alternate temperament formula?

hi there,

i'm new here. does anybody have any idea what formula the following temperaments use?

Bach (Klais) : Hz
262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18, 393.70, 415.30, 440.00, 467.18, 492.26
Just (Barbour) : Hz
264.00, 275.00, 297.00, 316.80, 330.00, 352.00, 371.25, 396.00, 412.50, 440.00, 475.20, 495.00
Schlick (H. Vogel) : cents
0, -14.5, -5.5, -5.5, -11, 3, -16, -2.5, -12.5, -8, 1, -13.5

Bach (Klais) and Just (Barbour) are from http://www.rdrop.com/users/tblackb/music/temperament/ site.
Schlick (H. Vogel) is used by CTS-5 at http://www.vogel-scheer.de/GB/CTS/Tuning%20Set.html site.

the info i need is something like the following so that i can calculate the exact pitch.

e.g.
Werkmeister III (circle of fifths, -1/4 pythagorean comma)
C -1/4P G -1/4P D -1/4P A 0 E 0 B -1/4P F# 0 C# 0 G# 0 Eb 0 Bb 0 F 0 C

i'm going to use these temperaments in my tuner application for Mac OS X.

http://homepage.mac.com/katsura/shareware.html#CHTN

also i'm looking for the info for Neidhardt I, II, and III temperaments. and i don't even know if these are the right naming for Neidhardt temperaments, some uses the name Neidhardt 1724, 1729. are there standardized names for each temperament, like ISO standard or something? it's very confusing...

oh, one more thing, which one is correct, Vallotti or Valotti? :)

regards,
katsura
Sunnyvale, CA
USA

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:
> hi there,
>
> i'm new here. does anybody have any idea what formula the following
> temperaments use?
>
> Bach (Klais) : Hz
> 262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18, 393.70,
415.30,
> 440.00, 467.18, 492.26
> Just (Barbour) : Hz
> 264.00, 275.00, 297.00, 316.80, 330.00, 352.00, 371.25, 396.00,
412.50,
> 440.00, 475.20, 495.00

These are Hz frequencies, beginning with C, calibrated to A-440.

> Schlick (H. Vogel) : cents
> 0, -14.5, -5.5, -5.5, -11, 3, -16, -2.5, -12.5, -8, 1, -13.5

Cents deviations from 12-equal, most likely beginning with C.

> the info i need is something like the following so that i can
calculate
> the exact pitch.
>
> e.g.
> Werkmeister III (circle of fifths, -1/4 pythagorean comma)
> C -1/4P G -1/4P D -1/4P A 0 E 0 B -1/4P F# 0 C# 0 G# 0 Eb 0 Bb 0
F 0 C

Hmm . . . I'm confused as to how this makes it easier to calculate
the exact pitch. Isn't a list of pitches more straighforward, since
then there's less to calculate. I honestly want to help make this as
simple for you as possible, so tell me a little more about the format
your application expects the tuning data to be in. I can then convert
the tunings above to that format for you.

> also i'm looking for the info for Neidhardt I, II, and III
> temperaments. and i don't even know if these are the right naming
for
> Neidhardt temperaments, some uses the name Neidhardt 1724, 1729.

Neidhardt used names like "village" and "small city" for his
temperaments -- you can find a lot of information on them here:

http://ourworld.compuserve.com/homepages/paulpoletti/T4D.PDF

Looking forward to serving you more usefully,
Paul

> > Bach (Klais) : Hz
> > Just (Barbour) : Hz
>
> These are Hz frequencies, beginning with C, calibrated to A-440.
>
> > Schlick (H. Vogel) : cents
>
> Cents deviations from 12-equal, most likely beginning with C.

sorry, i wasn't clear. i know what they are. i need to know what kind
of ratio or comma it uses. pythagorean or syntonic?

> > e.g.
> >   Werkmeister III (circle of fifths, -1/4 pythagorean comma)
> >   C -1/4P G -1/4P D -1/4P A 0 E 0 B -1/4P F# 0 C# 0 G# 0 Eb 0 Bb 0
> F 0 C
>
> Hmm . . . I'm confused as to how this makes it easier to calculate
> the exact pitch. Isn't a list of pitches more straighforward, since
> then there's less to calculate. I honestly want to help make this as
> simple for you as possible, so tell me a little more about the format
> your application expects the tuning data to be in. I can then convert
> the tunings above to that format for you.

i do know how to calculate pitches and cents from pythagorean and
syntonic comma. so i just need to know where and how much it needs to
be applied for these temperaments. and that's what "C -1/4P G -1/4P
..." is.

Bach (Klais), Just (Barbour), Schlick (H. Vogel), Neidhardt I, II,
III.

> Neidhardt used names like "village" and "small city" for his
> temperaments -- you can find a lot of information on them here:
>
> http://ourworld.compuserve.com/homepages/paulpoletti/T4D.PDF

i was aware of the T4D.PDF but some other sources use I, II, II, or
1724, 1729, and i have no clue which is which, or these are totally
different temperaments. it's very confusing...

thanks,
katsura

> Just (Barbour) : Hz
> 264.00, 275.00, 297.00, 316.80, 330.00, 352.00, 371.25, 396.00, > 412.50, 440.00, 475.20, 495.00

i figured this one out. it's a just intonation using the following ratios.

1/1, 25/24, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 25/16, 5/3, 9/5, 15/8

is there a correct or proper name for this just intonation? is "Just (Barbour)" okay?

katsura

hi katsura,

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:

> > Just (Barbour) : Hz
> > 264.00, 275.00, 297.00, 316.80, 330.00, 352.00, 371.25, 396.00,
> > 412.50, 440.00, 475.20, 495.00
>
> i figured this one out. it's a just intonation using the following
> ratios.
>
> 1/1, 25/24, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 25/16, 5/3, 9/5, 15/8
>
> is there a correct or proper name for this just intonation?
> is "Just (Barbour)" okay?
>
> katsura

the triangular lattice of this scale is as follows:

(to see this correctly on the Yahoo interface,
you'll have to click "Reply")

25/24--25/16
/ \ / \
5/3---5/4--15/8--45/32
/ \ / \ / \ /
4/3--1/1---3/2----9/8
\ / \ /
6/5---9/5

it's a 12-tone periodicity-block, defined by the
unison-vectors 81/80 (syntonic comma) and 128/125
(enharmonic diesis).

it is known as "Marpurg's monochord #1" tuning,
published by Marpurg in 1776 in his _Versuch uber
die musikalische Temperatur_, and called by Barbour
the "model form of just-intonation" (see p 99 in
Barbour).

-monz

hi katsura,

about your other scale ...

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:

> hi there,
>
> i'm new here. does anybody have any idea what formula the following
> temperaments use?
>
> Bach (Klais) : Hz
> 262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18,
> 393.70, 415.30, 440.00, 467.18, 492.26

i found a scale which is a circulating temperament,
using various fractions of a Pythagorean comma (P)
between the notes in its "circle of 5ths".

to the same accuracy of 2 decimal places which you
used, and arranged in the same format, here are the
frequencies:

262.81, 276.89, 294.33, 311.48, 328.72, 350.41, 369.19,
393.77, 415.34, 440.00, 467.22, 492.25

here is a schematic of the "circle of 5ths":

Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 2/7P E 1/8P B 0 F# 0 C# 0 G# 0

to an accuracy of 7 decimal places, here are the
frequencies resulting from this scheme:

B 492.2526789
Bb 467.2177848
A 440
G# 415.3381978
G 393.7700888
F# 369.1895091
F 350.4133386
E 328.7248013
Eb 311.4785232
D 294.328761
C# 276.8921319
C 262.8100039

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi katsura,
>
>
> about your other scale ...
>
>
> --- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...>
wrote:
>
> > hi there,
> >
> > i'm new here. does anybody have any idea what formula the
following
> > temperaments use?
> >
> > Bach (Klais) : Hz
> > 262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18,
> > 393.70, 415.30, 440.00, 467.18, 492.26
>
>
>
> i found a scale which is a circulating temperament,
> using various fractions of a Pythagorean comma (P)
> between the notes in its "circle of 5ths".
>
> to the same accuracy of 2 decimal places which you
> used, and arranged in the same format, here are the
> frequencies:
>
> 262.81, 276.89, 294.33, 311.48, 328.72, 350.41, 369.19,
> 393.77, 415.34, 440.00, 467.22, 492.25
>
>
> here is a schematic of the "circle of 5ths":
>
> Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 2/7P E 1/8P B 0 F# 0 C# 0 G# 0

Hi Monz,

If I interpret these as fifths *flattened* by *exactly* these
frections of a Pythagorean comma, something isn't adding up exactly
right.

I'm not sure if Katsura would be too concerned about this, but if you
tune these 12 fifths consecutively, and reduce by octaves, you end up
1/168 of a pythagorean comma from where you began.

Since you gave the scale supposedly to *seven* decimal places, I
didn't feel this was out of line to point out.

Cheers,
Paul

> > > Bach (Klais) : Hz
> > > 262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18,
> > > 393.70, 415.30, 440.00, 467.18, 492.26
> >
> > Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 2/7P E 1/8P B 0 F# 0 C# 0 G# 0
>
> I'm not sure if Katsura would be too concerned about this, but if you
> tune these 12 fifths consecutively, and reduce by octaves, you end up
> 1/168 of a pythagorean comma from where you began.

thanks monz and Paul. i'm looking for the exact ratios that "Bach (Klais)" (whatever it is) is using. monz's guess is very close to the given frequencies but not something i'm looking for, and of course something is probably wrong if the pythagorean commas don't add up to one.

i've figured most temperaments in Terry Blackburn's list by the name (and found a typo in Silbermann by the way) but i just cannot find any temperaments called "Bach (Klais)" (the "Bach" temperament as reported by Klais) anywhere on the internet. i asked Terry but he may have lost the original documents (or he is still looking all over his house...).

katsura

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:
> > > > Bach (Klais) : Hz
> > > > 262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18,
> > > > 393.70, 415.30, 440.00, 467.18, 492.26
> > >
> > > Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 2/7P E 1/8P B 0 F# 0 C#
0 G# 0
> >
> > I'm not sure if Katsura would be too concerned about this, but
if you
> > tune these 12 fifths consecutively, and reduce by octaves, you
end up
> > 1/168 of a pythagorean comma from where you began.
>
> thanks monz and Paul. i'm looking for the exact ratios that "Bach
> (Klais)" (whatever it is) is using.

Fractions of a pythagorean comma can't be expressed as exact
frequency ratios (i.e., in terms of Rational Intontation) to begin
with. Other temperaments can't be expressed even using exact
fractions of various commas. Most likely, the practical limitations
of whatever synth or tuning procedure you're planning to use make
such accuracy as Monz gave totally unnecessary anyway.

> monz's guess is very close to the
> given frequencies but not something i'm looking for,

I guess I need to understand better what it is exactly that you *are*
looking for . . .

Let me know,
Paul

> Fractions of a pythagorean comma can't be expressed as exact
> frequency ratios (i.e., in terms of Rational Intontation) to begin
> with. Other temperaments can't be expressed even using exact
> fractions of various commas. Most likely, the practical limitations
> of whatever synth or tuning procedure you're planning to use make
> such accuracy as Monz gave totally unnecessary anyway.

sorry i didn't mean fractions, i meant ratios like 1:1.059463094 = 1:2^(1/12). i just don't know the right English term.

i'm aware of the practical limitations. i'm interested in the correctness of the temperament i'm using in my tuner, and theory behind it.

> I guess I need to understand better what it is exactly that you *are*
> looking for . . .

i'm looking for the formula or data so that i can calculate the pitch. if it's a pythagorean, i just need to know how much comma i need to apply where. and i don't even know if "Bech (Klais)" is a pythagorean.

e.g.
Werkmeister III (1/4 pythagorean comma)
C -1/4P G -1/4P D -1/4P A 0 E 0 B -1/4P F# 0 C# 0 G# 0 Eb 0 Bb 0 F 0 C

Meantone (1/4 syntonic comma)
C -1/4S G -1/4S D -1/4S A -1/4S E -1/4S B -1/4S F# -1/4S C# -1/4S G# 7/4S Eb -1/4S Bb -1/4S F -1/4S C

katsura

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:
> > Fractions of a pythagorean comma can't be expressed as exact
> > frequency ratios (i.e., in terms of Rational Intontation) to
begin
> > with. Other temperaments can't be expressed even using exact
> > fractions of various commas. Most likely, the practical
limitations
> > of whatever synth or tuning procedure you're planning to use make
> > such accuracy as Monz gave totally unnecessary anyway.
>
> sorry i didn't mean fractions, i meant ratios like 1:1.059463094 =
> 1:2^(1/12). i just don't know the right English term.
>
> i'm aware of the practical limitations. i'm interested in the
> correctness of the temperament i'm using in my tuner, and theory
behind
> it.
>
> > I guess I need to understand better what it is exactly that you
*are*
> > looking for . . .
>
> i'm looking for the formula or data so that i can calculate the
pitch.
> if it's a pythagorean, i just need to know how much comma i need to
> apply where. and i don't even know if "Bech (Klais)" is a
>pythagorean.

Looking at some other websites, which give the tuning in a different
format (cents), I'm almost certain that the "2/7" Monz gave is really
a leftover figure and an approximation. So one could get the exact
values by rotating Monz's figures thus:

E 1/8P B 0 F# 0 C# 0 G# 0 Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A

The interval between A and E was a 'leftover' interval. This is a
very common predicament in tuning, even found today in the Modern
Indian Gamut . . .

Bach tuned his harpsichord by ear. It may have been within 1 cent of
the Bach-Barnes temperament, or the Bach-Kellner temperament, or
Werckmeister III, or something else (I'm not familiar with the Klais
variation, but it's probably in Scala). But specifying it more
accurately than 1 cent is probably meaningless. If you're using an
electronic tuner, you're going to run into far bigger errors just
because of the nature of such devices. Not to mention the historical
myths surrounding temperament. Please read this article:

http://ourworld.compuserve.com/homepages/paulpoletti/T4D.PDF

hi paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

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

> > <snip>
> > here is a schematic of the "circle of 5ths":
> >
> > Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 2/7P E 1/8P B 0 F# 0 C# 0
G# 0
>
> Hi Monz,
>
> If I interpret these as fifths *flattened* by *exactly*
> these frections of a Pythagorean comma, something isn't
> adding up exactly right.
>
> I'm not sure if Katsura would be too concerned about this,
> but if you tune these 12 fifths consecutively, and reduce
> by octaves, you end up 1/168 of a pythagorean comma from
> where you began.
>
> Since you gave the scale supposedly to *seven* decimal
> places, I didn't feel this was out of line to point out.
>
> Cheers,
> Paul

yes, you are absolutely right to point that out, and
thanks for doing so. i simply didn't have any more time
to spend on that post, but it was bothering me that
-- without taking the trouble to actually add up all of
those fraction-of-a-Pythagorean-comma temperings to
see if they formed a perfect 2:1 "8ve" -- i could sense
that the mixture of denominators in the fractions might
not add up.

also thanks for point out that i meant the "5ths" to
be flattened ... i just forgot to do that.

-monz

i probably should have included the cents as well.

what is this temperament? where did it come from? what comma or ratio does it use? does it look familiar to anyone?

Bach (Klais)
262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18, 393.70, 415.30, 440.00, 467.18, 492.26
0.000, -9.444, -3.750, -5.639, -12.369, -1.840, -11.306, 0.021, -7.511, -7.491, -3.721, -13.191

katsura

hi katsura and paul,

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:

> > > > Bach (Klais) : Hz
> > > > 262.76, 276.87, 294.30, 311.46, 328.70, 350.37, 369.18,
> > > > 393.70, 415.30, 440.00, 467.18, 492.26
> > >
> > > Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 2/7P E 1/8P B 0 F# 0 C#
0 G# 0
> >
> > I'm not sure if Katsura would be too concerned about this,
> > but if you tune these 12 fifths consecutively, and reduce
> > by octaves, you end up 1/168 of a pythagorean comma from
> > where you began.
>
> thanks monz and Paul. i'm looking for the exact ratios that
> "Bach (Klais)" (whatever it is) is using. monz's guess is
> very close to the given frequencies but not something i'm
> looking for, and of course something is probably wrong if
> the pythagorean commas don't add up to one.
>
> i've figured most temperaments in Terry Blackburn's list by
> the name (and found a typo in Silbermann by the way) but i
> just cannot find any temperaments called "Bach (Klais)"
> (the "Bach" temperament as reported by Klais) anywhere on
> the internet. i asked Terry but he may have lost the original
> documents (or he is still looking all over his house...).
>
> katsura

i wish i had had the time to elaborate further when i
first posted on this. anyway, simply juggling the numbers
in the fraction-of-a-Pythagorean-comma tempering until
they work out to form a perfect 2:1 "8ve" would do the
trick.

my guess is that this is one of Neidhardt's temperaments,
which were among the most mathematically complicated of
the circulating temperaments of the Baroque era. a little
research into his scales would probably show you the answer.
they're all listed in Barbour's book, and you can probably
find something if you search the internet -- i know for
sure that some of them were discussed in the archives of
this list.

anyway, this one works and is pretty close to what you
posted:

Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 1/4P E 1/6P B 0 F# 0 C# 0 G#

if i get time, i'll post the frequencies of those.

-monz

> Eb 0 Bb 0 F 0 C 1/12P G 1/4P D 1/4P A 1/4P E 1/6P B 0 F# 0 C# 0 G#

thanks again, monz. it's close but probably not the ratios that "Bach (Klais)" is using. it could be a mix of both pythagorean and syntonic like Kirnberger III.

i calculated it.

262.810003937, 276.869798387, 294.328760963, 311.478523185, 328.883931299, 350.413338583, 369.159731182, 393.770088773, 415.304697580, 440.000000000, 467.217784777, 492.212974910

0.000000000, -9.775004327, -3.910001731, -5.865002596, -11.730005192, -1.955000865, -11.730005192, 0.000000000, -7.820003462, -7.820003462, -3.910001731, -13.685006058

katsura

This one is indeed in the scale archive too: klais.scl.
Johannes Klais, Bach temperament
256/243 196.0900 32/27 387.2925 4/3 1024/729 700.0000
128/81 892.1800 16/9 4096/2187 2/1

See http://www.xs4all.nl/~huygensf/doc/scales.zip

Manuel

> This one is indeed in the scale archive too: klais.scl.
> Johannes Klais, Bach temperament
> 256/243 196.0900 32/27 387.2925 4/3 1024/729 700.0000
> 128/81 892.1800 16/9 4096/2187 2/1
>
> See http://www.xs4all.nl/~huygensf/doc/scales.zip

thanks Manuel. it's getting closer. it's using 1/24 pythagorean comma.

0, -9.775, -3.91, -5.865, -12.7075, -1.955, -11.73, 0, -7.82, -7.82, -3.91, -13.685

Eb 0 Bb 0 F 0 C -1/12P G -1/4P D -1/4P A -7/24P E -1/8P B 0 F# 0 C# 0 G# 0

does anybody know who created klais.scl and what is the source?

could someone confirm if this is in fact "Bach (Klais)" (or whatever it's called officially)?

katsura

--- In tuning@yahoogroups.com, Hidetomo Katsura <katsura@m...> wrote:
> > This one is indeed in the scale archive too: klais.scl.
> > Johannes Klais, Bach temperament
> > 256/243 196.0900 32/27 387.2925 4/3 1024/729 700.0000
> > 128/81 892.1800 16/9 4096/2187 2/1
> >
> > See http://www.xs4all.nl/~huygensf/doc/scales.zip
>
> thanks Manuel. it's getting closer. it's using 1/24 pythagorean
comma.
>
> 0, -9.775, -3.91, -5.865, -12.7075, -1.955, -11.73, 0, -7.82, -
7.82,
> -3.91, -13.685
>
> Eb 0 Bb 0 F 0 C -1/12P G -1/4P D -1/4P A -7/24P E -1/8P B 0 F# 0 C#
0
> G# 0

This is exactly as I surmised in a recent post. I went from E to A so
I omitted mention of the -7/24P fifth that necessarily results
between the ends of the chain.

> does anybody know who created klais.scl and what is the source?

It says above -- Johannes Klais is the source. Keller, Kellner,
Barnes and others also created (different) "Bach" tunings.

> > does anybody know who created klais.scl and what is the source?
>
> It says above -- Johannes Klais is the source. Keller, Kellner,
> Barnes and others also created (different) "Bach" tunings.

i know. i meant the .scl file. who entered the ratios and cents in the .scl file on a computer?

and what is the comma or ratios it's using? this is the main question i've been asking.

katsura