Eleven perfect fifths scale

To the tuning list,

Fourteen months took me to derive a well tempered scale that works with 11 perfect fifths and the twelve major thirds are the same of 12 tet (1.2599....). About two years ago I designed another 11 p. f. scale but with a stretched octave (2C = 2.003875) and coincidently the same stretched scale had been proposed by Mr. Cordier in Paris, 1945, which was consudered an event. This time, 2C = 2.

It could be useful that some members of the list evaluate the 11 perfect fifths scale and inform about its positive and negative properties, I don´t have experience for taking that job.

The scale data are given below

Thank you

Mario Pizarro
Lima, August 06
piagui@...

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To the tuning list,

The 12 tones of a new scale produce eleven perfect fifths and work in the progression of musical cells with exactness. The frequency tones are found within its 624 consonant frequencies comprised between C = 1 and (9/8)^6.

The tone frequencies and their position in the progression follows:

......................... Major Thirds

C = 1 (Cell # 0)...................1.2542

C# = 1.05349794236, (Cell # 46), 90.225 cents.....1.265625

D = 1.11488414335, (Cell # 96), 188.272 cents.....1.2599

Eb = 1.185185185, (Cell # 150), 294.135 cents....1.265625

E = 1.25424466127, (Cell # 200), 392.182 cents.....1.2599

F = 1.33333333333, (Cell # 254), 498.045 cents.....1.2542

F# = 1.40466392312, (Cell # 300), 588.27 cents.....1.265625

G = 1.5, (Cell # 358), 701.955 cents.........1.2542

Ab = 1.58024691358, (Cell # 404), 792.18 cents.....1.265625

A = 1.67232621503, (Cell # 454), 890.227 cents.....1.2599

Bb = 1.77777777777, (Cell # 508), 996 cents.....1.2542

B = 1.88136699191, (Cell # 558), 1094.137 cents...1.2599

2C =2

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On Mon, Aug 9, 2010 at 4:21 PM, Mario Pizarro <piagui@...> wrote:
>
> To the tuning list,
>
>  The 12 tones of a new scale produce eleven perfect fifths and work in the progression of musical cells with exactness. The frequency tones are found within its 624 consonant frequencies comprised between C = 1 and (9/8)^6.

Mario - is this scale one that's been generated just by chaining 11
3/2's together and allowing the D-A fifth to be flat by a pythagorean
comma? If so, then this isn't a new scale - this is the same
Pythagorean scale that has been around since antiquity, probably the
first scale ever created.

Perhaps I'm misunderstanding.

-Mike

Come to think of it, that's exactly what it looked like to me as well: it
looks like Pythagorean scale complete with the comma (only Mario's version is a
different "mode", placing the "commatic fifth" after the first fifth instead of
as the last fifth).
The "sad" thing is, in many ways, I don't see a whole lot of difference
between the "Pythagorean" scale and 12TET (so much for musical evolution!)

While I'm at it...I wonder what research has been done concerning taking a
circle of fifths and scaling them down to the period is not the 2/1 octave, but
within 8 cents of the octave (above or below). You would wonder if relieving
the octave constraint a bit would enable some of the other intervals to be a bit
more clear...

Mario>  The 12 tones of a new scale produce eleven perfect fifths and ...

Mike B> Mario - is this scale one that's been generated just by chaining 11 3/2's together and allowing the D-A fifth to be flat by a pythagorean comma?

Mario, did you mean eleven fifths of exactly 3/2 ratio? I don't think the set you presented has that; it looks like ten fifths of 3/2 (or very close to) and two others: G:D tempered by 2/3 PC and B:F# tempered by 1/3 PC. This gives smallest M3's on Bb F C G and Pythagorean M3's (or very close to) on F# C# G# Eb. Comparing it to other 12-note temperaments, the surprise (in my very limited experience) is the big jump in size between Eb:G and Bb:D - explained of course by the small fifth on G.

Steve M.

Steve, You are right, there are only 10 perfect fifths.
The tone frequencies and their position in the progression follows:
C = 1 (Cell # 0), major third: 1.2542
C# = 1.05349794236, (Cell # 46), 90.225 cents, major third: 1.265625
D = 1.11488414335, (Cell # 96), 188.272, major third: 1.2599
Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625
E = 1.25424466127, (Cell # 200), 392.182, major third: 1.2599
F = 1.33333333333, (Cell # 254), 498.045, major third: 1.2542
F# = 1.40466392312,
G = 1.5, (Cell # 358), 701.955, major third: 1.2542
Ab = 1.58024691358, (Cell # 404), 792.18, major third: 1.265625
A = 1.67232621503, (Cell # 454), 890.227, major third: 1.2599
Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542
B = 1.88136699191, (Cell # 558), 1094.137, major third: 1.2599
2C =2

C 1 X 1.5 = 1.5 (G)
C# 1.05349794236 X 1.5 = 1.58024691354 (Ab)
D 1.11488414335 X 1.5 = 1.67232621503 (A)
Eb 1.18518518518 X 1.5 = 1.77777777777 (Bb)
E 1.25424466127 X 1.5 = 1.88136699191 (B)
F 1.33333333333 X 1.5 = 2 ----- / 1 �
F# 1.40466392312 X 1.5 = 2.10699588468 / 2 gives 1.05349794234 (C#)
G 1.5 X 1.5 = 2.25000000000
Ab 1.58024691358 X 1.5 = 2.37037037037 / 2 = 1.18518518519 (Eb)
A 1.67232621503 X 1.5 = 2.50848932255 / 2 = 1.25424466127 (E.)
Bb 1.77777777777 X 1.5 = 2.66666666666 / 2 = 1.33333333333 (F)
B 1.88136699191 X 1.5 = 2.82205048787 / 2 = 1.41102524393

In your message you copied a phrase written by Mike Battaglia in a note he sent me.
Since I didn�t receive his message that contains the phrase I put below, please send me a copy of it.

"Mike B> Mario - is this scale one that's been generated just by chaining 11 3/2's together
and allowing the D-A fifth to be flat by a pythagorean comma?" (It is not clear to me Mike)

Steve, You can see in the progression that Cells # 51, 102, 153, 204, 255, 306, 357, 408,
459, 510, 561, 612 might be called "The consonant equal tempered scale". Spacing: 51 cells.
Regards
Mario
August 10

----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Tuesday, August 10, 2010 4:45 AM
Subject: [tuning] Re: Eleven perfect fifths scale

Mario> The 12 tones of a new scale produce eleven perfect fifths and ...

Mike B> Mario - is this scale one that's been generated just by chaining 11 3/2's together and allowing the D-A fifth to be flat by a pythagorean comma?

Mario, did you mean eleven fifths of exactly 3/2 ratio? I don't think the set you presented has that; it looks like ten fifths of 3/2 (or very close to) and two others: G:D tempered by 2/3 PC and B:F# tempered by 1/3 PC. This gives smallest M3's on Bb F C G and Pythagorean M3's (or very close to) on F# C# G# Eb. Comparing it to other 12-note temperaments, the surprise (in my very limited experience) is the big jump in size between Eb:G and Bb:D - explained of course by the small fifth on G.

Steve M.

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Steve, You are right, there are only 10 perfect fifths.
The tone frequencies and their position in the progression follows:
C = 1 (Cell # 0), major third: 1.2542
C# = 1.05349794236, (Cell # 46), 90.225 cents, major third: 1.265625
D = 1.11488414335, (Cell # 96), 188.272, major third: 1.2599
Eb = 1.185185185, (Cell # 150), 294.135, major third: 1.265625
E = 1.25424466127, (Cell # 200), 392.182, major third: 1.2599
F = 1.33333333333, (Cell # 254), 498.045, major third: 1.2542
F# = 1.40466392312, (Cell # 300), 588.27, major third: 1.265625
G = 1.5, (Cell # 358), 701.955, major third: 1.2542
Ab = 1.58024691358, (Cell # 404), 792.18, major third: 1.265625
A = 1.67232621503, (Cell # 454), 890.227, major third: 1.2599
Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542
B = 1.88136699191, (Cell # 558), 1094.137, major third: 1.2599
2C =2

C 1 X 1.5 gives 1.5 (G)
C# 1.05349794236 X 1.5 gives 1.58024691354 (Ab)
D 1.11488414335 X 1.5 gives 1.67232621503 (A)
Eb 1.18518518518 X 1.5 gives 1.77777777777 (Bb)
E 1.25424466127 X 1.5 gives 1.88136699191 (B)
F 1.33333333333 X 1.5 gives 2 ----- / 1 �
F# 1.40466392312 X 1.5 gives 2.10699588468 / 2 gives 1.05349794234 (C#)
G 1.5 X 1.5 gives 2.25000000000
Ab 1.58024691358 X 1.5 gives 2.37037037037 / 2 gives 1.18518518519 (Eb)
A 1.67232621503 X 1.5 gives 2.50848932255 / 2 gives 1.25424466127 (E.)
Bb 1.77777777777 X 1.5 gives 2.66666666666 / 2 gives 1.33333333333 (F)
B 1.88136699191 X 1.5 gives 2.82205048787 / 2 gives 1.41102524393

In your message you copied a phrase written by Mike Battaglia in a note he sent me.
Since I didn�t receive his message that contains the phrase I put below, please send me a copy of it.
I can get an 11 perfect fifths scale from the progression.
"Mike B> Mario - is this scale one that's been generated just by chaining 11 3/2's together
and allowing the D-A fifth to be flat by a pythagorean comma?" (It is not clear to me Mike)

Steve, You can see in the progression that Cells # 51, 102, 153, 204, 255, 306, 357, 408,
459, 510, 561, 612 might be called "The consonant equal tempered scale". Spacing: 51 cells .
Regards
Mario
August 10

----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Tuesday, August 10, 2010 4:45 AM
Subject: [tuning] Re: Eleven perfect fifths scale

Mario> The 12 tones of a new scale produce eleven perfect fifths and ...

Mike B> Mario - is this scale one that's been generated just by chaining 11 3/2's together and allowing the D-A fifth to be flat by a pythagorean comma?

Mario, did you mean eleven fifths of exactly 3/2 ratio? I don't think the set you presented has that; it looks like ten fifths of 3/2 (or very close to) and two others: G:D tempered by 2/3 PC and B:F# tempered by 1/3 PC. This gives smallest M3's on Bb F C G and Pythagorean M3's (or very close to) on F# C# G# Eb. Comparing it to other 12-note temperaments, the surprise (in my very limited experience) is the big jump in size between Eb:G and Bb:D - explained of course by the small fifth on G.

Steve M.

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--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
> Steve, You are right, there are only 10 perfect fifths.
...
> Bb = 1.77777777777, (Cell # 508), 996, major third: 1.2542
...
> In your message you copied a phrase written by Mike Battaglia in a
> note he sent me.
> Since I didn´t receive his message that contains the phrase I put
> below, please send me a copy of it.

Mario,
First, It was message #91620 from Mike B; in case you can't find it:
<< Mario - is this scale one that's been generated just by chaining 11
3/2's together and allowing the D-A fifth to be flat by a pythagorean
comma? If so, then this isn't a new scale - this is the same
Pythagorean scale that has been around since antiquity, probably the
first scale ever created. Perhaps I'm misunderstanding.
-Mike >>

Second, for the Bb I think 996.09 cents, rather than 996, corresponds to 1.777... and gives 3/2 fifths either side.

Third, I have looked at my list of historical temperaments and see that "Marpurg IV" has the same pattern of fifths as yours, but rotated around the circle of fifths.

Steve M.