Yahoo Tuning Groups Ultimate Backup tuning Formula to Place Frequencies into one Octave

Hi Members,

Been a long time. I haven't been here in a couple of years. Picking up studies again. Hope all is well with everyone.

I have been reviewing the origins of the 12-tone system and the 7-tone diatonic scale. I am primarily using the book titled, "Science and Music" by Sir James Jeans first paperback edition 1961. The price on the front cover says, $1.95 Imagine that! The book is worth its weight in gold and it sold for $1.95. Amazing

Using the cycle of 5ths, I started with "C" and assigned to it the frequency of "1". I multiplied "C" by 1.5 (1*1.5) and arrived at "G" which is 1.5. Them multiplied "G" by 1.5 (1.5*1.5) to arrive at "D" which results in 2.25

I continued on multiplying each 5th by 1.5 until I have arrived at the octave of "C". See results for each step of the cycle of 5ths below please.

C - 1

G - 1.5

D 2.25

A - 3.375

E 5.0625

B 7.59375

F# 11.390625

C# - 17.085937

G# - 25.628905

Eb - 38.443357

Bb 57.665035

F 86.497552

C 129.74632

Please help me to find the formula to reduce all the above frequencies to ONE OCTAVE. The answer should match the table on page 167 in the book, "Science and Music". I'm not sure what the proper term is for this procedure. The table lists the frequencies for the Pythagorean seven tone scale and its octave (all in one octave). The frequencies are listed below.

Pythagorean 7 Tone Scale

C 1/1 = 1.0000
D 9/8 = 1.1250
E 81/64 = 1.2656
F 4/3 = 1.3333
G 3/2 = 1.5000
A 27/16 = 1.6875
B 243/128 = 1.8984
C 2/1 = 2

In other words, using the frequencies from the cycle of 5ths in the first table above, I start with

C = 1.0000 (no problem)

C# = from the cycle of 5ths list above, the "C# "would be equal to 17.0859. It's too large. It is way above "2" (the octave).

D = from the cycle of 5ths list above would be equal to 2.25. Still too large as it is greater than "2" (the octave). How do I reduce this "D" to 1.1250 as listed in the Pythagorean table above?

Etc. with the rest of the remaining chromatic frequencies listed in the first table above.

Thank you for your time
Wally Lepore

🔗Mike Battaglia <battaglia01@...>

🔗Wolf Peuker <wolfpeuker@...>

🔗clamengh <clamengh@...>

Wally> > Please help me to find the formula to reduce all the above frequencies to ONE OCTAVE.

Mike Battaglia> Just keep dividing each value through by 2 until you end up at a
> number that's between 1 and 2

Yes, of course; or, if you want to do that in an automated fashion, (for instance using Excel) do the following:
"Take logarithms in base 2, subtract integer parts, then exponentiate again".
For instance:

D=1.5*1.5=2.25; log_2 2.25 = 1.169925001; integer part of 1.169925001=1, so please sebtract 1; (log_2 2.25)-1 = 0.169925001; 2^(0.169925001)=1.125 as you wish.

Suppose you have your frequencies in an Excel worksheet, column B (note names in column A).
Just copy the following formula in C1 (including the "=" sign):
=2^(LOG(B1;2)-INT(LOG(B1;2)))
Then please copy this formula in as many cells in column C as you need (i.e., as many cells as you have in column B). Results will fit into an octave.
Now, you could want to reorder them: first,"copy" all cells, then "paste special" them as "values". Select all cells with content and "order" them according to column C. Done!.

Example:
Gb 0,087791495 1,404663923
Db 0,131687243 1,053497942
Ab 0,197530864 1,580246914
Eb 0,296296296 1,185185185
Bb 0,444444444 1,777777778
F 0,666666667 1,333333333
C 1 1
G 1,5 1,5
D 2,25 1,125
A 3,375 1,6875
E 5,0625 1,265625
B 7,59375 1,8984375
F# 11,390625 1,423828125
C# 17,0859375 1,067871094
G# 25,62890625 1,601806641
D# 38,44335938 1,20135498
A# 57,66503906 1,802032471
E# 86,49755859 1,351524353
B# 129,7463379 1,013643265

Reordered:
C 1 1
B# 129,7463379 1,013643265
Db 0,131687243 1,053497942
C# 17,0859375 1,067871094
D 2,25 1,125
Eb 0,296296296 1,185185185
D# 38,44335938 1,20135498
E 5,0625 1,265625
F 0,666666667 1,333333333
E# 86,49755859 1,351524353
Gb 0,087791495 1,404663923
F# 11,390625 1,423828125
G 1,5 1,5
Ab 0,197530864 1,580246914
G# 25,62890625 1,601806641
A 3,375 1,6875
Bb 0,444444444 1,777777778
A# 57,66503906 1,802032471
B 7,59375 1,8984375

So, voilà a Pythagorean tuning with 19 fifths and C originally in seventh position.
Beware! It is off-key :-))) so you will need to narrow your fifths :-)))

See also:
http://www.squidoo.com/tuning-systems
Best wishes
Claudi Meneghin

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Sep 13, 2011 at 12:06 PM, Wally <soundmaker@...> wrote:
> >
> > Please help me to find the formula to reduce all the above frequencies to ONE OCTAVE. The answer should match the table on page 167 in the book, "Science and Music". I'm not sure what the proper term is for this procedure. The table lists the frequencies for the Pythagorean seven tone scale and its octave (all in one octave). The frequencies are listed below.
>
> Just keep dividing each value through by 2 until you end up at a
> number that's between 1 and 2.
>
> Hope that helps,
> Mike
>

🔗Wally <soundmaker@...>

Thank you all for the detailed replies. Yes I soon realized (after a quick google search) that I had to keep dividing by "2" until each frequency resulted in a answer between "1" and "2".

Claudi, thank you for the very detailed response. I haven't used excel in awhile and need to go over your response a few times to understand how to go about this procedure. I'm not sure if you own the book I originally mentioned below titled, "Science and Music"

http://www.amazon.com/Science-Music-Sir-James-Jeans/dp/0486619648

but in the table on page 167 it displays the Pythagorean frequency interval of a fourth as 1.3333 which is actually the "just intonation frequency for 4/3". However when I multiplied each frequency starting with "1" (labeling it "C") by a fifth (i.e. 1 * 1.5 etc. and proceeding in fifths as he supposedly accomplished), I arrived at the answer of 1.3515 for the frequency of "E" (after reducing 86.497552 by dividing it by "2" until I arrived at 1.3515).

You can see the table showing 86.497552 which represents the frequency of a fourth (or the letter "E#" which obviously is "F").

Is there a misprint in the book or am I missing something. Which is the correct answer? 1.3333 or 1.3515 ?

Thank you very much for your time.
Wally

--- In tuning@yahoogroups.com, "clamengh" <clamengh@...> wrote:
>
>
>
> Wally> > Please help me to find the formula to reduce all the above frequencies to ONE OCTAVE.
>
> Mike Battaglia> Just keep dividing each value through by 2 until you end up at a
> > number that's between 1 and 2
>
> Yes, of course; or, if you want to do that in an automated fashion, (for instance using Excel) do the following:
> "Take logarithms in base 2, subtract integer parts, then exponentiate again".
> For instance:
>
> D=1.5*1.5=2.25; log_2 2.25 = 1.169925001; integer part of 1.169925001=1, so please sebtract 1; (log_2 2.25)-1 = 0.169925001; 2^(0.169925001)=1.125 as you wish.
>
> Suppose you have your frequencies in an Excel worksheet, column B (note names in column A).
> Just copy the following formula in C1 (including the "=" sign):
> =2^(LOG(B1;2)-INT(LOG(B1;2)))
> Then please copy this formula in as many cells in column C as you need (i.e., as many cells as you have in column B). Results will fit into an octave.
> Now, you could want to reorder them: first,"copy" all cells, then "paste special" them as "values". Select all cells with content and "order" them according to column C. Done!.
>
> Example:
> Gb 0,087791495 1,404663923
> Db 0,131687243 1,053497942
> Ab 0,197530864 1,580246914
> Eb 0,296296296 1,185185185
> Bb 0,444444444 1,777777778
> F 0,666666667 1,333333333
> C 1 1
> G 1,5 1,5
> D 2,25 1,125
> A 3,375 1,6875
> E 5,0625 1,265625
> B 7,59375 1,8984375
> F# 11,390625 1,423828125
> C# 17,0859375 1,067871094
> G# 25,62890625 1,601806641
> D# 38,44335938 1,20135498
> A# 57,66503906 1,802032471
> E# 86,49755859 1,351524353
> B# 129,7463379 1,013643265
>
> Reordered:
> C 1 1
> B# 129,7463379 1,013643265
> Db 0,131687243 1,053497942
> C# 17,0859375 1,067871094
> D 2,25 1,125
> Eb 0,296296296 1,185185185
> D# 38,44335938 1,20135498
> E 5,0625 1,265625
> F 0,666666667 1,333333333
> E# 86,49755859 1,351524353
> Gb 0,087791495 1,404663923
> F# 11,390625 1,423828125
> G 1,5 1,5
> Ab 0,197530864 1,580246914
> G# 25,62890625 1,601806641
> A 3,375 1,6875
> Bb 0,444444444 1,777777778
> A# 57,66503906 1,802032471
> B 7,59375 1,8984375
>
> So, voilà a Pythagorean tuning with 19 fifths and C originally in seventh position.
> Beware! It is off-key :-))) so you will need to narrow your fifths :-)))
>
> See also:
> http://www.squidoo.com/tuning-systems
> Best wishes
> Claudi Meneghin
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Tue, Sep 13, 2011 at 12:06 PM, Wally <soundmaker@> wrote:
> > >
> > > Please help me to find the formula to reduce all the above frequencies to ONE OCTAVE. The answer should match the table on page 167 in the book, "Science and Music". I'm not sure what the proper term is for this procedure. The table lists the frequencies for the Pythagorean seven tone scale and its octave (all in one octave). The frequencies are listed below.
> >
> > Just keep dividing each value through by 2 until you end up at a
> > number that's between 1 and 2.
> >
> > Hope that helps,
> > Mike
> >
>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "Wally" <soundmaker@...> wrote:
>
>
> Thank you all for the detailed replies. Yes I soon realized (after a quick google search) that I had to keep dividing by "2" until each frequency resulted in a answer between "1" and "2".
>
> Claudi, thank you for the very detailed response. I haven't used excel in awhile and need to go over your response a few times to understand how to go about this procedure. I'm not sure if you own the book I originally mentioned below titled, "Science and Music"
>
> http://www.amazon.com/Science-Music-Sir-James-Jeans/dp/0486619648
>
> but in the table on page 167 it displays the Pythagorean frequency interval of a fourth as 1.3333 which is actually the "just intonation frequency for 4/3". However when I multiplied each frequency starting with "1" (labeling it "C") by a fifth (i.e. 1 * 1.5 etc. and proceeding in fifths as he supposedly accomplished), I arrived at the answer of 1.3515 for the frequency of "E" (after reducing 86.497552 by dividing it by "2" until I arrived at 1.3515).
>
> You can see the table showing 86.497552 which represents the frequency of a fourth (or the letter "E#" which obviously is "F").

Why do you say that E# is obviously the same as F? They're the same on a standard keyboard of fretboard, but those instruments are tuned differently than the tuning you're working out now.

> Is there a misprint in the book or am I missing something. Which is the correct answer? 1.3333 or 1.3515 ?

Wally, it looks to me like you've just re-discovered the Pythagorean comma!

Join the club. It was started over 2300 years ago.

Keenan

🔗martinsj013 <martinsj@...>

🔗Wally <soundmaker@...>

>>> Why do you say that E# is obviously the same as F? They're the same on a standard keyboard of fretboard, but those instruments are tuned differently than the tuning you're working out now.

You are correct. Thanks for bringing that to my attention. I must have had my mind stuck on 12tEQ.

Wally

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "Wally" <soundmaker@> wrote:
> >
> >
> > Thank you all for the detailed replies. Yes I soon realized (after a quick google search) that I had to keep dividing by "2" until each frequency resulted in a answer between "1" and "2".
> >
> > Claudi, thank you for the very detailed response. I haven't used excel in awhile and need to go over your response a few times to understand how to go about this procedure. I'm not sure if you own the book I originally mentioned below titled, "Science and Music"
> >
> > http://www.amazon.com/Science-Music-Sir-James-Jeans/dp/0486619648
> >
> > but in the table on page 167 it displays the Pythagorean frequency interval of a fourth as 1.3333 which is actually the "just intonation frequency for 4/3". However when I multiplied each frequency starting with "1" (labeling it "C") by a fifth (i.e. 1 * 1.5 etc. and proceeding in fifths as he supposedly accomplished), I arrived at the answer of 1.3515 for the frequency of "E" (after reducing 86.497552 by dividing it by "2" until I arrived at 1.3515).
> >
> > You can see the table showing 86.497552 which represents the frequency of a fourth (or the letter "E#" which obviously is "F").
>
> Why do you say that E# is obviously the same as F? They're the same on a standard keyboard of fretboard, but those instruments are tuned differently than the tuning you're working out now.
>
> > Is there a misprint in the book or am I missing something. Which is the correct answer? 1.3333 or 1.3515 ?
>
> Wally, it looks to me like you've just re-discovered the Pythagorean comma!
>
> Join the club. It was started over 2300 years ago.
>
> Keenan
>

🔗Wally <soundmaker@...>

>>>Wally, in case Keenan's hints are not quite enough, try this:

* calculate the answer for B#
* divide the answer for E# by the answer for B#

Does this help? Steve M.

Hi Steve,

I was confused as to what you are recommending. However I discovered an error in my post dated Sept. 21 message # 101676.

I originally said ..

>>> -begin quote- However when I multiplied each frequency starting with "1" (labeling it "C") by a fifth (i.e. 1 * 1.5 etc. and proceeding in fifths as he supposedly
accomplished), I arrived at the answer of 1.3515 for the frequency of "E"
(after reducing 86.497552 by dividing it by "2" until I arrived at 1.3515). end quote-

I should have originally labeled "1" as "F" and NOT "C". By multiplying each frequency by 1.5, (starting on "F" as 1/1) the resulting frequency of 1.3515 should have been labeled as the note "Bb" and NOT "E".

Let's start over please..I hope this makes better sense.

On page 166, in the book titled, "Science and Music" by Sir James Jeans, it says, " He assigned to his "C" exactly 1.5 times the frequency of "F", to his "G" exactly 1.5 times the frequency of "C", and so on, thus arriving at a scale with the ratios shown in the following table.

In my original post, I started with the note "C" as 1/1. However I should have started with the note "F" as 1/1 to be more in-line with what the book is explaining.

Therefore starting on "F" as 1/1 and proceeding in 5ths we have the following:

F - 1

C - 1.5

G 2.25

D - 3.375

A 5.0625

E 7.59375

B 11.390625

F# - 17.085937

Db - 25.628905

Ab - 38.443357

Eb 57.665035

Bb 86.497552

F 129.74632

However, the results they printed in the table on page 167 in the book titled, "Science and
Music", show the results as starting on "C" as 1/1 and not "F". For some reason they "switched gears" and started with "C" as 1/1. They should have started with "F" as 1/1 in keeping in-line with the previous sentence (before the table listing the scale). Thus to be in-line with the results listed in the table, I proceeded in 5ths starting with "C" as 1/1. The following results are listed below:

C - 1

G - 1.5

D 2.25

A - 3.375

E 5.0625

B 7.59375

F# 11.390625

Db - 17.085937

Ab - 25.628905

Eb - 38.443357

Bb 57.665035

F 86.497552

C 129.74632

All frequencies above are reduced to one octave by continuously dividing each frequency by "2" until the number falls between "1"(tonic) and "2" (octave). My results are listed in numeric order below.

Pythagorean 7 Tone Scale

C 1/1 = 1.0000
D 9/8 = 1.1250
E 81/64 = 1.2656
F 4/3 = 1.3515
G 3/2 = 1.5000
A 27/16 = 1.6875
B 243/128 = 1.8984
C 2/1 = 2

My results above should have matched the table on page 167 in the book titled, "Science and Music" but they do not.

You can see "my" table above showing the frequency of "F" as 1.3515 by continuously dividing "F" (86.497552 ) by "2" until I arrived at a frequency between "1" and "2".

However, in the table under the column titled, "Pythagorean Frequency Ratio" in the table on page 167 in the book titled, "Science and Music", the result for "F" (4/3) is listed as 1.3333 (a "just" fourth). It should have been listed as 1.3515 if one has proceeded in 5ths as Pythagoras supposedly did and as mentioned in the above book.

Therefore, coming back to my original question, is there a misprint in the column (in the book) or am I missing something? Which is the correct answer for "F", 1.3333 or 1.3515?

Thank you
Wally

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > > Is there a misprint in the book or am I missing something. Which is the correct answer? 1.3333 or 1.3515 ?
> >
> > Wally, it looks to me like you've just re-discovered the Pythagorean comma!
>
> Wally, in case Keenan's hints are not quite enough, try this:
>
> * calculate the answer for B#
> * divide the answer for E# by the answer for B#
>
> Does this help?
>
> Steve M.
>

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Wally" <soundmaker@...> wrote:
> ... Let's start over please..I hope this makes better sense.
> ... starting on "F" as 1/1 and proceeding in 5ths we have the following:
> F - 1
> C - 1.5
> G 2.25
> D - 3.375
> A 5.0625
> E 7.59375
> B 11.390625
> F# - 17.085937
> Db - 25.628905
> Ab - 38.443357
> Eb 57.665035
> Bb 86.497552
> F 129.74632
<< and a similar table starting on C instead >>

Wally, happy (to try!) to help.

First, you will have noticed (and it is kind of obvious) that your table starting on C is identical to the one starting on F, apart from the names of the notes. They both start at 1.0 and end on 129.74632. Have you looked upon this as a discrepancy? For if you divide 129.74632 repeatedly by 2, you do not reach 1.0. Please try it! and then try dividing 1.3515 by 1.3333 and see what you notice (that was what I meant in my previous response). The two discrepancies are caused by the same thing.

Second, the note that is F#*3/2, I would not call Db but C#. And so on (G#, D#, A#, E#) - IIRC you got this right in your previous message that I responded to. This is good because it means that the (corrected) table above would not give two different values for F, but one for F and a different one for E#.

Third, have you tried starting from C and going in the opposite direction (i.e. dividing by 3/2 to get F, then again to get Bb, Eb, Ab, etc - note you then need to multiply repeatedly by 2 to get back to a number between 1 and 2)?

>
> Pythagorean 7 Tone Scale
>
> C 1/1 = 1.0000
> D 9/8 = 1.1250
> E 81/64 = 1.2656
> F 4/3 = 1.3515
> G 3/2 = 1.5000
> A 27/16 = 1.6875
> B 243/128 = 1.8984
> C 2/1 = 2
> My results above should have matched the table on page 167 in the book titled, "Science and Music" but they do not. ... Therefore, coming back to my original question, is there a misprint in the column (in the book) or am I missing something? Which is the correct answer for "F", 1.3333 or 1.3515?

It is not a misprint. My explanation for this is that the major scale starting on C, although it goes up from C in steps of 3/2 for the notes G, D, A, E, B, it goes down from C by 3/2 to get to F. I'm not sure I can explain why without getting into a debate. One way to look at it is, if you start from F and play F, G, A, B, C, D, E, F as defined, it doesn't sound like a major scale; you "need" a Bb instead. And how do you find the ratio of that Bb? Why, you go another step in the "downward" direction from F. And similarly, when starting from C, you need to include F ("in the downward direction"), and not F# (which would be B*3/2). Also, have you considered that with your short table above, it would be a little odd to include the note E#=1.3515, while omitting other notes from the larger table (F#, Db, etc)? For, as I said before, yours is not really F, but E#.

There is a lot more to all of this, e.g. Pythagorean tuning says E=81/64 but others say E=5/4. Does the Jeans book go on to cover that? And does it cover "cents", as Keenan mentioned in your other thread?

Hope this helps,
Steve M.