A twelve-tone JI scale derived from a series

To tuning yahoogroups

The common fraction (1/60) = 0.016666... = 1 / [(3)(4)(5)] contains the factors 3, 4, 5; which are linked to the JI scale: Note G = (3 / 2); C = (4 / 4) = 1 and D = (5 / 4). Their product (60) is also linked with music since 5 x 12 = 60 equals to five octaves.

If we use 0.016666... as a musical interval there is not much to do with it, however this number can be reduced by dividing by 2^N where N is a positive number like 2, 3, 4,....without cancelling its link with music. The new smaller interval could be used as an adder to derive a series. The very small adder might reveal new and useful values along the series. By using the divider 2^4 we get the following:

[1/(60) (2^4)] = [1 / (2^6)(3)(5)] = (1/960) = 0.00104166666.....

Part of the series whose expansion depended on the small musical adder is given below.

It was a surprise to notice that all the JI tones appeared in the developed series. I think it is an important finding no matter if JI scales are not most welcome by some musicians.

Adder = 0.00104166666..... Starting from C = 1 a JI scale variant emerged as well as other known frequencies which were derived from the seven-tone JI scale proposed before our era:

DELTA .......HZ......CENTS......Nº........FREQUENCY / C ...........NOTE

CENTS

0.............261.62 ......0.............0.........1.00000 ....................= C

..............................................1..........1.001041666 = 1 + Adder

..............................................2..........1.002083333 = Former value+ Adder

..............................................3..........1.003125

..............................................4..........1.004166666

..............................................5..........1.005208333

..............................................6..........1.00625

..............................................12........1.0125

..............................................24........1.025

..............................................36........1.0375

..............................................48........1.05

4.955.......277.97......104.95.....60....1.0625 = (17/16) .....=.C# = C x (17/16).....M

..............................................64.....1.0666... = (16/15)

3.91........294.32......203.91....120...1.125 = (9/8).............= D = C# x (18/17).....P

............................................165...1.171875 = (75/64)

--2.487....310.68......297.51....180....1.1875 = (19/16)......= Eb = D x (19/18).....Q

............................................192.....1.2 = (6/5)

--13.68....327.03......386.31....240...1.25 = (5/4)................= E = Eb x (20/19).....R

............................................255.....1.265625 = (81/64)

............................................290.......1.30208333 = (125/96)

--1.955...348.83.......498.04....320......1.33333... = (4/3)......= F = E x (16/15)

............................................390.....1.40625 = (45/32)

3...........370.63.......603.........400...1.41666... = (17/12) = F# = F x (17/16)......M

1.955.....392.43......701.95.....480.....1.5 = (3/2).................. = G = F# x (18/17).....P

............................................540....1.5625 = (25/16)

-- 4.44....414.24......795.56.....560.....1.58333... = (19/12) = Ab = G x (19/18).....Q

--15.64...436.04......884.36.....640.....1.6666... = (5/3).........= A = Ab x (20/19).....R

............................................660....1.6875 = (27/16)

--10.69...463.29.....989.31......740...1.770833 = (85/48) = Bb = A x (17/16)....M

--11.73...490.54....1088.3.......840.....1.875 = (15/8)...........= B = Bb x (18/17)......P

0...........523.25....1200..........960...2................................= 2C = B x (16/15)

Semitone factors (Frequency ratios between two consecutive tones), comply with the following sequence:

M P Q R (16/15) M P Q R M P (16/15).

The number of adders comprised in ranges C/C#, C#/D, D/Eb, Eb/E equals 60 while from E to A it is 80; from Bb to B there are 100 adders (840 - 740) and from B to 2C there are 120 adders, (960 - 840) in this case there is a total increase of 120 x 0.001041666.. = 0.125 regarding B = 1,875. Hence 0.125 + 1.875 = 2 = 2C.

Similarly, since in the semitone interval C to C# there are 60 adders and C = 1, the frequency increase is given by 60 x 0.001041666... = 0.0625. Hence, C# = 1 + 0,0625 = 1.0625 = (17/16) and so forth.

Thanks

Mario Pizarro

piagui@...

Lima, July 16, 2009

Dear friends,

for some reason Mario's message does not go through and he has asked me to
send it, including my comments which I have added below.

Kind regards,

Claudio
__________________________________________

From: Mario Pizarro <mailto:piagui@...>

To: tuning yahoogroups <mailto:tuning@yahoogroups.com>
Sent: Thursday, July 16, 2009 3:43 PM
Subject: A twelve-tone JI scale derived from a series

To tuning yahoogroups

The common fraction (1/60) = 0.016666... = 1 / [(3)(4)(5)] contains the
factors 3, 4, 5; which are linked to the JI scale: Note G = (3 / 2); C = (4
/ 4) = 1 and D = (5 / 4). Their product (60) is also linked with music since
5 x 12 = 60 equals to five octaves.

If we use 0.016666... as a musical interval there is not much to do with it,
however this number can be reduced by dividing by 2^N where N is a positive
number like 2, 3, 4,....without cancelling its link with music. The new
smaller interval could be used as an adder to derive a series. The very
small adder might reveal new and useful values along the series. By using
the divider 2^4 we get the following:

[1/(60) (2^4)] = [1 / (2^6)(3)(5)] = (1/960) = 0.00104166666.....

Part of the series whose expansion depended on the small musical adder is
given below.

It was a surprise to notice that all the JI tones appeared in the developed
series. I think it is an important finding no matter if JI scales are not
most welcome by some musicians.

Adder = 0.00104166666..... Starting from C = 1 a JI scale variant emerged as
well as other known frequencies which were derived from the seven-tone JI
scale proposed before our era:

DELTA .......HZ......CENTS......Nº........FREQUENCY / C ...........NOTE

CENTS

0.............261.62 ......0.............0.........1.00000
....................= C

..............................................1..........1.001041666 = 1 +
Adder

..............................................2..........1.002083333 =
Former value+ Adder

..............................................3..........1.003125

..............................................4..........1.004166666

..............................................5..........1.005208333

..............................................6..........1.00625

..............................................12........1.0125

..............................................24........1.025

..............................................36........1.0375

..............................................48........1.05

4.955.......277.97......104.95.....60…...1.0625 = (17/16) .....=.C# = C x
(17/16).....M

..............................................64…....1.0666... = (16/15)

3.91........294.32......203.91....120…..1.125 = (9/8).............= D = C# x
(18/17).....P

............................................165…….1.171875 = (75/64)

--2.487....310.68......297.51....180…...1.1875 = (19/16)......= Eb = D x
(19/18).....Q

............................................192…....1.2 = (6/5)

--13.68....327.03......386.31....240…..1.25 = (5/4)................= E = Eb
x (20/19).....R

............................................255…....1.265625 = (81/64)

............................................290.......1.30208333 = (125/96)

--1.955...348.83.......498.04....320......1.33333... = (4/3)......= F = E x
(16/15)

............................................390…....1.40625 = (45/32)

3...........370.63.......603.........400…..1.41666... = (17/12) = F# = F x
(17/16)......M

1.955.....392.43......701.95.....480.....1.5 = (3/2).................. = G =
F# x (18/17).....P

............................................540…...1.5625 = (25/16)

-- 4.44....414.24......795.56.....560.....1.58333... = (19/12) = Ab = G x
(19/18).....Q

--15.64...436.04......884.36.....640.....1.6666... = (5/3).........= A = Ab
x (20/19).....R

............................................660…...1.6875 = (27/16)

--10.69...463.29.....989.31......740…..1.770833 = (85/48) = Bb = A x
(17/16)....M

--11.73...490.54....1088.3.......840.....1.875 = (15/8)...........= B = Bb x
(18/17)......P

0...........523.25....1200..........960…..2................................=
2C = B x (16/15)

Semitone factors (Frequency ratios between two consecutive tones), comply
with the following sequence:

M P Q R (16/15) M P Q R M P (16/15).

The number of adders comprised in ranges C/C#, C#/D, D/Eb, Eb/E equals 60
while from E to A it is 80; from Bb to B there are 100 adders (840 - 740)
and from B to 2C there are 120 adders, (960 - 840) in this case there is a
total increase of 120 x 0.001041666.. = 0.125 regarding B = 1,875. Hence
0.125 + 1.875 = 2 = 2C.

Similarly, since in the semitone interval C to C# there are 60 adders and C
= 1, the frequency increase is given by 60 x 0.001041666... = 0.0625. Hence,
C# = 1 + 0,0625 = 1.0625 = (17/16) and so forth.

Thanks

Mario Pizarro

piagui@...

Lima, July 16, 2009

__________________________

Claudio's comment:

Mario's calculations are interesting in that instead of the ET nnn
logarithmic subdivisions (from Newton onwards), he returns to the
traditional arithmetic subdivisions.

Quite obviously, since the column Nº in your list, in the "important" lines
shows all numbers multiple of 20, we can divide the original subdivisor
"960" by 20 and thus get the same results with the much smaller "Adder" 48.
The results by Mario below are simplified thus:

Adder = [1 / (2^4)(3)] = (1/48) = 0.020833333333.....

DELTA .......HZ......CENTS......Nº........FREQUENCY / C ...........NOTE

CENTS

0.............261.62 ......0...............0.......1.00000
....................= C

................................................1.......1.02083333 = 1 +
Adder

4.955.......277.97......104.95...... 3.…..1.0625 = (17/16) .....=.C# = C x
(17/16).....M

3.91........294.32......203.91........6.…..1.125 = (9/8).............= D =
C# x (18/17).....P

--2.487....310.68.....297.51.... 9…...1.1875 = (19/16)......= Eb = D x
(19/18).....Q

--13.68....327.03......386.31...12… .1.25 = (5/4)................= E = Eb x
(20/19).....R

--1.955...348.83.......498.04...16......1.33333... = (4/3)......= F = E x
(16/15)

3...........370.63.......603......... 20…..1.41666... = (17/12) = F# = F x
(17/16)......M

1.955.....392.43......701.95.....24.....1.5 = (3/2).................. = G =
F# x (18/17).....P

...............................................27…...1.5625 = (25/16)

-- 4.44....414.24......795.56.....28.....1.58333... = (19/12) = Ab = G x
(19/18).....Q

--15.64...436.04......884.36.....32.....1.6666... = (5/3).........= A = Ab x
(20/19).....R

............................................... 33…...1.6875 = (27/16)

--10.69...463.29.....989.31......37…..1.770833 = (85/48) = Bb = A x
(17/16)....M

--11.73...490.54....1088.3.......42......1.875 = (15/8)...........= B = Bb x
(18/17)......P

0...........523.25....1200..............48…..2..............................
..= 2C = B x (16/15)

That the number 48 is the minimum one with the above properties is quite
obvious, as some of the necessary "Nº", e.g. 32 and 9, are coprime.
However, if we use the generator

Adder = [1 / (2^3)(3)] = (1/24) = 0.0416666......

we still visit most consonant intervals, though not a complete chromatic
scale:

DELTA .......HZ......CENTS......Nº........FREQUENCY / C ...........NOTE

CENTS

0.............261.62 ......0...............0.......1.00000
....................= C

................................................1.......1.0416666 = 1 +
Adder

3.91........294.32......203.91........3.…..1.125 = (9/8).............= D =
C# x (18/17).....P

--13.68....327.03......386.31.....6… .1.25 = (5/4)................= E = Eb x
(20/19).....R

--1.955...348.83.......498.04.....8......1.33333... = (4/3)......= F = E x
(16/15)

3...........370.63.......603......... 10…..1.41666... = (17/12) = F# = F x
(17/16)......M

1.955.....392.43......701.95.....12.....1.5 = (3/2).................. = G =
F# x (18/17).....P

-- 4.44....414.24......795.56.....14.....1.58333... = (19/12) = Ab = G x
(19/18).....Q

--15.64...436.04......884.36.....16.....1.6666... = (5/3).........= A = Ab x
(20/19).....R

--11.73...490.54....1088.3.......21......1.875 = (15/8)...........= B = Bb x
(18/17)......P

0...........523.25....1200..............24…..2..............................
..= 2C = B x (16/15)

At this point it is clear WHY Mario got those interesting numbers starting
precisely from 960.

I am sure to have read successions like the above (though obviously without
the Cents) in ancient theoreticians: surely in late German ones. Have no
idea which is the earliest theoretician in history who did such an analysis:
I am sure somebody in the list knows!

Thanks Mario for your interesting Table.

Claudio

Dear Claudio and Mario,

FYI: I got Mario's mail the day before yesterday via this list. No need to resend it!

Best
Torsten