Marpurg (1776) and his irregular systems of temperament

1) The three variants of Piagui octaves I deduced were analyzed by Mike Battaglia who found less than four cents of difference between any equal tempered tone and the corresponding tone of any of the three Piagui scale variants (Piagui I, Piagui II and Piagui III).

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Here I reproduce the values calculated by Mike Battaglia for the three versions of Piagui intonations:

PIAGUI I SEMITONE SEQUENCE: (C K K P K K P K K P K K P 2)

K = 1.06066017178 = 101.955 cents
P = 1.0570729911 = 96.090 cents

PIAGUI I TONE CENTS
C - 0...................cents
C# - 101,955...... cents
D - 203,91 .........cents
Eb - 300 ...............cents
E - 401,955 .......cents
F - 503,91 .........cents
F# - 600 ..............cents
G - 701,955 ......cents
Ab - 803,91 ........cents
A - 900 ..............cents
Bb - 1001,955 ....cents
B - 1103,91 .......cents
C - 1200 ............cents
-----------------------

PIAGUI I TONE CENTS MINUS NOMINAL TEMPERED CENTS
C .......... 0 ..............cents
C# ..........+ 1,955 cents
D .......... + 3,91 cents
Eb.......... 0 .............. cents
E ............+ 1,955 cents
F ............+ 3,91 cents
F#........... 0 ............. cents
G.............+ 1,955 cents
Ab ..........+ 3,91 cents
A .............0 ............. cents
Bb...........+ 1,955 cents
B .......... + 3,91 cents
2C ......... 0 .............. cents

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--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:> Here is the second layout he mentioned:> PIAGUI II SEMITONE SEQUENCE: (Do=1) x K x P x K x K x P x K x K x P x K x K x P x K = 2> K = 1.06066017178 = 101.955 cents> P = 1.0570729911 = 96.090 centsPIAGUI II TONE CENTS > C - 0 cents ..............................0 > C# - 101.955 cents .................+1,955 > D - 198.045 cents.................-1,955 > Eb - 300 cents ........................ 0> E - 401.955 cents..................+1,955 > F - 498.045 cents...................-1,955 > F# - 600 cents .........................0> G - 701.955 cents..................+1,955 > Ab - 798.045 cents.................-1,955 > A - 900 cents........................ 0 > Bb - 1001.955 cents..............+1,955 > B - 1098.045 cents.............. -1,955 > C - 1200 cents..................... 0xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxThe third layout he mentioned: PIAGUI III = (Do=1) x P x K x K x P x K x K x P x K x K x P x K x K = 2> K = 1.06066017178 = 101.955 cents> P = 1.0570729911 = 96.090 centsPIAGUI III TONE CENTS -> C - 0 cents 0 > C# - 96.090 cents - 3,91 - > D - 198.045 cents -1,955 > Eb - 300 cents 0> E - 396.090 cents - 3,91 > F - 498.045 cents - 1,955 > F# - 600 cents 0> G - 696.090 cents - 3,91 > Ab - 798.045 cents - 1,955 > A - 900 cents 0> Bb - 996.090 cents - 3,91 > B - 1098.045 cents - 1,955 > C - 1200 cents 0

2) I read in tuning Yahoogroups that tone discrepances like +/- 4 cents referred to the corresponding tempered tones make almost the same sound of the referred tempered tone. I agree with that. However, the equal tempered scale is also an almost perfect scale and consequently it is not a 100 % good reference; therefore we should not establish 5 or 4 or 3,,,,,, cents as a minimum acceptable discrepance; before that we should know and acknowledge the scale able to produce flawless harmony.

3) Mr. Brad Lehman stated that one of the three Piagui versions was proposed in 1776 by F. W. MARPURG, born in Breslau as detailed in the article "IRREGULAR SYSTEMS OF TEMPERAMENT" published in 1948 by .J. Murray Barbour. I got the 8 pages article.

Brad Lehman wrote that in the mentioned article, page 23 row H, can be found the coincident proposal of MARPURG. I got the article. In that page 23, his eleven proposals of "Sophisticated Systems" -- "Temperament of fifths in cents" don't show any mathematical support and the supposed coincident scale is given by the difference between the tone cents and the twelve nominal cents of tempered octave (0, 100, 200, .............1000, 1100, 1200).

(As printed in article): ................................................................................(Ordained):
......... c ......g.....d......a......e......b......f#......c#.....g#.....eb....bb......f ------- C ......C#.....D ....Eb......E......F.....F#....G....Ab......A Bb.....B......C A ....................................................................................................................................................................................................................................
B ...................................................................................................................................................................................................................................
C ...................................................................................................................................................................................................................................
D ..................................................................................................................................................................................................................................
E .................................................................................................................................................................................................................................
F ...................................................................................................................................................................................................................................
G ..................................................................................................................................................................................................................................
H ...... 0 .... 6 .... 0 .... 0..... 6 ..... 0 .... 0 ..... 6 ..... 0 ..... 0 ..... 6 .....0............0....... 6....... 0 .......0.......6......0......0......6.......0........0.......6.......0......0
J ..........................................................................................................................................................................................................................................
K .........................................................................................................................................................................................................................................
L ..........................................................................................................................................................................................................................................

We can see that the cents shown In row H are quite different if compared with the ones detailed below where eight values per octave establish the authentic sets of perfect scales no matter these values are lower than 4.

NOTE : ......... C ........ C# ............ D ........ Eb ...... E .............. F ........ F# ...... G ................ Ab ...... A ......... Bb .......... B ....... C

I) PIAGUI I : ....0......+1,955....... +3,91...... 0......+1,955.......+ 3,91...... 0......+1,955........+ 3,91.......0......+ 1,955.....+ 3,91......0

II) PIAGUI II :...0......+1,955....... - 1,955.....0......+1,955....... - 1,955.....0......+1,955....... - 1,955......0......+1,955...... - 1,955....0

III) PIAGUI III : 0......- 3,91.......... - 1,955.....0......- 3,91......... - 1,955.....0......- 3,91..........- 1,955...... 0..... - 3,91....... - 1,955.....0

The figures +/- 1,955, +/- 3,91 and 0 cents work in the three Piagui variants,

Consequently, Brad Lehman misunderstood the data contained in row H proposed by MARPURG and stated that the irregular set (row H) containing only 4 discrepances per octave with a value of 6 each, coincides with one of the Piagui variants. You can see that there is no any aspect that could even show similitude between the MARPURG proposal and any of the three Piagui variants.

4) I realized that the idea is to disapprove the scales whose tone frequencies are about the equal tempered ones. In other words, the three piaguis could be considered as second quality intonations for having less than 4 cents of descrepance referred to the nominal tempered tone cents (100, 200, 300. ,,,,,,).

5) Recently, I sent "harmony photos" to some of the member list. Graphs of tempered and Piagui triads were printed .on the same page. In my book (internet) 34 tempered and Piagui triad graphs are shown. The aesthetic and periodic displays of the Piagui intonations contrast with the disordered and chaotic responses of the equal tempered graphs. Nobody said a word about it so I will send again the graphs, perhaps this time the Piaguis will be acknowledged as a better system when compared to the equal tempered.

MARIO PIZARRO

ELECTRONIC ENGINEER

< piagui@... >

I'm not sure what you mean when you say "perfect" harmony, but there
will definitely be noticeable beating for any major triad in your
scale, as will there be in equal temperament. Some of your fifths are
6 cents flat of a "just" third, and your thirds are either 16 cents
sharp or 12 cents sharp, which will cause pronounced beating in any
case. What exactly do you mean by "perfect" or "flawless" harmony?

-Mike

On Sun, Jun 15, 2008 at 8:19 PM, Mario Pizarro <piagui@...> wrote:
> 1) The three variants of Piagui octaves I deduced were analyzed by Mike
> Battaglia who found less than four cents of difference between any equal
> tempered tone and the corresponding tone of any of the three Piagui scale
> variants (Piagui I, Piagui II and Piagui III).
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> Here I reproduce the values calculated by Mike Battaglia for the three
> versions of Piagui intonations:
>
> PIAGUI I SEMITONE SEQUENCE: (C K K P K K P K K P K K P 2)
>
> K = 1.06066017178 = 101.955 cents
> P = 1.0570729911 = 96.090 cents
>
> PIAGUI I TONE CENTS
> C - 0...................cents
> C# - 101,955...... cents
> D - 203,91 .........cents
> Eb - 300 ...............cents
> E - 401,955 .......cents
> F - 503,91 .........cents
> F# - 600 ..............cents
> G - 701,955 ......cents
> Ab - 803,91 ........cents
> A - 900 ..............cents
> Bb - 1001,955 ....cents
> B - 1103,91 .......cents
> C - 1200 ............cents
> -----------------------
>
> PIAGUI I TONE CENTS MINUS NOMINAL TEMPERED CENTS
> C .......... 0 ..............cents
> C# ..........+ 1,955 cents
> D .......... + 3,91 cents
> Eb.......... 0 .............. cents
> E ............+ 1,955 cents
> F ............+ 3,91 cents
> F#........... 0 ............. cents
> G.............+ 1,955 cents
> Ab ..........+ 3,91 cents
> A .............0 ............. cents
> Bb...........+ 1,955 cents
> B .......... + 3,91 cents
> 2C ......... 0 .............. cents
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
>> Here is the second layout he mentioned:
>
>> PIAGUI II SEMITONE SEQUENCE: (Do=1) x K x P x K x K x P x K x K x P x K x
>> K x P x K = 2
>
>> K = 1.06066017178 = 101.955 cents
>
>> P = 1.0570729911 = 96.090 cents
>
> PIAGUI II TONE CENTS
>
>> C - 0 cents ..............................0
>
>> C# - 101.955 cents .................+1,955
>
>> D - 198.045 cents.................-1,955
>
>> Eb - 300 cents ........................ 0
>
>> E - 401.955 cents..................+1,955
>
>> F - 498.045 cents...................-1,955
>
>> F# - 600 cents .........................0
>
>> G - 701.955 cents..................+1,955
>
>> Ab - 798.045 cents.................-1,955
>
>> A - 900 cents........................ 0
>
>> Bb - 1001.955 cents..............+1,955
>
>> B - 1098.045 cents.............. -1,955
>
>> C - 1200 cents..................... 0
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> The third layout he mentioned:
>
> PIAGUI III = (Do=1) x P x K x K x P x K x K x P x K x K x P x K x K = 2
>
>> K = 1.06066017178 = 101.955 cents
>
>> P = 1.0570729911 = 96.090 cents
>
> PIAGUI III TONE CENTS -
>
>> C - 0 cents 0
>
>> C# - 96.090 cents - 3,91
>
> -
>
>> D - 198.045 cents -1,955
>
>> Eb - 300 cents 0
>
>> E - 396.090 cents - 3,91
>
>> F - 498.045 cents - 1,955
>
>> F# - 600 cents 0
>
>> G - 696.090 cents - 3,91
>
>> Ab - 798.045 cents - 1,955
>
>> A - 900 cents 0
>
>> Bb - 996.090 cents - 3,91
>
>> B - 1098.045 cents - 1,955
>
>> C - 1200 cents 0
>
>
>
> 2) I read in tuning Yahoogroups that tone discrepances like +/- 4 cents
> referred to the corresponding tempered tones make almost the same sound of
> the referred tempered tone. I agree with that. However, the equal tempered
> scale is also an almost perfect scale and consequently it is not a 100 %
> good reference; therefore we should not establish 5 or 4 or 3,,,,,, cents as
> a minimum acceptable discrepance; before that we should know and acknowledge
> the scale able to produce flawless harmony.
>
> 3) Mr. Brad Lehman stated that one of the three Piagui versions was proposed
> in 1776 by F. W. MARPURG, born in Breslau as detailed in the article
> "IRREGULAR SYSTEMS OF TEMPERAMENT" published in 1948 by .J. Murray Barbour.
> I got the 8 pages article.
>
> Brad Lehman wrote that in the mentioned article, page 23 row H, can be found
> the coincident proposal of MARPURG. I got the article. In that page 23, his
> eleven proposals of "Sophisticated Systems" -- "Temperament of fifths in
> cents" don't show any mathematical support and the supposed coincident scale
> is given by the difference between the tone cents and the twelve nominal
> cents of tempered octave (0, 100, 200, .............1000, 1100, 1200).
>
> (As printed in article):
> ................................................................................(Ordained):
> .........
> c ......g.....d......a......e......b......f#......c#.....g#.....eb....bb......f -------
> C ......C#.....D ....Eb......E......F.....F#....G....Ab......A
> Bb.....B......C A
> ....................................................................................................................................................................................................................................
> B
> ...................................................................................................................................................................................................................................
> C
> ...................................................................................................................................................................................................................................
> D
> ..................................................................................................................................................................................................................................
> E
> .................................................................................................................................................................................................................................
> F
> ...................................................................................................................................................................................................................................
> G
> ..................................................................................................................................................................................................................................
> H ...... 0 .... 6 .... 0 .... 0..... 6 ..... 0 .... 0 ..... 6 ..... 0 .....
> 0 ..... 6 .....0............0....... 6.......
> 0 .......0.......6......0......0......6.......0........0.......6.......0......0
> J
> ..........................................................................................................................................................................................................................................
> K
> .........................................................................................................................................................................................................................................
> L
> ..........................................................................................................................................................................................................................................
>
> We can see that the cents shown In row H are quite different if compared
> with the ones detailed below where eight values per octave establish the
> authentic sets of perfect scales no matter these values are lower than 4.
>
> NOTE : ......... C ........ C# ............ D ........ Eb ...... E
> .............. F ........ F# ...... G ................ Ab ...... A .........
> Bb .......... B ....... C
>
> I) PIAGUI I : ....0......+1,955....... +3,91...... 0......+1,955.......+
> 3,91...... 0......+1,955........+ 3,91.......0......+ 1,955.....+
> 3,91......0
>
> II) PIAGUI II :...0......+1,955....... - 1,955.....0......+1,955....... -
> 1,955.....0......+1,955....... - 1,955......0......+1,955...... - 1,955....0
>
> III) PIAGUI III : 0......- 3,91.......... - 1,955.....0......-
> 3,91......... - 1,955.....0......- 3,91..........- 1,955...... 0..... -
> 3,91....... - 1,955.....0
>
> The figures +/- 1,955, +/- 3,91 and 0 cents work in the three Piagui
> variants,
>
> Consequently, Brad Lehman misunderstood the data contained in row H proposed
> by MARPURG and stated that the irregular set (row H) containing only 4
> discrepances per octave with a value of 6 each, coincides with one of the
> Piagui variants. You can see that there is no any aspect that could even
> show similitude between the MARPURG proposal and any of the three Piagui
> variants.
>
> 4) I realized that the idea is to disapprove the scales whose tone
> frequencies are about the equal tempered ones. In other words, the three
> piaguis could be considered as second quality intonations for having less
> than 4 cents of descrepance referred to the nominal tempered tone cents
> (100, 200, 300. ,,,,,,).
>
> 5) Recently, I sent "harmony photos" to some of the member list. Graphs of
> tempered and Piagui triads were printed .on the same page. In my book
> (internet) 34 tempered and Piagui triad graphs are shown. The aesthetic and
> periodic displays of the Piagui intonations contrast with the disordered and
> chaotic responses of the equal tempered graphs. Nobody said a word about it
> so I will send again the graphs, perhaps this time the Piaguis will be
> acknowledged as a better system when compared to the equal tempered.
>
> MARIO PIZARRO
>
> ELECTRONIC ENGINEER
>
> < piagui@... >
>
>

Mario Pizarro wrote:
> 3) Mr. Brad Lehman stated that one of the three Piagui versions was
> proposed in 1776 by F. W. MARPURG, born in Breslau as detailed in
> the article "IRREGULAR SYSTEMS OF TEMPERAMENT" published in 1948 by
> .J. Murray Barbour. I got the 8 pages article.
> > Brad Lehman wrote that in the mentioned article, page 23 row H, can
> be found the coincident proposal of MARPURG. I got the article. In
> that page 23, his eleven proposals of "Sophisticated Systems" --
> "Temperament of fifths in cents" don't show any mathematical support
> and the supposed coincident scale is given by the difference between
> the tone cents and the twelve nominal cents of tempered octave (0,
> 100, 200, .............1000, 1100, 1200).
(...)
> I) PIAGUI I : ....0......+1,955....... +3,91......
> 0......+1,955.......+ 3,91...... 0......+1,955........+
> 3,91.......0......+ 1,955.....+ 3,91......0
>
> II) PIAGUI II :...0......+1,955....... -
> 1,955.....0......+1,955....... - 1,955.....0......+1,955....... -
> 1,955......0......+1,955...... - 1,955....0
>
> III) PIAGUI III : 0......- 3,91.......... - 1,955.....0......-
> 3,91......... - 1,955.....0......- 3,91..........- 1,955......
> 0..... - 3,91....... - 1,955.....0
>
> The figures +/- 1,955, +/- 3,91 and 0 cents work in the three
> Piagui variants,
>
> Consequently, Brad Lehman misunderstood the data contained in row > H
> proposed by MARPURG and stated that the irregular set (row H)
> containing only 4 discrepances per octave with a value of 6 each,
> coincides with one of the Piagui variants. You can see that there > is
> no any aspect that could even show similitude between the MARPURG
> proposal and any of the three Piagui variants.

Mario,

I stand by what I said the first time. You've misunderstood me, *and* you've misunderstood both Barbour's article and Marpurg's temperament "H".

The way Barbour presents that temperament, it is *not* as deviations from equal temperament (which is how you're mistakenly reading it). It is, rather, as cent offsets from a series of pure 3:2 5ths. Barbour's explanatory text there on page 23, especially his discussion of row "L" in his table, makes it perfectly clear that he's talking about offsets from pure 5ths, not from equal temperament.

Barbour's table arranges all the notes by 5ths:
c, g, d, a, e, b, f#, c#, g#, eb, bb, f.
And under each one, in its column, Barbour puts a cent value that indicates how much that 5th gets tempered when moving forward to the next note. In row "H" (there are "A" through "L" of them), his numbers are:
0, 6, 0, 0, 6, 0, 0, 6, 0, 0, 6, 0.

That is:
C-G is pure. G-D is narrow by 6 cents. D-A is pure. A-E is pure. E-B is narrow by 6 cents. B-F# is pure. F#-C# is pure. C#-G# is narrow by 6 cents. G#-Eb is pure. Eb to Bb is pure. Bb to F is narrow by 6 cents. F to C is pure.

If we arrange all that in chromatic sequence, and express it as offsets from equal temperament (*your* way), we get:

C = 0 (000)
C# = +2 (102)
D = -2 (198) [1398]
Eb = 0 (300)
E = +2 (402)
F = -2 (498)
F# = 0 (600)
G = +2 (702)
G# = -2 (798)
A = 0 (900)
Bb = +2 (1002)
B = -2 (1098)

C to G is pure: 702 cents. G to D is 6 cents narrow: 696 cents. D to A is pure: 702 cents. Etc, etc, etc.

That is the same thing as "PIAGUI II". That's what I said the first time. The layout you're calling PIAGUI II (one of the three rotations of your idea) was already published by Marpurg as his temperament "H" in 1776.

Here again is that June 10th posting of mine, in its entirety:

Tom Dent wrote:
> > What I *think* the answer is:
> >
> > D-A, B-F#, G#-Eb, F-C are tempered 1/4 Pythagorean comma; all the
> > other fifths are pure.
> >
> > (Of course one can transpose this temperament up or down a fifth
> > to get three possibilities in total.)
> >
> > I would not be surprised if someone had already written this
> > temperament down in the 18th century... anyway it is generally a
> > good thing to be able to describe your result in a simple and
> > obvious way rather than taking up hundreds of lines of
> > mathematical obfuscation.

One of those two rotations is F W Marpurg's temperament "H" from
_Versuch �ber die musicalische Temperatur_ (Breslau, 1776):

G-D, E-B, C#-G#, Bb-F each narrow by 1/4 PC; all other 5ths pure.

Available in J Murray Barbour's article "Irregular systems of
temperament", _Journal of the American Musicological Society_ 1:3 (Fall
1948), page 23.

Brad Lehman