Yahoo Tuning Groups Ultimate Backup tuning The optimal Peruvian scale

To the tuning yahoogroup members,

Lately, I got interesting information from Dr. Bradley Lehman regarding two scales I constructed in the past year. You can remember that in September/October 2008 I presented to the tuning members the 11 perfect fifths plus a narrow one scale whose deducing method was quite interesting. Some days later, Miss Margo Schulter sent me a message to inform that curiously the same scale was discovered by Mr. Cordier in Paris in the past century, about 1945, and it was an event. She was a participant of a meeting where the Cordier scale was discussed and copied for me the details that were published in Paris concerning this scale, so I might say that I am "a second inventor" of the Cordier scale.

Brad said that when tuning his harpsichord to a scale that has a narrow fifth of 6 or more cents, the sound is badly so I guess that this could happen with the Cordier scale where the stretched octave is having 1203.3514 cents. All the Pythagorean comma was concentrated on a single narrow fifth. The twelft tone of this scale (2C), if it is given in a decimal number relation with respect to the basic note C = 1 is the stretched 2.00387547376...I have called this twelft tone as the acoustical octave.

Now, I present another stretched scale whose twelft tone is slightly lower than the Cordier scale, that is, 2.00386, an exact number. You can deduce that we could use any other stretched tone to produce different sets of scales where note G can be 1.5.

The advantage of the Optimal Peruvian Scale regarding the Cordier scale is that instead of having a single narrow fifth associated with 23.460010383 cents of the Pythagorean comma, the O. P. S. works with six equal narrow fifths (1.49709708923 or 698.60134271 cents) plus six pure fifths (1.5). So the Pythagorean comma ( 23.460010383 cents) is divided in six parts of 3.35365815 each since (701.95500086** - 698.60134271) = 3.35365815 cents.

THE OPTIMAL PERUVIAN SCALE

C -- 1----------------------- 0 cents
C#-- 1.05962856769----- 100.27037345
D -- 1.12282281692------ 200.55634357
Eb -- 1.18977513326----- 300.82671702
E -- 1.2607310782-------- 401.11268716
F -- 1.33590666666-------501.383056
F# -- 1.4155776206------ 601.66903073
G -- 1.5-------------------- 701.95500086**
Ab -- 1.58944285153---- 802.22537431
A -- 1.68423422538----- 902.51134444
Bb -- 1.78466269989---- 1002.78171789
B -- 1.89109661729----- 1103.06768802
(2C)* -- 2.00386--------- 1203.33806152 (Acoustical Octave)

FIFTHS
(G/C) = 1.5 ---------Pure fifth
(D/G) = 1.49709708923 ---------3.35365815 cents-- Narrow fifth
(A/D) = 1.5 ---------Pure fifth
(2E/A) = 1.49709708923 ---------3.35365815 cents-- Narrow fifth
(B/E) = 1.5 ---------Pure fifth
(2F#/B) = 1.49709708923 ---------3.35365815 cents-- Narrow fifth
(2C#/F#) = 1.49709708923 ---------3.35365815 cents-- Narrow fifth
(Ab/C#) = 1.5 ---------Pure fifth
(2Eb/Ab) = 1.49709708923 ---------3.35365815 cents-- Narrow fifth
(Bb/Eb) = 1.5 ---------Pure fifth
(2F/Bb) = 1.49709708923 ---------3.35365815 cents-- Narrow fifth
[(2C)*/F) = 1.5 ---------Pure fifth

Note that the six narrow fifths have the same value and 3.35365815 x 6 = 20.1219489 cents. The Pythagorean comma is the total of 20.1219489 + 1203.33806152 - 1200 = 23.460010383 cents.

I would like to get your viewpoints regarding this new scale which cancels the negative effect of narrow fifths greater than six cents as explained by Brad since the O.P.S. works with six narrow fifths of only 3.35365815 cents each.

Thanks

Mario Pizarro
piagui@...

Lima, January 29, 2009

🔗Michael Sheiman <djtrancendance@...>

I had made a post a while ago concerning "why do we use scales like 12TET where some intervals are perfect yet others are off by 20+ cents?!...why don't we make a scale where every interval is off by 5-or-less cents (better to have many unnoticeable errors than perfection on some intervals and VERY noticeable problems with others)?"

It sounds like your scale at least indirectly, if not intentionally, aims at that problem. I will have to actually try using it to be sure...but it sounds quite promising.

-Michael

--- On Thu, 1/29/09, Mario Pizarro <piagui@...> wrote:

From: Mario Pizarro <piagui@...>
Subject: [tuning] The optimal Peruvian scale
To: "tuning yahoogroups" <tuning@yahoogroups.com>
Cc: "Fernando de Lucchi" <dirgen@...>, "Director E. M. J. Echecopar" <esmusica@pucp.edu.pe>
Date: Thursday, January 29, 2009, 12:24 PM

To the tuning yahoogroup members,

Lately, I got interesting information from Dr. Bradley
Lehman regarding two scales I constructed in the past year. You can
remember that in September/October 2008 I presented to the tuning members the 11
perfect fifths plus a narrow one scale whose deducing method was quite
interesting. Some days later, Miss Margo Schulter sent me a message to
inform that curiously the same scale was discovered by Mr. Cordier in Paris
in the past century, about 1945, and it was an event. She was a participant
of a meeting where the Cordier scale was discussed and copied for me the details
that were published in Paris concerning this scale, so I might say that I am "a
second inventor" of the Cordier scale.

Brad said that when tuning his harpsichord to a scale
that has a narrow fifth of 6 or more cents, the sound is badly so I guess
that this could happen with the Cordier scale where the stretched octave is
having 1203.3514 cents. All the Pythagorean comma was concentrated on
a single narrow fifth. The twelft tone of this scale (2C), if it is given
in a decimal number relation with respect to the basic note C = 1 is the
stretched 2.00387547376. ..I have called this twelft tone as the acoustical
octave.

Now, I present another stretched scale whose twelft
tone is slightly lower than the Cordier scale, that is, 2.00386, an exact
number. You can deduce that we could use any other stretched tone to
produce different sets of scales where note G can be 1.5.

The advantage of the Optimal Peruvian Scale regarding
the Cordier scale is that instead of having a single narrow fifth associated
with 23.460010383 cents of the Pythagorean comma, the O. P. S. works
with six equal narrow fifths (1.49709708923 or 698.60134271
cents) plus six pure fifths (1.5). So the Pythagorean comma ( 23.460010383
cents) is divided in six parts of 3.35365815 each since (701.95500086* * -
698.60134271) = 3.35365815 cents.

THE OPTIMAL PERUVIAN SCALE

C -- 1----------- --------- --- 0 cents
C#-- 1.05962856769- ---- 100.27037345
D -- 1.12282281692- ----- 200.55634357
Eb -- 1.18977513326- ---- 300.82671702
E -- 1.2607310782- ------- 401.11268716
F -- 1.33590666666- ------501. 383056
F# -- 1.4155776206- ----- 601.66903073
G --
1.5--------- --------- -- 701.95500086* *
Ab -- 1.58944285153- --- 802.22537431
A -- 1.68423422538- ---- 902.51134444
Bb -- 1.78466269989- --- 1002.78171789
B -- 1.89109661729- ---- 1103.06768802
(2C)* -- 2.00386----- ---- 1203.33806152 (Acoustical
Octave)

FIFTHS
(G/C) = 1.5 ---------Pure fifth
(D/G) = 1.49709708923 ---------3.35365815 cents--
Narrow fifth
(A/D) = 1.5 ---------Pure fifth
(2E/A) = 1.49709708923 ---------3.35365815 cents--
Narrow fifth
(B/E) = 1.5 ---------Pure fifth
(2F#/B) = 1.49709708923 ---------3.35365815 cents--
Narrow fifth
(2C#/F#) = 1.49709708923 ---------3.35365815 cents--
Narrow fifth
(Ab/C#) = 1.5 ---------Pure fifth
(2Eb/Ab) = 1.49709708923 ---------3.35365815 cents--
Narrow fifth
(Bb/Eb) = 1.5 ---------Pure fifth
(2F/Bb) = 1.49709708923 ---------3.35365815 cents--
Narrow fifth
[(2C)*/F) = 1.5 ---------Pure fifth

Note that the six narrow fifths have the same value and
3.35365815 x 6 = 20.1219489 cents. The Pythagorean comma is the total of
20.1219489 + 1203.33806152 - 1200 = 23.460010383 cents.

I would like to get your viewpoints regarding this new
scale which cancels the negative effect of narrow fifths greater than six cents
as explained by Brad since the O.P.S. works with six narrow fifths of only
3.35365815 cents each.

Thanks

Mario Pizarro
piagui@ec-red. com

Lima, January 29, 2009

🔗Mario Pizarro <piagui@...>

Michael,

Thanks for your viewpoints. I believe this time I did my best. Let us be in contact.

Regards

Mario

piagui@...

Lima, January 29, 2009
----- Original Message -----
From: Michael Sheiman
To: tuning@yahoogroups.com
Sent: Thursday, January 29, 2009 3:56 PM
Subject: Re: [tuning] The optimal Peruvian scale

It sounds like your scale at least indirectly, if not intentionally, aims at that problem. I will have to actually try using it to be sure...but it sounds quite promising.

-Michael

--- On Thu, 1/29/09, Mario Pizarro <piagui@...> wrote:

To the tuning yahoogroup members,

Brad said that when tuning his harpsichord to a scale that has a narrow fifth of 6 or more cents, the sound is badly so I guess that this could happen with the Cordier scale where the stretched octave is having 1203.3514 cents. All the Pythagorean comma was concentrated on a single narrow fifth. The twelft tone of this scale (2C), if it is given in a decimal number relation with respect to the basic note C = 1 is the stretched 2.00387547376. ..I have called this twelft tone as the acoustical octave.

THE OPTIMAL PERUVIAN SCALE

C -- 1----------- --------- --- 0 cents
C#-- 1.05962856769- ---- 100.27037345
D -- 1.12282281692- ----- 200.55634357
Eb -- 1.18977513326- ---- 300.82671702
E -- 1.2607310782- ------- 401.11268716
F -- 1.33590666666- ------501. 383056
F# -- 1.4155776206- ----- 601.66903073
G -- 1.5--------- --------- -- 701.95500086* *
Ab -- 1.58944285153- --- 802.22537431
A -- 1.68423422538- ---- 902.51134444
Bb -- 1.78466269989- --- 1002.78171789
B -- 1.89109661729- ---- 1103.06768802
(2C)* -- 2.00386----- ---- 1203.33806152 (Acoustical Octave)

Note that the six narrow fifths have the same value and 3.35365815 x 6 = 20.1219489 cents. The Pythagorean comma is the total of 20.1219489 + 1203.33806152 - 1200 = 23.460010383 cents.

Thanks

Mario Pizarro
piagui@ec-red. com

Lima, January 29, 2009

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