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To the tuning list,

The progression of musical cells was derived in 1983, it is a chain of an unlimited number of octaves each containing 612 cell frequencies. Any cell frequency is the product of M, J or U commas by the preceding cell frequency. To reduce product errors 11 decimal digit cells were used, common fractions had to be set aside.

The progression is able to produce 12 tone scales according to a pair of adders like 50/52, 49/53 provided their sum gives 102. Their values should be about 50.

When the pair is 49/53, the C# frequency with values comprised within 1 and 2 to be detected in the progression is given by the first number (49), it means that C# frequency corresponds to cell # 49 whose consonant frequency is 1.05707299111. The second adder 53 plus the preceding 49 makes 102 so cell # 102 = 1.12246370821 is D.frequency.

The first adder 49 works again, it is summed to 102 to give (102 + 49) = 151 so cell # 151 (1.1865234375) is the Eb note frequency.

Similarly, 151 summed to the second adder 53 gives 204, so cell # 204 = 1.25991918672 is the

E tone frequency.

F tone frequency corresponds to cell # (204 + 49) = Cell # 253 = 1.3318294975

and so forth.

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

As an example, the scale shown below corresponds to pair 51/51. Note that Eb, F# and A tone frequencies of 12 TET and the corresponding cell tones that work in the progression coincide.

The remaining tone comparisons show very small differences which can be neglected.

Therefore, a consonant 12 TET cell scale has been drawn from the progression. Should M, J or U values or their sequence are not the ones that work in this set, a useless progression would be the result. How this pair (51/51) succeeded in constructing the 12 TET. I think the job was done by the progression exactness

May be the tuning list is not interested on this subject since only a few members requested the progression. The 12 TET cell scale shown below is a reason to investigate on this matter. By now, there is no explanation on what principles were used by the M, J and U commas to settle such abilities.

I would be pleased to send you the progression (7 pages) as soon as you request it.

Thanks

Mario Pizarro

piagui@...

Lima, May 05

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

Pair 51/51..........N0TE --PROGRESSION ----12 TET TONE ----- CELL FREQ. / 12 TET

ADDER -----Cell Number ----------- CELL FREQUENCIES---FREQUENCIES---TONE FREQUENCIES

.........# 0.....C...1.............1...........1

51.......# 51.....C#......1.05946387770....1.05946309436...1.000000739

51.......# 102..... D....1.12246370821......1.12246204831.....1.000001479

51.......# 153..... Eb..1.1892071150...1.18920711500 .......1......

51.......# 204..... E....1.25991918672....1.25992104989.....0.999998521

51.......# 255..... F....1.33483886719....1.33483985415......0.999999261

51.......# 306..... F#.......1.41421356237....1.41421356237.....1

51.......# 357......G....1.49830818473....1.49830707687.....1.000000739

51.......# 408.... Ab...1.58740339942....1.58740105196.....1.000001479

51.......# 459..... A....1.68179283050.....1.68179283050.....1

51.......# 510..... Bb..1.78179480135...1.78179743628....0.999998521

51.......# 561..... B....1.88774722956...1.88774862535.....0.999999261

51.......# 612......2C..2.00000000000...2.00000000000.....1

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🔗genewardsmith <genewardsmith@...>

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@> wrote:
>
> > May be the tuning list is not interested on this subject since only a few members requested the progression. The 12 TET cell scale shown below is a reason to investigate on this matter. By now, there is no explanation on what principles were used by the M, J and U commas to settle such abilities.
>
> I can't figure out what your point is. You seem to be saying you can reproduce 12 equal within 612 equal, which is obvious since 12 divides 612.
>
-------------------------
To genewardsmith:

The roots of authentic musical elements and scales, that is,the smallest interval M that can be distinguished by ear which was called "schisma" and the new J and U ones I detected, define the Natural Progression of Musical Cells, an able set of 624 relative frequencies from note Do = 1 up to (9/8)^6. This interval and commas are cell generators. If any of them is multiplied by a cell frequency, the product equals to the following cell frequency that is the basic element of the progression.

The sequence and positions of M, J, U along the numbered 624 cells are as follows:
Positions of M and J, from cell # 1 up to Nº 22: MMJJMM MMJJMM -...
Positions of U: Nº23 and Nº24.
From cell # 25 up to cell # 74 operate MMJJ generators.
Positions of U: Nº75 and Nº76.
From cell # 77 up to cell # 104 operate MMJJ generators.

It was noticed that cell frequency #104 equals (9/8)= 1.125 (First segment)that is the sixth root of (9/8)^6.

The second segment ends on [(9/8)^2]= 1.265625
The third one ends on [(9/8)^3] = 1.423828125 and so forth.

Twelve tones before (9/8)^6, is found Cell Nº612 = 2 and frequency ratio (Cell Nº624 / Cell Nº612)= Pythagorean comma.

As you can see, I didn´t need neither applied number 12 to develope the Progression of Musical Cells.

Since the 12TET cell scale also named "progression scale" looks disordered in my message I´d better send an arranged scale.

BTW,details on this subject are given in my book:"The Piagui Musical Scale: Perfecting Harmony" by C.Mario Pizarro.(Internet).

Thanks

Mario Pizarro
piagui@...
Lima, May 05

🔗martinsj013 <martinsj@...>

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@> wrote:
> > > May be the tuning list is not interested on this subject since only a few members requested the progression. The 12 TET cell scale shown below is a reason to investigate on this matter. By now, there is no explanation on what principles were used by the M, J and U commas to settle such abilities.
> >
> > I can't figure out what your point is. You seem to be saying you can reproduce 12 equal within 612 equal, which is obvious since 12 divides 612.
> >
> As I understand it, the 612 notes of the system are not quite equally spaced, in such a way that (for example) cell 197 is exactly 5/4 and 208 is exactly 81/64. But the cell spacings are all around 1.96 cents so a number of well temperaments and meantones can be approximated. e.g. Marpurg (exactly), Vallotti, Young, Lehman. And 12-tET.
>
> Steve M.
>
--------------------------------------
Regarding "genewardsmith" I seem to be saying "I can reproduce 12 equal within 612 equal, which is obvious since 12 divides 612",

Sorry, this is not the case, I believe that at this time "genewardsmith" understood that the analysis and cell progression development had nothing to do with number 12 . Since 624 is the number of cells per cycle, the only point where 12 appeared is that the progression "decided" 12 cells between cell #612 (=2) and last cell of this first cycle that ends on [(9/8)^6]= 2.02728652954098. Let me add that mysteriously the ratio between the just mentioned cell#624 frequency and (cell # 612)= 2 is the pythagorean comma (1.01364326477051. The mystery extends to cell # 12 which has also the pythagorean frequency and more: Cell # 6 equals to the square root of that comma.

Steve Martins is correct about the cell frequency values of #197 and #208. It is also correct that frequency intervals between two contiguos cells are not equally spaced due to the different values of M,J,U cell factors: [M = (32805/32768)]=Schisma= 1.001129150390625; J =(8/9)*2^(1/4)= 1.0011313711..;U = [(2^12)*(5^2)*(3^0.5)]/(3^11).

Steve, I would appreciate it if you could give me the number of well temperaments considered by Marpurg, Vallotti, Young and Dr. Bradley Lehman.

I would be pleased to give you, "genewardsmith" and members of the list more information regarding the system. It is time to interchange viewpoints regarding this matter.

Thank you

Mario Pizarro
piagui@...

Lima, May 06, 2010

🔗genewardsmith <genewardsmith@...>

🔗Mario Pizarro <piagui@...>

Re: Information and data

--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:

> Sorry, this is not the case, I believe that at this time "genewardsmith"
understood that the analysis and cell progression development had nothing to do
with number 12 .

Genewardsmith still doesn't have much of a clue. You gave 12 notes, each
separated by precisely 51 small skisma sized intervals, and that still looks
like 12et in 612 to me. It's true you can define other circulating temperaments
which work in 612, but that isn't what I saw nor does it seem to be what you say
you are doing, which remains a mystery.

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

To genewardsmith

Now I see what is the point that confuses you.

In my first messages I gave you two or three examples of scales that were drawn from the progression. There I used the adders 49/53, 50/52 and said that the sum of these 2 numbers must give 102. (The adder is not a frequency, it is only a number)

For a scale with adders 49/53 the tone frequencies are:

---Numerator 49 ---(Freq. of cell # 49) = 1.05707299111 = C#

Denominator 53---(Freq. of cell # 53) = 1.06185781662 = D

Subtotal -----102

--[(Numerator 49)+102] = 151---(Freq. of cell # 151) = 1.12246370821 = Eb

(Denominator 53)+151] = 204---(Freq. of cell # 204).= 1.25991918672 = E

New subtotal = 204 and so forth.

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

The adders you have are 51/51 which are represented by a single 51 in the first column..

51 is not a schisma , you must operate as the above two examples.

I would like to know from you that the mystery is gone OK???

Thanks

Mario Pizarro

Lima, May 06

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🔗genewardsmith <genewardsmith@...>

🔗Mario Pizarro <piagui@...>

Re: Information and data

To genewardsmith: D and Eb frequencies given in my preceding are wrong; the corrected ones are underlined below. Sorry, Mario Pizarro.

For a scale with adders 49/53 the tone frequencies are:

---Numerator 49 ---(Freq. of cell # 49) = 1.05707299111 = C#

Denominator 53---(Freq. of cell # 53 + 49 = # 102) = 1.12246370821 = D

Subtotal -----102

--[(Numerator 49)+102] = 151---(Freq. of cell # 151) = 1.1865234375 = Eb

(Denominator 53)+151] = 204---(Freq. of cell # 204).= 1.25991918672 = E

New subtotal = 204 and so forth.

Thanks

Mario Pizarro

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🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@> wrote:
>
> > ---Numerator 49 ---(Freq. of cell # 49) = 1.05707299111 = C#
>
> This is very close to 49 steps of 612, but not exactly. What it is exactly I don't know, and why you don't simply make it 49 steps of 612 I don't know either. I also don't know if this is supposed to result in a system for constructing circulating temperaments. Also, what do you mean by "numerator"?
--------------------------------

To Geneward Smith,

I explained that the progression cells can provide the twelve tone frequencies of a scale by using two numbers, both near to 50 which I called numerator and denominator despite that they are not common fraction elements.

I think that nobody has derived a scale by using a progression. this is the main reason of explaining my viewpoints in tuning.

Most of classic consonances if not all, like 3/2,4/3,9/8,5/4,32/27,45/32 take part of the cell progression. This feature could improve all sorts of harmony and this improvement is not achieved by the 12 TET scale.

You are talking about 12 TET equal steps; are you sure that a musical octave deserves to be divided by equal intervals?.

"Numerator" was explained above. BTW, according to the Cuyas dictionary this word means "Numerator" in mathematics.

Thanks

Mario Pizarro

Lima, May 26

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
> ... I think that nobody has derived a scale by using a progression. this is the main reason of explaining my viewpoints in tuning. Most of classic consonances if not all, like 3/2,4/3,9/8,5/4,32/27,45/32 take part of the cell progression. This feature could improve all sorts of harmony and this improvement is not achieved by the 12 TET scale.

It may be that nobody has derived a scale in this way, but the Indian scale of 22 notes is often described in this way, with three different "sruti" sizes deployed in the correct order to give 22 notes with many JI or Pythagorean intervals appearing. In effect you have three much smaller "sruti" generating many more notes. It is notable that your three are all very similar in size so that the system closely approximates 612-equal (whereas the Indian system does not closely approximate 22-equal). BTW I think all 22 Indian notes are among your 612.

> ... are you sure that a musical octave deserves to be divided by equal intervals?.

I can't answer that, but it does seem to me that most people here are thinking that way. I think that's because of the advantages, including simplicity, of "tempering" intervals by an amount that does not destroy all their quality. On the other hand, your system retains the precise intervals but is large and complex.

Steve M.

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
> ... Steve, I would appreciate it if you could give me the number of well temperaments considered by Marpurg, Vallotti, Young and Dr. Bradley Lehman. ...

I assume you mean the "cell numbers". Using "C" as the baseline. Best viewed with fixed-width font. NB these cells only approximate the tuning, except for the Marpurg which happens to be exact.

12-tET
0 51 102 153 204 255 306 357 408 459 510 561 (612)

Marpurg VIII or H
0 52 101 153 205 254 306 358 407 459 511 560 (612)

Vallotti
0 48 100 152 200 256 302 356 406 456 510 556 (612)

Young I
0 48 100 152 200 255 302 356 406 456 510 557 (612)

Young II
0 46 100 150 200 254 300 356 404 456 508 556 (612)

Barnes
0 48 100 152 200 256 302 356 406 456 510 558 (612)

Lehman
0 50 100 152 200 256 304 356 407 456 509 558 (612)

Here's hoping I made no "typo"s.
Steve M.

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
> > ... Steve, I would appreciate it if you could give me the number of well temperaments considered by Marpurg, Vallotti, Young and Dr. Bradley Lehman. ...
>
> I assume you mean the "cell numbers". Using "C" as the baseline. Best viewed with fixed-width font. NB these cells only approximate the tuning, except for the Marpurg which happens to be exact.
>
> 12-tET
> 0 51 102 153 204 255 306 357 408 459 510 561 (612)
>
> Marpurg VIII or H
> 0 52 101 153 205 254 306 358 407 459 511 560 (612)
>
> Vallotti
> 0 48 100 152 200 256 302 356 406 456 510 556 (612)
>
> Young I
> 0 48 100 152 200 255 302 356 406 456 510 557 (612)
>
> Young II
> 0 46 100 150 200 254 300 356 404 456 508 556 (612)
>
> Barnes
> 0 48 100 152 200 256 302 356 406 456 510 558 (612)
>
> Lehman
> 0 50 100 152 200 256 304 356 407 456 509 558 (612)
>
> Here's hoping I made no "typo"s.
> Steve M.
>
----------------------
Thanks Steve,

What a coincidence, Marpurg VIII cell numbers also coincides with
my scale I named Piagui II. In my book "The Piagui Musical Scale: Perfecting Harmony" (US)- 2004. the Piagui I, II and III scales were deduced mathematically. Here you have the information:

Each sequence begins with keynote C making successive groups of KKP, KPK and PKK. It follows that if keynote is changed to C# whose frequency is 277.4958 Hz, the chromatic scale is made up by sequence KPK. Likewise, when it is changed to D whose frequency is 294.3287 Hz, the PKK sequence works in this new chromatic scale. Thus, chromatic scales of the three sequences work simultaneously.

TABLE XI - SECOND SEQUENCE OR PIAGUI II
NOTE..SF..CELL#..RELATIVE...............................FREQUENCY
.................FREQUENCY..............................(Hz)
C....1...0......1.......................................261.6255*
C#...K...52.....1.06066017178 =K...(9/8)^0.5............277.4958
D....P...101....1.12119522034 =KP..(4/3*2^(1/4).........293.3333
Eb...K...153....1.189207115 =(K^2)*P..2^(1/4)...........311.127 *
E....K...205....1.26134462288 =(K^3)*P..3/2^(5/4).....330
F....P...254....1.333333... =(K^3)*P*2....(4/3).....348.834 **
F#...K...306....1.41421356237=(K^4)*P^2....[2^0.5)......369.9944*
G....K...358....1.5...........= (K^5)*P^2....(3/2)......392.4383**
Ab..P..407..1.585609486=(K^5)*P^3..((8/9)^0.5))*2^0.75..414.836
A....K...459....1.68179283051= (K^6)*P^3....(2^0.75)....440 *
Bb.K..511..1.7838106725=(K^7)*P^3..((9/8)^0.5)*2^0.75...466.6905
B....P...560....1.88561808316= (K^7)*P^4..2*(8/9)^0.5...493.3259
2C...K...612....2 = (K^8)*P^4.............2.............523.2511*

* E.Tempered and Piagui II frequencies
** Pythagorean frequencies
SF = Semitone Factor

Regards

Mario Pizarro

Lima. May 07

🔗genewardsmith <genewardsmith@...>

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
>
> > I can't answer that, but it does seem to me that most people here are thinking that way. I think that's because of the advantages, including simplicity, of "tempering" intervals by an amount that does not destroy all their quality. On the other hand, your system retains the precise intervals but is large and complex.
>
> What are these precise intervals? I can't seem to get an answer to the basic questions, such as this one. Using three independent intervals, rather than three step sizes of 612, suggests a rank three system, but I don't even know if that system is the 5-limit or something else.
>
--------------
To Genewardsmith":

Since "martinsj013" didn´t respond yet, in the mean time I will say the following:

It is true that between 12 TET constant step size and the variable progression cell size there is no much difference, however the whole set of 12 TET steps doesn´t show "comma fraction consonances" like (25/24)= 1.041666...., (9/8)= 1.125 due to the step size of [2^(1/612)]= 1.00113323506. In no case (1.00113323506)^N gives a number that can be exactly equivalent to a common fraction provided N is a round number.

Two examples demonstrate the 12 TET step inexactness:

12 TET steps................. Progression of cells

Step # 36 = 1.04161601....... Cell # 36 = 1.04166666... = (25/24)

Step # 104 = 1.125007516..... Cell # 104 = 1.125 = (9/8)

Thanks

Mario Pizarro

Lima, May 07

🔗martinsj013 <martinsj@...>

> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > What are these precise intervals? I can't seem to get an answer to the basic questions, such as this one. Using three independent intervals, rather than three step sizes of 612, suggests a rank three system, but I don't even know if that system is the 5-limit or something else.
--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
> Since "martinsj013" didn´t respond yet, in the mean time I will say the following ...

Mario,
You want me to answer? OK then: the three intervals are:

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

(see /tuning/topicId_75743.html#75743
or /tuning/topicId_85276.html#85324)

and the order in which they occur is:
MMJJMM MMJJMM MMJJMM JJMMUU MMJJMM MMJJ MMJJMM MMJJMM MMJJMM.
(52 cells)
(see /tuning/topicId_85498.html#85498)

then the same thing reversed, which takes us to 104 cells. If Cell 0 is 1/1, then Cell 104 is 9/8. Then the whole thing repeated 6 times to get 624 cells. Cell 624 is 2/1*PC and Cell 612 is 2/1. Cell 197 is 5/4, etc etc.

As a matter of interest I calculated (by spreadsheet) that 258 of the 624 cells contain rationals. Or 253 of the first 612. I am not sure if it is supposed to repeat after 612 or 624.

Gene,
I think it is clear that Mario is not familiar with terminology such as "rank three system" - actually I am not that clear either - is this enough for you to determine if this is one or not?

I am able to answer mathematical questions about the system, but I don't understand the motivation. It is no doubt clever of Mario to have found the three intervals, but after all it is very close indeed to 612-equal as you say.

I see that A.S. mentioned (a while back) that Farey had found something similar.
(see /tuning-math/message/16962
or /tuning/topicId_75743.html#75773)

Steve M.

🔗martinsj013 <martinsj@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> Gene,
> I think it is clear that Mario is not familiar with terminology such as "rank three system" - actually I am not that clear either - is this enough for you to determine if this is one or not?

Thanks. There are three independent multiplying factors, so it is rank 3.

> I am able to answer mathematical questions about the system, but I don't understand the motivation. It is no doubt clever of Mario to have found the three intervals, but after all it is very close indeed to 612-equal as you say.

All I can see at the moment is that they all are close to a schisma, and if you square U and take J to the fourth power, you get three independent 5-limit commas which together suffice to notate (that is, serve as a basis for) the 5-limit. They are dual to the three 5-limit vals for 376, 53, and 12; that is, the inverse of the unimodular matrix they define gives those vals in that order.

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> All I can see at the moment is that they all are close to a schisma, and if you square U and take J to the fourth power, you get three independent 5-limit commas which together suffice to notate (that is, serve as a basis for) the 5-limit. They are dual to the three 5-limit vals for 376, 53, and 12; that is, the inverse of the unimodular matrix they define gives those vals in that order.

I can't remember what a val is, but in my spreadsheet I can see that M^376 * J4^53 * U2^12 = 2/1 which must be related. (where J4 means your J^4, U2 means U^2). I found this by inverting a matrix - I suppose this is like changing the basis.

Similarly, with reference to Farey's parameters (see /tuning-math/message/16962) Skh, f and m, I find that for example
Skh^612 * f^12 * m^53 = 2/1, and
Skh^104 * f^2 * m^9 = 9/8, and
Skh^197 * f^4 * m^17 = 5/4.

i.e. the power of Skh in such an expression seems to be precisely the Cell number as given by Mario. Not sure if this helps.

Steve M.

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
>
> > Gene,
> > I think it is clear that Mario is not familiar with terminology such as "rank three system" - actually I am not that clear either - is this enough for you to determine if this is one or not?
>
> Thanks. There are three independent multiplying factors, so it is rank 3.
>
> > I am able to answer mathematical questions about the system, but I don't understand the motivation. It is no doubt clever of Mario to have found the three intervals, but after all it is very close indeed to 612-equal as you say.
>
> All I can see at the moment is that they all are close to a schisma, and if you square U and take J to the fourth power, you get three independent 5-limit commas which together suffice to notate (that is, serve as a basis for) the 5-limit. They are dual to the three 5-limit vals for 376, 53, and 12; that is, the inverse of the unimodular matrix they define gives those vals in that order.
>
---------------
Geneward:

You wrote the following: "THERE ARE THREE INDEPENDENT MULTIPLYING FACTORS, SO IT IS RANK 3."
I guess you are talking about the three M,J,U cell generator factors (cgf). As you know, 612 cgf work in the range 1 up to 2(2C)= [(M^376)*(J^212)*(U^24)].

About (U^2)*(J^4),I never worked with 5-limit. Would you please explain me the meaning of "three independent 5-limit commas?- I might understand you.

Since the progression was derived by using eleven-place factors (cgf),I noticed the coherent first 104 cells so I decided to continue the progressión mainly because along the way (9/8)^0.5 ...(9/8)...
....(9/8)^6 plus many well known common fraction consonances appeared. No doubt the cell intervals and 12 TET steps show about the same value. The cell progression was useful to me, it let me to attain the PIAGUI scales.

HOW and WHAT FOR YOU USE THE 12 TET STEPS.

Steve said that cell progression is large if compared with 12 tet steps. Howmany steps per octave you have in 12 tet?. I tell you that it would be a nonsense if I reduce the 612 cells of the octave because all of them are working to get the 2 and the (9/8)^6 frequencies. It is a chain of 612 inseparable cgf per octave.

Steve also wrote that progression of cells is complex. I am sure that if I give to both of you this progression, you would understand it in less than one minute. Should you request it I would send it by e-mail. The six page table can be sectioned in 4 parts to reduce its weight.

IMO, the progression of cells suffice any kind of uses, all its elements have the exact values needed to obtain a large group of common fraction consonances. Hope you agree with me.

Thanks,

Mario
Lima, May 08

🔗genewardsmith <genewardsmith@...>

🔗martinsj013 <martinsj@...>

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
> > ... You wrote the following: "THERE ARE THREE INDEPENDENT MULTIPLYING FACTORS, SO IT IS RANK 3."
> > ... HOW and WHAT FOR YOU USE THE 12 TET STEPS. ... How many steps per octave you have in 12 tet?
> > IMO, the progression of cells suffice any kind of uses, all its elements have the exact values needed to obtain a large group of common fraction consonances. Hope you agree with me.
>
> Mario, you made some of these points in an email to me (offlist) - have you had time to consider my response?
>
> Steve M.
>

--------------------------
Yes, These questions were already discussed except one;

Which are (is) the case (s) where you and the tuning list use the 12 TET steps?.

As an example, I used the progresion of cells to derive mathematically the three Piagui scales. Details in my book "The Piagui Musical Scale: Perfecting Harmony". Here it was detailed how six cells of the progresion let to solve 2 equations with 4 variables.

Regards

Mario

Sunday 09

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
> Geneward:
> About (U^2)*(J^4),I never worked with 5-limit. Would you please explain me the meaning of "three independent 5-limit commas?- I might understand you.

I'll leave this to Gene to answer - but one key point is that only powers of 2, 3 and 5 are found in your system. Powers of 7, 11, or higher primes are not there (they may be reasonably approximated, I have not checked).

> ... No doubt the cell intervals and 12 TET steps show about the same value. The cell progression was useful to me, it let me to attain the PIAGUI scales.

The point I was trying to make, is that the progression does not seem to be an efficient way to derive the common ratios, or indeed the tempered values found in 12 TET, Vallotti etc. You need 612 cells to finally derive just 12 notes of Piagui II.

> HOW and WHAT FOR YOU USE THE 12 TET STEPS. ... How many steps per octave you have in 12 tet?.

12 TET is just one system with (by definition) 12 steps per octave. It was not derived from a progression, but can be described as such - with 12 cells, and one step size of 2^(1/12). There are other TETs, some of which are multiples of 12 but many more are not. Gene mentioned 612-TET only because it seems very close to your system; it is not used by many musicians as far as I know. These are not new ideas - e.g. (I hope I have remembered this right) 31 TET was described by Huygens, 53 TET by Newton, and 612 TET by both of them.

The other systems like Vallotti and Marpurg are not derived from a progression, nor from 12 TET.

> I am sure that if I give to both of you this progression, you would understand it in less than one minute.

I do understand what it is (but not how you derived it); I still say it is complex.

> IMO, the progression of cells suffice any kind of uses, ....

I would say "many" not "any". There is a 1.96 cent gap between cells on average, so there may be some intervals that cannot be sufficiently closely approximated for some purposes. e.g. the 5th of the equal-beating meantone, the 5th of the Lucy temperament, arguably also the 7/4 and 7/6 ratios.

> ... all its elements have the exact values needed to obtain a large group of common fraction consonances. Hope you agree with me.

Yes, but it is not the only way to obtain them, and almost certainly not the best way. And many good judges think that there are other things that are equally important. Even Piagui II uses non-rational cells from the progression.

Steve M.

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
>
> > You wrote the following: "THERE ARE THREE INDEPENDENT MULTIPLYING FACTORS, SO IT IS RANK 3."
> > I guess you are talking about the three M,J,U cell generator factors (cgf). As you know, 612 cgf work in the range 1 up to 2(2C)= [(M^376)*(J^212)*(U^24)].
> >
> > About (U^2)*(J^4),I never worked with 5-limit. Would you please explain me the meaning of "three independent 5-limit commas?- I might understand you.
>
> Let U2 be U^2 and J4 be J^4. Together with M, each is a rational number which can be factored using only the primes 2, 3, and 5. On the other hand, we have
>
> 2 = M^356 * J4^53 * U2^12
> 3 = M^596 * J4^84 * U2^19
> 5 = M^873 * J4^123 * U2^28
>
> Hence every product of the primes 2, 3 and 5 with integer exponents can be written in terms of M, J4 and U2.
>
Geneward,

It is not advisable to create other bases like J4 and U2. The original ones
M,J,U should be the only ones for any number due to the confused terms like your
J4^53 as well as the increased number of terms. In fact you had to use 8 terms
for 2 = M^356 * J4^53 * U2^12.
The 8 terms are M,356,J,4,53,U,2,12.

In one of my messages I wrote 2 = M^376 * J^212 * U^24 that contains only 6
terms. This way the operation symbols are clear and the number of operations is
lower. This better expression is given in my book.

Thanks

Mario Pizarro

Lima, May 09

🔗genewardsmith <genewardsmith@...>

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
> > Geneward:
> > About (U^2)*(J^4),I never worked with 5-limit. Would you please explain me the meaning of "three independent 5-limit commas?- I might understand you.
>
> I'll leave this to Gene to answer - but one key point is that only powers of 2, 3 and 5 are found in your system. Powers of 7, 11, or higher primes are not there (they may be reasonably approximated, I have not checked).
>
> > ... No doubt the cell intervals and 12 TET steps show about the same value. The cell progression was useful to me, it let me to attain the PIAGUI scales.
>
> The point I was trying to make, is that the progression does not seem to be an efficient way to derive the common ratios, or indeed the tempered values found in 12 TET, Vallotti etc. You need 612 cells to finally derive just 12 notes of Piagui II.
>
> > HOW and WHAT FOR YOU USE THE 12 TET STEPS. ... How many steps per octave you have in 12 tet?.
>
> 12 TET is just one system with (by definition) 12 steps per octave. It was not derived from a progression, but can be described as such - with 12 cells, and one step size of 2^(1/12). There are other TETs, some of which are multiples of 12 but many more are not. Gene mentioned 612-TET only because it seems very close to your system; it is not used by many musicians as far as I know. These are not new ideas - e.g. (I hope I have remembered this right) 31 TET was described by Huygens, 53 TET by Newton, and 612 TET by both of them.
>
> The other systems like Vallotti and Marpurg are not derived from a progression, nor from 12 TET.
>
> > I am sure that if I give to both of you this progression, you would understand it in less than one minute.
>
> I do understand what it is (but not how you derived it); I still say it is complex.
>
> > IMO, the progression of cells suffice any kind of uses, ....
>
> I would say "many" not "any". There is a 1.96 cent gap between cells on average, so there may be some intervals that cannot be sufficiently closely approximated for some purposes. e.g. the 5th of the equal-beating meantone, the 5th of the Lucy temperament, arguably also the 7/4 and 7/6 ratios.
>
> > ... all its elements have the exact values needed to obtain a large group of common fraction consonances. Hope you agree with me.
>
> Yes, but it is not the only way to obtain them, and almost certainly not the best way. And many good judges think that there are other things that are equally important. Even Piagui II uses non-rational cells from the progression.
>
> Steve M.
>
-----------------------
Steve, Would you please explain me what do you mean by "Even Piagui II uses non-rational cells from the progression" Particularly about "non-rational cells from the progression."

I asked Geneward to use the original symbols I used in my book since the terms he use produce confusion, uses a greater number of symbols and forces to do a greater number of operations. It is advisable that numbers 2,3,5 should be expressed as follows:

2 = M^376 * J^212 * U^24

3 = M^596 * J^336 * U^38

5 = M^873 * J^492 * U^56

Thank you

Mario

Lima, May 10

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
>
> > Geneward,
> >
> > It is not advisable to create other bases like J4 and U2. The original ones
> > M,J,U should be the only ones for any number due to the confused terms like your
> > J4^53 as well as the increased number of terms. In fact you had to use 8 terms
> > for 2 = M^356 * J4^53 * U2^12.
> > The 8 terms are M,356,J,4,53,U,2,12.
> >
> > In one of my messages I wrote 2 = M^376 * J^212 * U^24 that contains only 6
> > terms.
>
> Both are six "terms", consisting of three numbers and three exponents, so your argument fails.
>
-----------------------------
Gene, The exponents you have are the following:
356, 4, 53, 2, 12,
The numbers represented by letters are the following:
M, J, N ---------- TOTAL : 8
It would be incredible that you try to convince me that J4 is one number and U2 is also only one number. Don´t you realize that your conclusion is against the elemental logic?

Thanks
Mario

Lima, May 10

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:

> -----------------------------
> Gene, The exponents you have are the following:
> 356, 4, 53, 2, 12,
> The numbers represented by letters are the following:
> M, J, N ---------- TOTAL : 8

The 4 and the 2 are not exponents. Since it seems to produce confusion, let's call "J4" by K, and "U2" by V. Now you have
three numbers, and three exponents. You haven't shown any purpose to your choice of numbers as yet, so an explanation of why M, K, and V aren't a better choice (since they clearly are a simpler one) would be a place to start.

> It would be incredible that you try to convince me that J4 is one number and U2 is also only one number. Don´t you realize that your conclusion is against the elemental logic?

I'm a mathematician. You are not likely to win a math argument with me, and you would be better advised not to try and instead attempt instead to follow what I am saying, which isn't hard, involving counting to six.

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
>
> > -----------------------------
> > Gene, The exponents you have are the following:
> > 356, 4, 53, 2, 12,
> > The numbers represented by letters are the following:
> > M, J, N ---------- TOTAL : 8
>
> The 4 and the 2 are not exponents. Since it seems to produce confusion, let's call "J4" by K, and "U2" by V. Now you have
> three numbers, and three exponents. You haven't shown any purpose to your choice of numbers as yet, so an explanation of why M, K, and V aren't a better choice (since they clearly are a simpler one) would be a place to start.
>
> > It would be incredible that you try to convince me that J4 is one number and U2 is also only one number. Don´t you realize that your conclusion is against the elemental logic?
>
> I'm a mathematician. You are not likely to win a math argument with me, and you would be better advised not to try and instead attempt instead to follow what I am saying, which isn't hard, involving counting to six.
>
--------------------
Geneward,
We are not in a competition, I only want to preserve the suitable rules on this matter which are given in my book. Respects your insolence to dare to say "You are not likely to win a math argument" despite I am acting politely, I tell you that I won´t lower myself should T inform you about my career. Too much noise from a few coconuts.

Mario Pizarro

Lima, May 10

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
> > > It would be incredible that you try to convince me that J4 is one number and U2 is also only one number. Don´t you realize that your conclusion is against the elemental logic? ...
> Geneward,
> ... I only want to preserve the suitable rules on this matter which are given in my book. Respects your insolence to dare to say "You are not likely to win a math argument" ...

Mario, steady on, old chap. J4 is one identifer and U2 is also only one identifier. I might have called them JUPITER and URANUS and they would still each be one identifier.

If you wish to argue is that Gene's equations look less clear than yours, that is OK but really not a mathematical reason to reject them.

If, as I guess, you wish to argue that your equations have a good mathematical derivation behind them, and Gene's do not, then please say so instead of mentioning the irrelevant fact that his equation is longer than yours.

Steve M.

🔗martinsj013 <martinsj@...>

🔗Mario <piagui@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Mario" <piagui@> wrote:
> > > > It would be incredible that you try to convince me that J4 is one number and U2 is also only one number. Don´t you realize that your conclusion is against the elemental logic? ...
> > Geneward,
> > ... I only want to preserve the suitable rules on this matter which are given in my book. Respects your insolence to dare to say "You are not likely to win a math argument" ...
>
> Mario, steady on, old chap. J4 is one identifer and U2 is also only one identifier. I might have called them JUPITER and URANUS and they would still each be one identifier.
>
> If you wish to argue is that Gene's equations look less clear than yours, that is OK but really not a mathematical reason to reject them.
>
> If, as I guess, you wish to argue that your equations have a good mathematical derivation behind them, and Gene's do not, then please say so instead of mentioning the irrelevant fact that his equation is longer than yours.
>
> Steve M.
>
-------------------------------
Steve,

I am a peaceful man since I was a boy. Even now after Geneward attack I forgive him and would like to share a sincere friendship with him.
Still I don´t know his full name. Geneward Smith?, Gene Wardsmith?.

I had forgotten to inform you about another important characteristic of the Progression of Musical Cells. It is a compendium of unexpected abilities. The new follows:

"If any cell frequency is multiplied by 1.5 to get its perfect fifth, the frequency obtained equals the frequency of another cell.
Similarly, the perfect fourth (x 1.333333...)of any cell frequency equals with exactness another cell frequency."

These features and those explained before gave a reason to write in my book about the following: " The Progression of Musical Cells is a scientific set that deserves to be included in the acoustical chapter of Physics".

Regards

Mario

Lima, May 10