margo, manuel -- was it kirnberger?

> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
>
> > Viking [161, -84, -12] .015361 cents
>
> this is the difference between 11 pythagorean commas and 12
syntonic
> commas. i'm going to call it "atomic" instead, unless someone comes
> up with a better name . . .

was it kirnberger who proposed foreshortening each fifth by a schisma
to approximate 12-equal? a chain of 12 such fifths would fail to
close on itself by a mere "atom", or .015361 cents . . .

note that this tiny difference between 11 pythagorean commas and 12
syntonic commas was the subject of quite a lot of conversation on
this list a while ago, mainly between monz and bob wendell . . .

Paul asked:
>was it kirnberger who proposed foreshortening each fifth by a schisma
>to approximate 12-equal? a chain of 12 such fifths would fail to
>close on itself by a mere "atom", or .015361 cents . . .

That I don't know, but he was at least the first to publish a temperament
with a fifth flattened by a schisma, in the 1780's. Then Prinz did so
around 1810.
Found this on http://www.kirnberger.fsnet.co.uk

Manuel

--- In tuning@yahoogroups.com, manuel.op.de.coul@e... wrote:
> Paul asked:
> >was it kirnberger who proposed foreshortening each fifth by a
schisma
> >to approximate 12-equal? a chain of 12 such fifths would fail to
> >close on itself by a mere "atom", or .015361 cents . . .
>
> That I don't know, but he was at least the first to publish a
temperament
> with a fifth flattened by a schisma, in the 1780's.

just one fifth so flattened?

> Then Prinz did so
> around 1810.

again, just one fifth?

> Found this on http://www.kirnberger.fsnet.co.uk

can't find them there. where do i look?

--- In tuning@yahoogroups.com, manuel.op.de.coul@e... wrote:
> Paul asked:
> >was it kirnberger who proposed foreshortening each fifth by a
schisma
> >to approximate 12-equal? a chain of 12 such fifths would fail to
> >close on itself by a mere "atom", or .015361 cents . . .
>
> That I don't know,

i searched the archives and margo has apparently referenced this
numerous times.

message #6240:

"For example, in 1766, Johann Philipp Kirnberger proposed a method for
approximating 12-tet by finding a series of pure fifths and thirds
together forming a fifth of 16384:10935, an interval repeated 11 times
to complete the system. Here all intervals are based on (very large!)
integer ratios, and the system provides pure 2:1 octaves, but not pure
3:2 fifths or 4:3 fourths. Thus Kirnberger's "quasi-12-tet" is not a
JI system in our sense."

see also message #2832, #12459, and #16007 . . .

i'm considering naming the .015361 cent interval the "atom of
kirnberger" or something like that -- i would appreciate the feedback
of margo and others . . . gene and i found that it's a really nice
stopping point when making a list of 5-limit unison vectors.

Paul wrote:
> just one fifth so flattened?

Yes, F#-C#.

>> Then Prinz did so
>> around 1810.

> again, just one fifth?

Yes, B-F#.

See http://www.kirnberger.fsnet.co.uk/TempsII.htm

> i'm considering naming the .015361 cent interval the "atom of
> kirnberger" or something like that

Sounds good to me.

Manuel