Yahoo Tuning Groups Ultimate Backup tuning IMO

I see that some new scale proposals decide on G = 1.5 = 3/2. Piagui scales also work with this value which was not selected when they were derived but was the result of a mathematical process started with the Piagui basic equations K^m P^n = 2 and m + n = 12.

Recently, I knew that Piagui D major chord wave peak graph shows a moderate distortion and not desired peaks on it so I extended the research.

Since K and P semitone factors developed good but imperfect Piagui scales, both were the basis of the complementary research because they came from a rigorous mathematical analysis. Curiously, the 1.01364326477 pythagorean comma appeared as a key factor in the analysis and finally a new scale system (three variants too) was derived. Here, the G frequency is 1.496614954 and this value has a solid connection with the remaining 11 semitone factors therefore it would be a nonsence to replace this narrow fifth by 1.5.

I would appreciate it if you give me your opinion on these results. It seems to me that 1.5 is just a perfect consonance and not the G tone frequency.

Thanks

Mario Pizarro
piagui@ec-red.com

Lima, November 20, 2009

🔗Michael <djtrancendance@...>

To me 1.4966 and 1.5 sound essentially the same just by themselves.
Perhaps the more important questions to me (about Piagui or any other scale somewhat resembling 12TET) are
******1. Does the scale enable any consonant chords impossible in 12TET?*********
******2. How many consonant chords from 12TET are lost?***********************

Achieve a lot of #1 with very little of #2 and, IMVHO, you have a very good scale.
How would you do this? One idea is to provide sound examples of chords that are impossible or sour in 12TET, but sweet sounding in your own scale.

-Michael

________________________________
From: Mario Pizarro <piagui@...>
To: tuning yahoogroups <tuning@yahoogroups.com>
Sent: Fri, November 20, 2009 5:48:46 PM
Subject: [tuning] IMO

I see that some new scale proposals decide on
G = 1.5 = 3/2. Piagui scales also work with this value which was not selected
when they were derived but was the result of a mathematical process started
with the Piagui basic equations K^m P^n = 2 and m + n = 12.

Recently, I knew that Piagui D major chord wave
peak graph shows a moderate distortion and not desired peaks on it so I
extended the research.

Since K and P semitone factors developed good
but imperfect Piagui scales, both were the basis of the complementary research
because they came from a rigorous mathematical analysis. Curiously, the
1.01364326477 pythagorean comma appeared as a key factor in the analysis
and finally a new scale system (three variants too) was derived. Here, the G
frequency is 1.496614954 and this value has a solid connection with the
remaining 11 semitone factors therefore it would be a nonsence to replace
this narrow fifth by 1.5.

I would appreciate it if you give me your opinion
on these results. It seems to me that 1.5 is just a perfect consonance and
not the G tone frequency.

Thanks

Mario Pizarro
piagui@ec-red. com

Lima, November 20,
2009

🔗Mario <piagui@...>

Michael,
One of the three new scales has the following cents/12ET- I don´t have at hand the exact values. They are about the following values:
As you can see,they do not resemble to 12ET.
(+4)
(+2)
About 0

(greater than +3.5 cents)
(greater than +2.4 cents)
About 0

<< 1. Does the scale enable any consonant chords impossible in 12TET?
!!!Somebody has to do it; a complex task for me!!!!!
> 2. How many consonant chords from 12TET are lost?
!!! This is a more complex task for me.

Right now we are processing the chord wave peaks of(C,D,E major; C,D,E minor; C major 7, C min (major 7), Cª(major 7), C+ (major 7) and minor sevenths: C7,Cª7, C+7.

¿IMVHO?

One idea is to provide sound examples of chords that are impossible or sour in 12TET, but sweet sounding in your own scale.

¿SOUND EXAMPLES VIA E-MAIL?- ¿WHAT IS THAT?-FIRST I WOULD NEED A LIST OF SOUR CHORDS IN 12 ET. HOW I CAN BE SURE THAT MY CHORDS ARE SWEET SOUNDING CHORDS. THIS TASK IS FOR A SKILLED MUSICIAN AT LEAST A PHD IN MUSIC. Probably you are guessing that I have a grand piano that can tuned to the new scale. No Sir, I just have an old guitar.

Thanks
Mario Pizarro
Lima, November 20, 2009

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> To me 1.4966 and 1.5 sound essentially the same just by themselves.
> Perhaps the more important questions to me (about Piagui or any other scale somewhat resembling 12TET) are
> ******1. Does the scale enable any consonant chords impossible in 12TET?*********
> ******2. How many consonant chords from 12TET are lost?***********************
>
> Achieve a lot of #1 with very little of #2 and, IMVHO, you have a very good scale.
> How would you do this? One idea is to provide sound examples of chords that are impossible or sour in 12TET, but sweet sounding in your own scale.
>
> -Michael
>
>
>
>
> ________________________________
> From: Mario Pizarro <piagui@...>
> To: tuning yahoogroups <tuning@yahoogroups.com>
> Sent: Fri, November 20, 2009 5:48:46 PM
> Subject: [tuning] IMO
>
>
> I see that some new scale proposals decide on
> G = 1.5 = 3/2. Piagui scales also work with this value which was not selected
> when they were derived but was the result of a mathematical process started
> with the Piagui basic equations K^m P^n = 2 and m + n = 12.
>
> Recently, I knew that Piagui D major chord wave
> peak graph shows a moderate distortion and not desired peaks on it so I
> extended the research.
>
> Since K and P semitone factors developed good
> but imperfect Piagui scales, both were the basis of the complementary research
> because they came from a rigorous mathematical analysis. Curiously, the
> 1.01364326477 pythagorean comma appeared as a key factor in the analysis
> and finally a new scale system (three variants too) was derived. Here, the G
> frequency is 1.496614954 and this value has a solid connection with the
> remaining 11 semitone factors therefore it would be a nonsence to replace
> this narrow fifth by 1.5.
>
> I would appreciate it if you give me your opinion
> on these results. It seems to me that 1.5 is just a perfect consonance and
> not the G tone frequency.
>
> Thanks
>
> Mario Pizarro
> piagui@ec-red. com
>
> Lima, November 20,
> 2009
>

🔗Charles Lucy <lucy@...>

Well stated Michael;

I would also like to add;

********3. Can this tuning system infinitely modulate and transpose?******
********4. Can it emulate all imaginable musical scales?*********
********5. Can it produce and control both consonance and dissonance?******
********6. Are the harmonic "rules" it uses "practical" and easily understood? *******

On 21 Nov 2009, at 01:35, Michael wrote:

>
> To me 1.4966 and 1.5 sound essentially the same just by themselves.
> Perhaps the more important questions to me (about Piagui or any other scale somewhat resembling 12TET) are
> ******1. Does the scale enable any consonant chords impossible in 12TET?*********
> ******2. How many consonant chords from 12TET are lost?***********************
>
> Achieve a lot of #1 with very little of #2 and, IMVHO, you have a very good scale.
> How would you do this? One idea is to provide sound examples of chords that are impossible or sour in 12TET, but sweet sounding in your own scale.
>
> -Michael
>
> From: Mario Pizarro <piagui@...>
> To: tuning yahoogroups <tuning@yahoogroups.com>
> Sent: Fri, November 20, 2009 5:48:46 PM
> Subject: [tuning] IMO
>
>
>
> I see that some new scale proposals decide on G = 1.5 = 3/2. Piagui scales also work with this value which was not selected when they were derived but was the result of a mathematical process started with the Piagui basic equations K^m P^n = 2 and m + n = 12.
>
> Recently, I knew that Piagui D major chord wave peak graph shows a moderate distortion and not desired peaks on it so I extended the research.
>
> Since K and P semitone factors developed good but imperfect Piagui scales, both were the basis of the complementary research because they came from a rigorous mathematical analysis. Curiously, the 1.01364326477 pythagorean comma appeared as a key factor in the analysis and finally a new scale system (three variants too) was derived. Here, the G frequency is 1.496614954 and this value has a solid connection with the remaining 11 semitone factors therefore it would be a nonsence to replace this narrow fifth by 1.5.
>
> I would appreciate it if you give me your opinion on these results. It seems to me that 1.5 is just a perfect consonance and not the G tone frequency.
>
> Thanks
>
> Mario Pizarro
> piagui@ec-red. com
>
> Lima, November 20, 2009
>
>

Charles Lucy
lucy@lucytune.com

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

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🔗a_sparschuh <a_sparschuh@...>

🔗martinsj013 <martinsj@...>

🔗a_sparschuh <a_sparschuh@...>

🔗Mario Pizarro <piagui@...>

Steve,

As I said, when I knew that Piagui D major chord wave peak graph showed unexpected peaks, started the complementary analysis which uses K and P semitone factors.
K = (9/8)^(1/2) and P = (8/9)*2^(1/4) are cells of a geometric progression where the first 612 cells cover the octave 1 up to 2. Here, commas M, J and U are the factors which work in sequence. K for instance, is also given by (M^32)*(J^18)*(U^2)= 1.06066017178...-- 32 + 18 + 2 = 52, so cell # 52 equals K. Similarly, P = 1.0570729911.... = (M^30)*(J^17)*(U^2) = Cell # 49.

Neither K nor P involves powers of 3 and of 2, as you can see. However, commas M, J, U do it:

M = [(38 x 5) / 215] = [(32805 / 32768)] = 1.00112915039062

J = [(225x21/4)/(313x52)] = [(33554432x21/4)/39858075]=1.001131371103

U = [(212x 52 x 31/2) / 311] = [(102400 x 31/2)/177147] = 1.0012136965066

<If the mathematical analysis involves powers of 3 and of 2 then I don't think it <"curious" that the Pythagorean Comma is involved. The 1.496614954 ratio looks <very close indeed to the "1/6 PC tempered 5th" - is it actually identical in your <system?
----
ï¿½Would you explain me how can I calculate "1/6 PC tempered 5th"?. This way I could answer your question. Is "tempered 5th" related to the equal tempered scale ?-----

Last week, after many years the progression was deduced, I noticed the interesting ratio:
[(K^4)/(P^4)] = 1.01364326477 = Pythagorean comma which is also given by the twelft cell of the progression and the sixth cell is equal to the square root of the PC:
It took me years to derive the progression where the first cell is the schisma. Cell # 0 is note C. Below I give you the first 20 cells.

THE PROGRESSION OF MUSICAL CELLS

FIRST SEGMENT

CELL RELATIVE FREQUENCY

No. COMMA F(M,J,U) DECIMAL VALUE

- - 1 = C

1 M M 1.00112915039062 ZC

2 M M2 1.00225957576060

3 J M2J 1.00339350328355

4 J M2J2 1.00452871369807

5 M M3J2 1.00566297768753

6 M M4J2 1.00679852243163 (Its square root)

7 M M5J2 1.00793534937651

8 M M6J2 1.00907345996998

SET 9 J M6J3 1.01021509652337

Q 10 J M6J4 1.01135802469136 ZC

11 M M7J4 1.0125 ZC

12 M M8J4 1.01364326477051 * Pythag. comma

13 M M9J4 1.01478782045888

14 M M10J4 1.01593366852275

15 J M10J5 1.01708306651785

16 J M10J6 1.01823376490863

17 M M11J6 1.01938350396202

18 M M12J6 1.02053454124372

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

19 J M12J7 1.02168914453326

20 J M12J8 1.02284505410760

21 M M13J8 1.024 ZC

22 M M14J8 1.02515625

Thanks
Mario Pizarro
piagui@...

Lima, November 23

----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Monday, November 23, 2009 12:30 PM
Subject: [tuning] Re: IMO

> --- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
>> > I see that some new scale proposals decide on
>> > G = 1.5 = 3/2. Piagui scales also work with this value which was not >> > selected
>> > when they were derived but was the result of a mathematical process >> > started
>> > with the Piagui basic equations K^m P^n = 2 and m + n = 12.
> Curiously, the
>> > 1.01364326477 pythagorean comma appeared as a key factor in the >> > analysis
>> > and finally a new scale system (three variants too) was derived. Here, >> > the G
>> > frequency is 1.496614954 and this value has a solid connection with the
>> > remaining 11 semitone factors therefore it would be a nonsence to >> > replace
>> > this narrow fifth by 1.5.
>
> If the mathematical analysis involves powers of 3 and of 2 then I don't > think it "curious" that the Pythagorean Comma is involved. The > 1.496614954 ratio looks very close indeed to the "1/6 PC tempered 5th" - > is it actually identical in your system?
>
> Steve M.
>
>
>
> ------------------------------------
>
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🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
> As I said, when I knew that Piagui D major chord wave peak graph showed
> unexpected peaks, started the complementary analysis which uses K and P
> semitone factors. ... K for instance, is also given by
> (M^32)*(J^18)*(U^2)= 1.06066017178...-- 32 + 18 + 2 = 52, so cell # 52
> equals K. Similarly, P = 1.0570729911.... = (M^30)*(J^17)*(U^2) = Cell # 49.
> Neither K nor P involves powers of 3 and of 2, as you can see.

Well, yes they do:
K = 2^(-3/2) x 3^1
P = 2^(13/4) x 3^(-2)

Your formulae for M, J, U are missing some ^ and () I think

M = [(3^8 x 5) / 2^15] = [(32805 / 32768)] = 1.00112915039062

J = [(2^25 x 2^(1/4))/(3^13 x 5^2)] = [(33554432x21/4)/39858075]=1.001131371103

U = [(2^12 x 5^2 x 3^(1/2)) / 3^11] = [(102400 x 3^(1/2))/177147] = 1.0012136965066

> ¿Would you explain me how can I calculate "1/6 PC tempered 5th"?. This way I
> could answer your question. Is "tempered 5th" related to the equal tempered
> scale ?-----

PC is 3^12 / 2^19, and "1/6 PC tempered 5th" means "3/2 / PC^(1/6)"
i.e. 2^(13/6) / 3
It is used in a meantone and in several circulating temperaments.

Equal tempered scale has "1/12 PC tempered 5th", this means "3/2 / PC^(1/12)"
i.e. 2^(7/12)

>
> Last week, after many years the progression was deduced, I noticed the
> interesting ratio:
> [(K^4)/(P^4)] = 1.01364326477 = Pythagorean comma which is also given by the
> twelfth cell of the progression

Yes, starting with:
K = 2^(-3/2) x 3^1
P = 2^(13/4) x 3^-2

we can see that (K^4)/(P^4) = 2^(-19) x 3^12 QED.

Steve M.

🔗martinsj013 <martinsj@...>

🔗Mario Pizarro <piagui@...>

Steve, 1.496614954 is a rounded figure. The attachment explains, starting from the G frequency of F#(PM), how cell # 356 is linked to the accurate G frequency. Ratio J/M always establishes an associated consonant cell. I see that you bought my book otherwise you wouldnï¿½t know the value of cell # 356. isnï¿½t?

Cell # 356
1.49661827761514

M
1.00112915039062
(Schisma)

J
1.00113137110290
(Progression comma factor)

P
1.05707299111353
(Piagui scales semitone factor)

F#(PM)
1.49661495400000
The rounded G.

(For the three Justharm scales)

F#
1.41421356237310
Equals to 2^(0.5)

F#(PM)
1.49661495781239
Fourteen decimals (G)

(The accurate G frequency)

J/M
1.00000221820759
This ratio gives the associated
consonant cell.

F#(PM)(J/M)
1.49661827761505
The accurate frequency of Cell
# 356

The M and J comma factors of the progression are the two options that any cell can take for establishing the following cell and the J/M ratio applied to any cell determines another consonant frequency that could be called "associated cell".

The associated
cell (J/M)(Cell # 356)
equals to the "accurate G".

-------------------------------------------------------------------------------
IT IS A ROUNDING ERROR, ABOVE YOU HAVE THE EXPLANATION.

REGARDS,

Mario Pizarro
piagui@...
Lima, November 25, 2009
-------------------------------------------------------------------------------
----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Wednesday, November 25, 2009 6:01 AM
Subject: [tuning] Re: IMO

>I have realised I should be able to answer my own question:
>
> Well first of all I can't find 1.496614954 precisely in the system.
> Cell 356 is 1.4966182776 (this is 3/2/M^2)
>
> No point looking in any other cell since M ("schisma"), J, U and PC^(1/12) > ("grad") are all about the same size approx 2 cents.
>
> 3/2/(MxJ) is 1.496614958 - very close indeed, perhaps the difference from > your value is a rounding error? This can be found as the ratio between > Cell 359 and Cell 3, for example. But I may have misunderstood something > about the system.
> YES STEVE, IT IS A ROUNDING ERROR, THIS TIME I USED EXCEL.
> By the way, Cell 357 is 3/2/M = 2^14/3^7/5 (schisma tempered 5th)
>
> Steve M.
>
>
>
> ------------------------------------
>
>

🔗Mario Pizarro <piagui@...>

Steve,

It was a rounding error.

Cell # 356
1.49661827761514

M
1.00112915039062
(Schisma)

J
1.00113137110290
(Progression comma factor)

P
1.05707299111353
(Piagui scales semitone factor)

F#(PM)
1.49661495400000
The rounded G.

(For the three Justharm scales)

F#
1.41421356237310
Equals to 2^(0.5)

F#(PM)
1.49661495781239
Fourteen decimals (G)

(The accurate G frequency)

J/M
1.00000221820759
This ratio gives the associated
consonant cell.

F#(PM)(J/M)
1.49661827761505
The accurate frequency of Cell
# 356

Regards,

Mario Pizarro

Lima, November 25

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

----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Wednesday, November 25, 2009 6:01 AM
Subject: [tuning] Re: IMO

>I have realised I should be able to answer my own question:
>
> Well first of all I can't find 1.496614954 precisely in the system.
> Cell 356 is 1.4966182776 (this is 3/2/M^2)
>
> No point looking in any other cell since M ("schisma"), J, U and PC^(1/12) > ("grad") are all about the same size approx 2 cents.
>
> 3/2/(MxJ) is 1.496614958 - very close indeed, perhaps the difference from > your value is a rounding error? This can be found as the ratio between > Cell 359 and Cell 3, for example. But I may have misunderstood something > about the system.
>
> By the way, Cell 357 is 3/2/M = 2^14/3^7/5 (schisma tempered 5th)
>
> Steve M.
>
>
>
> ------------------------------------