tone chain, which
> > he manifestly did (and explained), it would have
> > been obvious that one more step in the chain
> > would result in a pitch very close to the one
> > he started with.
>
> i, too, would like to think it's obvious. but he did
> start with a ji scale and reduced it, rather than going
> through the process of generating the scale one fifth
> at a time. so maybe -- just maybe -- the thought never
> occured to him.

funny that you're off in this direction now ...
i was just wondering this afternoon if perhaps Salinas
realized the closure aspect of the 19-note set of
1/3-comma meantone first and *then* chose that
particular 24-note JI set for the purpose of his
demonstration of tempering JI.

... i'd love to have the time to work on that
as a research project! ... oh well ...

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > Salinas certainly knew the measurement of
> > > the "5th" (generator) of 1/3-comma meantone,
> > > and after calculating a 19-tone chain, which
> > > he manifestly did (and explained), it would have
> > > been obvious that one more step in the chain
> > > would result in a pitch very close to the one
> > > he started with.
> >
> > i, too, would like to think it's obvious. but he did
> > start with a ji scale and reduced it, rather than going
> > through the process of generating the scale one fifth
> > at a time. so maybe -- just maybe -- the thought never
> > occured to him.
>
>
> funny that you're off in this direction now ...
> i was just wondering this afternoon if perhaps Salinas
> realized the closure aspect of the 19-note set of
> 1/3-comma meantone first and *then* chose that
> particular 24-note JI set for the purpose of his
> demonstration of tempering JI.

it's more likely, but unfortunately we may never know.

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, March 31, 2003 10:45 PM
> Subject: [tuning] Re: Salinas and 19edo (was:: Dictionary updates:
2/7-comma ...)
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > funny that you're off in this direction now ...
> > i was just wondering this afternoon if perhaps Salinas
> > realized the closure aspect of the 19-note set of
> > 1/3-comma meantone first and *then* chose that
> > particular 24-note JI set for the purpose of his
> > demonstration of tempering JI.
>
> it's more likely, but unfortunately we may never know.

ah, but you know how *i* love to speculate!! :)

-monz