Werckmeister question

Hello all,

I was looking at the Werckmeister III scale that comes with Scala, and in the process of "playing" with some other scales, I noticed something that left me a little perplexed, and curious. I was wondering if I was just "seeing something" or if there is some connection.

This is the scale for Werckmeister III, and below is another scale (the description I used to create it is below as well).

0: 1/1 0.000 unison, perfect prime
1: 256/243 90.225 limma, Pythagorean minor second
2: 192.180 cents 192.180
3: 32/27 294.135 Pythagorean minor third
4: 390.225 cents 390.225
5: 4/3 498.045 perfect fourth
6: 1024/729 588.270 Pythagorean diminished fifth
7: 696.090 cents 696.090
8: 128/81 792.180 Pythagorean minor sixth
9: 888.270 cents 888.270
10: 16/9 996.090 Pythagorean minor seventh
11: 1092.180 cents 1092.180
12: 2/1 1200.000 octave

|
0: 1/1 0.000 unison, perfect prime
1: 256/243 90.225 limma, Pythagorean minor second
2: 9/8 203.910 major whole tone
3: 32/27 294.135 Pythagorean minor third
4: 81/64 407.820 Pythagorean major third
5: 4/3 498.045 perfect fourth
6: 1024/729 588.270 Pythagorean diminished fifth
7: 3/2 701.955 perfect fifth
8: 128/81 792.180 Pythagorean minor sixth
9: 27/16 905.865 Pythagorean major sixth
10: 16/9 996.090 Pythagorean minor seventh
11: 243/128 1109.775 Pythagorean major seventh
12: 2/1 1200.000 octave

I used the following "scala" commands to create this scale:

equal 12
approximate 3

The differences between the two scales lies in the "diatonic" notes (2, 4, 7, 9, & 11). I was curious how all the "chromatic notes" are the same in both scales (just coincidence???). How did Werchmeister arrive at these other values for his scale? The pitches that he selects don't seem to have a "common name" (as defined in Scala), and those pitches (in the scale file) are presented in Cent values and not in a ratio (not that this really means anything).

Also, if this is non-edo scale, what pitch was used (by Bach, and others of the time) as the base pitch? What was the frequency used to the base pitch (ie A440, A420, etc.)?

Thanks,

Mike

hi Michael,

> From: "Michael J McGonagle" <fndsnd@rcnchicago.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, November 02, 2002 8:25 PM
> Subject: [tuning] Werckmeister question
>
>
> Hello all,
>
> I was looking at the Werckmeister III scale that comes with Scala, and
> in the process of "playing" with some other scales, I noticed something
> that left me a little perplexed, and curious. I was wondering if I was
> just "seeing something" or if there is some connection.
>
> <snipped tables of scales...>
>
> The differences between the two scales lies in the "diatonic" notes (2,
> 4, 7, 9, & 11). I was curious how all the "chromatic notes" are the same
> in both scales (just coincidence???). How did Werchmeister arrive at
> these other values for his scale? The pitches that he selects don't seem
> to have a "common name" (as defined in Scala), and those pitches (in the
> scale file) are presented in Cent values and not in a ratio (not that
> this really means anything).

i have a webpage in the Tuning Dictionary which explains how
Werckmeister devised Werckmeister III tuning.

http://sonic-arts.org/dict/werckmeister.htm

basically, Werckmeister created a closed 12-tone unequal tuning,
distributing the Pythagorean comma by narrowing four of the "5ths"
(C:G:D:A and B:F#) by 1/4 of a Pythagorean comma, and tuning all
the rest of the "5ths" as 3:2s.

since he divided a rational interval (the Pythagorean comma)
into logarithmically equal parts, the notes affected by that
division are irrational numbers, so they cannot be described
as ratios. but they *can* be described with a {2,3}-prime-factor
vector notation which gives their exact values (... cents values
for this scale are by necessity approximations). here it is:

prime-factor vector
note 2 3

B [9&1/4 -4]
Bb [2 -2]
A [11&1/4 -6]
G# [11 -4]
G [3&3/4 -2]
F# [13 -6]
F [1 -1]
E [10&1/4 -5]
Eb [3 -3]
D [7&1/2 -4]
C# [12 -5]
C [0 0]

note that the Ab [4 -4] is only different from G# [11 -4]
by the difference in prime-factor 2, which, since it is an
integer value (namely, 7), only describes "8ves" and thus
does not affect the "8ve"-equivalent cents-value of the
note; therefore, Ab [4 -4] is the same as G# [11 -4] and
the tuning is thus closed.

as for your other question, regarding the reference pitch,
Ellis's appendix to his translation of Helmholtz's
_On the sensations of tone..._ gives a "History of pitch
in Europe" which should help answer your question.

-monz
"all roads lead to n^0"

--- In tuning@y..., Michael J McGonagle <fndsnd@r...> wrote:

> I used the following "scala" commands to create this scale:
>
> equal 12
> approximate 3
>
>
> The differences between the two scales lies in the "diatonic" notes
(2,
> 4, 7, 9, & 11). I was curious how all the "chromatic notes" are the
same
> in both scales (just coincidence???).

not coincidence at all!

> How did Werchmeister arrive at
> these other values for his scale?

the chromatic notes, as you've observed, form a pythagorean scale.
since the pythagorean comma is very close in size to the syntonic
comma, and has the correct sign, werckmeister saw that he could close
the circle of fifths by distributing out the pythagorean comma much
as aron's meantone temperament distributes the syntonic comma. you
can get a sense of how well werckmeister acheived his goals by
comparing several major thirds in the "whiter" keys against the just
ratio 5/4, and several minor thirds in those keys against the just
ratio 6/5. so, relative to your scale, werckmeister achieves two
goals: closing the circle of fifths, and providing the more common
key areas with much purer thirds and sixths.

Hi Monz and wallyesterpaulrus (I did look for something that might look like a name, but did not find one...),

Thanks for your explainations. In the past few weeks from trying to tune my piano to "pure fifths", and am really starting to realize just how "subtle" (and utterly amazing) the art of tuning is.

I guess I need to pick up a copy of "On the sensation of..." by Helmholtz (I just reread "A Genesis of a Music", and Harry talks of how he was influenced by it).

Thanks again,

Mike

wallyesterpaulrus wrote:
> --- In tuning@y..., Michael J McGonagle <fndsnd@r...> wrote:
> > >>I used the following "scala" commands to create this scale:
>>
>>equal 12
>>approximate 3
>>
>>
>>The differences between the two scales lies in the "diatonic" notes > > (2, > >>4, 7, 9, & 11). I was curious how all the "chromatic notes" are the > > same > >>in both scales (just coincidence???).
> > > not coincidence at all!
> > >>How did Werchmeister arrive at >>these other values for his scale?
> > > the chromatic notes, as you've observed, form a pythagorean scale. > since the pythagorean comma is very close in size to the syntonic > comma, and has the correct sign, werckmeister saw that he could close > the circle of fifths by distributing out the pythagorean comma much > as aron's meantone temperament distributes the syntonic comma. you > can get a sense of how well werckmeister acheived his goals by > comparing several major thirds in the "whiter" keys against the just > ratio 5/4, and several minor thirds in those keys against the just > ratio 6/5. so, relative to your scale, werckmeister achieves two > goals: closing the circle of fifths, and providing the more common > key areas with much purer thirds and sixths.
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