Marpurg-Lehman

If somebody deduces the ideal scale that overcomes the equal tempered, most people would probably be unable to distinguish the new chords from those played in an instrument tuned to the equal temperament scale. Since the imperfection of this scale is small, only the people that are accustomed to listening to works where the music expressions are notable creations will distinguish the sound coming from each instrument.

Small alterations of keyboard tone frequencies will do the change. We cannot expect noticeable pitches deviations because part of them would be altered only a few cents to comply with the new intonation; others would probably keep their equal tempered tones.

Recently, Brad Lehman wrote that my proposal of a new musical scale that was deduced by rigorous mathematical analysis (The Piagui scale, variant II ) coincides with other scale proposed by F. W. Marpurg in 1776. The Marpurg scale features rely on the number and how the deviations were inserted in the octave as showed for this scale in the layout given below.

C Pure G NarrowD Pure A Pure E NarrowB Pure F# Pure C#NarrowG# Pure Eb Pure Bb NarrowF

0......... 6 ........0.........0..........6.........0..........0..........6...........0.............0............6...........0

The number 6 indicates the cents deducted from the pure fifths (702 cents) to produce narrow fifths with only 696 cents. The layout given above shows four alterations of 6 cents each, which establish the features of the Marpurg scale.

Now, I will present the Piagui II scale to compare it with the Marpurg intonation.

The third column shows how the cents of deviations are distributed along the octave; here we can see that the Piagui II scale works with eight deviations, four positives and four negatives which are applied to different notes, while Marpurg used only four.

A +2 deviation means that the tone frequency to which the deviation is applied increases its frequency in 2 cents.

PIAGUI II........................ CENTS ............ DEVIATION (CENTS)..........PIAGUI II (HZ).....................................................

C = 1 ---------------------------......... 0 -------------------.........---------------- 0 ------------------...................... 261,6255 .................................................................................

C# = 1,06066017178 -----...------... 101,95500075 -----------------........ + 2 --------------------................... 277,4958 .................................................................................

D = 1,12119522034 ----------...... 198,04499909----------------............ -- 2 ...--------------------................ 293,3333..............................................................................

Eb = 1,189207115 ............... ..300 ---------------------------------.......... 0 ------------------------................ 311,127....................................................................................

E = 1,26134462288-----.......------.401,95500075---------------..............+ 2 -------------------.................... 330..........................................................................................

F = 1,33333333333-----------........498,04499909..............--------------- -- 2 -------------------.....................348,8341 .................................................................................

F# = 1,41421356237------------........600----------------------------------............0.-------------------..................... 369,9944 ..................................................................................

G = 1,5 ........ ------------------------- 701,95500075..............----------------+ 2...-------------------...................392,4383 .................................................................................

Ab = 1,58560948667-----------....... 798,04499909..............--------------- -- 2-----------------........................414,836................................................................................

A = 1,68179283051 ----------- 900.......... --------------------------------- 0 --------------------....................440 .........................................................................................

Bb = 1,7838106725........-----------.1001,95500075..........-----------------..+ 2 ---------------------.................. 466,6905..................................................................................

B = 1,88561808316 ------.......... 1098.04499909...................--------- ? 2.....---------------.................... 493,3259................................................................................

2C= 2 ........---------------------------- 1200 -------------------------------............- 0 -----------------..................... 523.2511 ................................................................................

I have demonstrated that the Piagui Musical scale, variant II, considerably differs from the F. W. Marpurg scale cited on page 23, row H of "Journal of the American Musicological Society, Vol.1 Nº. 3".

BRAD: Some of the tuning list perhaps believe that untrue new is true and that I copied the Marburg scale. ¿ Could you assist me to clarify this confused situation?

-----------------------------------------------------------------------

Let us see some words of Brad Lehman and Mike:

> > Piagui I is the first layout, Piagui II the second, and Piagui III
> >,
> > It is some interesting kind of diminished temperament. Some of the keys have

> > slightly better than 12-equal at the cost of the major thirds, some of the keys

> > > have slightly better thirds than12-equal at the cost of the fifths. Either way,

> > none of the notes are any more than 4 cents away from 12-equal, so I'd be

> > surprised if the two could really be distinguished in practice.
> > Thanks for clearing this up, Mike. Until now I had no idea what Piagui was

> > talking about.
> >-----------------------------------------------------------------------------------
> Piagui, what's your response to this statement that your scale is
> practically indistinguishable from equal temperament? (I´SHOULD GIVE YOU SOME INFO/ M. P.

Or a response to the observation that one of those three versions was already published by at least two other people (Marpurg in 1776, Barbour in 1948)? It's a possibly useful temperament; it's just not new. But, the lack of novelty doesn't necessarily diminish its value..

On harpsichord, the two different types of 5ths in it (pure vs 1/4 PC narrow) can *certainly* be distinguished in practice. When 5ths or 12ths that severely tempered get played directly, in music with two-voiced texture, the impurity is obvious. The vigorous vibrato of the beats stands out, in contrast with the steadiness of the pure 5ths.

As Mike mentioned, there are two different sizes of major 3rds in it. Eight of them are 8/11 SC sharp, and the other four are only 5/11 SC sharp.

There are two different sizes of semitone in it: eight of them are 102 cents, and the other four are 96 cents.

Brad Lehman

Lima, July 4, 2008

MARIO PIZARRO

< piagui@ec-red.com >

Mario,

In your presentation of "Piagui II" below you're still giving it as
deviations of +2 or -2 from EQUAL temperament. But, the Marpurg line
you give here below, in that same posting, shows deviations from a
line of PURE 5ths.

If you'd line up things so you're comparing the same measurement
scheme in both sections of your posting (make them *both* be
deviations from equal, or make them *both* be deviations from pure,
instead of one of each), you'd see that the two temperaments you're
comparing are the same. (In English we have a phrase, "comparing
apples with oranges....")

Here's what that Marpurg "H" temperament looks like, laid out
chromatically as cents:

C 0.00
C# 101.96
D 198.04
Eb 300.00
E 401.96
F 498.04
F# 600.00
G 701.96
G# 798.04
A 900.00
Bb 1001.96
B 1098.04

Now do you see the coincidence with your "Piagui II" yet?

Brad Lehman

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>
> If somebody deduces the ideal scale that overcomes the equal
tempered, most people would probably be unable to distinguish the new
chords from those played in an instrument tuned to the equal
temperament scale. Since the imperfection of this scale is small, only
the people that are accustomed to listening to works where the music
expressions are notable creations will distinguish the sound coming
from each instrument.
>
>
>
> Small alterations of keyboard tone frequencies will do the change.
We cannot expect noticeable pitches deviations because part of them
would be altered only a few cents to comply with the new intonation;
others would probably keep their equal tempered tones.
>
>
>
> Recently, Brad Lehman wrote that my proposal of a new musical scale
that was deduced by rigorous mathematical analysis (The Piagui scale,
variant II ) coincides with other scale proposed by F. W. Marpurg in
1776. The Marpurg scale features rely on the number and how the
deviations were inserted in the octave as showed for this scale in the
layout given below.
>
>
>
> C Pure G NarrowD Pure A Pure E NarrowB Pure F# Pure
C#NarrowG# Pure Eb Pure Bb NarrowF
>
>
>
> 0......... 6
........0.........0..........6.........0..........0..........6...........0.............0............6...........0
>
>
>
> The number 6 indicates the cents deducted from the pure fifths (702
cents) to produce narrow fifths with only 696 cents. The layout given
above shows four alterations of 6 cents each, which establish the
features of the Marpurg scale.
>
>
>
> Now, I will present the Piagui II scale to compare it with the
Marpurg intonation.
>
> The third column shows how the cents of deviations are distributed
along the octave; here we can see that the Piagui II scale works with
eight deviations, four positives and four negatives which are applied
to different notes, while Marpurg used only four.
>
> A +2 deviation means that the tone frequency to which the deviation
is applied increases its frequency in 2 cents.
>
>
>
> PIAGUI II........................ CENTS ............ DEVIATION
(CENTS)..........PIAGUI II
(HZ).....................................................
>
> C = 1 ---------------------------......... 0
-------------------.........---------------- 0
------------------...................... 261,6255
.................................................................................
>
> C# = 1,06066017178 -----...------... 101,95500075
-----------------........ + 2 --------------------...................
277,4958
.................................................................................
>
> D = 1,12119522034 ----------......
198,04499909----------------............ -- 2
...--------------------................
293,3333..............................................................................
>
> Eb = 1,189207115 ............... ..300
---------------------------------.......... 0
------------------------................
311,127....................................................................................
>
> E =
1,26134462288-----.......------.401,95500075---------------..............+
2 -------------------....................
330..........................................................................................
>
> F =
1,33333333333-----------........498,04499909..............---------------
-- 2 -------------------.....................348,8341
.................................................................................
>
> F# =
1,41421356237------------........600----------------------------------............0.-------------------.....................
369,9944
..................................................................................
>
> G = 1,5 ........ -------------------------
701,95500075..............----------------+
2...-------------------...................392,4383
.................................................................................
>
> Ab = 1,58560948667-----------.......
798,04499909..............--------------- --
2-----------------........................414,836................................................................................
>
> A = 1,68179283051 ----------- 900..........
--------------------------------- 0
--------------------....................440
.........................................................................................
>
> Bb =
1,7838106725........-----------.1001,95500075..........-----------------..+
2 ---------------------..................
466,6905..................................................................................
>
> B = 1,88561808316 ------..........
1098.04499909...................--------- ?
2.....---------------....................
493,3259................................................................................
>
> 2C= 2 ........---------------------------- 1200
-------------------------------............- 0
-----------------..................... 523.2511
................................................................................
>
>
>
>
>
> I have demonstrated that the Piagui Musical scale, variant II,
considerably differs from the F. W. Marpurg scale cited on page 23,
row H of "Journal of the American Musicological Society, Vol.1 Nº. 3".
>
>
>
> BRAD: Some of the tuning list perhaps believe that untrue new is
true and that I copied the Marburg scale. ¿ Could you assist me to
clarify this confused situation?
>
> -----------------------------------------------------------------------
>
> Let us see some words of Brad Lehman and Mike:
>
>
>
> > > Piagui I is the first layout, Piagui II the second, and Piagui III
> > >,
> > > It is some interesting kind of diminished temperament. Some of
the keys have
>
> > > slightly better than 12-equal at the cost of the major thirds,
some of the keys
>
> > > > have slightly better thirds than12-equal at the cost of the
fifths. Either way,
>
> > > none of the notes are any more than 4 cents away from 12-equal,
so I'd be
>
> > > surprised if the two could really be distinguished in practice.

> > > Thanks for clearing this up, Mike. Until now I had no idea what
Piagui was
>
> > > talking about.
> >
>-----------------------------------------------------------------------------------
> > Piagui, what's your response to this statement that your scale is
> > practically indistinguishable from equal temperament? (I´SHOULD
GIVE YOU SOME INFO/ M. P.
>
>
>
> Or a response to the observation that one of those three versions
was already published by at least two other people (Marpurg in 1776,
Barbour in 1948)? It's a possibly useful temperament; it's just not
new. But, the lack of novelty doesn't necessarily diminish its value..
>
>
>
> On harpsichord, the two different types of 5ths in it (pure vs 1/4
PC narrow) can *certainly* be distinguished in practice. When 5ths or
12ths that severely tempered get played directly, in music with
two-voiced texture, the impurity is obvious. The vigorous vibrato of
the beats stands out, in contrast with the steadiness of the pure 5ths.
>
> As Mike mentioned, there are two different sizes of major 3rds in
it. Eight of them are 8/11 SC sharp, and the other four are only 5/11
SC sharp.
>
> There are two different sizes of semitone in it: eight of them are
102 cents, and the other four are 96 cents.
>
> Brad Lehman
>
>
>
> Lima, July 4, 2008
>
>

>
> MARIO PIZARRO
>
>
>
> < piagui@... >
>

On Sat, Jul 5, 2008 at 4:57 AM, Mario Pizarro <piagui@...> wrote:
> If somebody deduces the ideal scale that overcomes the equal tempered, most
> people would probably be unable to distinguish the new chords from those
> played in an instrument tuned to the equal temperament scale. Since the
> imperfection of this scale is small, only the people that are accustomed to
> listening to works where the music expressions are notable creations will
> distinguish the sound coming from each instrument.

Well, it depends. A lot of scales that are considered "perfect," like
12-note subsets of JI, for example, are very distinguishably different
from 12-tet equal temperament.

> Small alterations of keyboard tone frequencies will do the change. We cannot
> expect noticeable pitches deviations because part of them would be altered
> only a few cents to comply with the new intonation; others would probably
> keep their equal tempered tones.

> Recently, Brad Lehman wrote that my proposal of a new musical scale that was
> deduced by rigorous mathematical analysis (The Piagui scale, variant II )
> coincides with other scale proposed by F. W. Marpurg in 1776. The Marpurg
> scale features rely on the number and how the deviations were inserted in
> the octave as showed for this scale in the layout given below.
>
>
>
> C Pure G NarrowD Pure A Pure E NarrowB Pure F# Pure C#NarrowG#
> Pure Eb Pure Bb NarrowF
>
>
>
> 0.........
> 6 ........0.........0..........6.........0..........0..........6...........0.............0............6...........0
>
>
>
> The number 6 indicates the cents deducted from the pure fifths (702 cents)
> to produce narrow fifths with only 696 cents. The layout given above shows
> four alterations of 6 cents each, which establish the features of the
> Marpurg scale.

This is the same as your scale, but in a different format.