Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

I think I'm getting lost here. 
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes? 
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

Original Message:
Sent: 10-28-2019 10:13
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)
Original Message:
Sent: 10-30-2019 00:42
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-28-2019 10:13
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

Paul,

Only 12 Fifths are needed to calculate the value of the Octave.

Therefore, a Circle of 12 Fifths is all that is required to build the whole scale of a temperament.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-30-2019 02:02
From: Paul Klaus
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)
Original Message:
Sent: 10-30-2019 00:42
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-28-2019 10:13
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

Thanks for info. Cents inform me not at all. All about beats.
Original Message:
Sent: 10-30-2019 12:39
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Paul,

Only 12 Fifths are needed to calculate the value of the Octave.

Therefore, a Circle of 12 Fifths is all that is required to build the whole scale of a temperament.

------------------------------
Roshan Kakiya

Original Message:
Sent: 10-30-2019 02:02
From: Paul Klaus
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)
Original Message:
Sent: 10-30-2019 00:42
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-28-2019 10:13
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

 

      I apologize for another off base post. (#5)

      I now get the figures. Thank you Kent and Roshan for how the stack of expanded octaves balances that of shrunken fifths.  2/1 octaves and 3/2 fifths?  Matters or not? I was taught and naively believed for quite a while that octaves by definition piano or otherwise have 1200 cents each. Good to know not really.

      Kent now that you have heard the Roshan Kayika P12 Stretched Young 1 and like it, please outline how to tune it without a visual aid. Got anything for us painters w/o a camera?

     Roshan please post a composition so that we may hear how you use it.

    David glad to hear more ideas from Mr. Gamble .He had some very good posts here. I remember failing to find anything by him about "quadrant tuning" but I was interested.

Original Message:
Sent: 10-30-2019 12:39
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Paul,

Only 12 Fifths are needed to calculate the value of the Octave.

Therefore, a Circle of 12 Fifths is all that is required to build the whole scale of a temperament.

------------------------------
Roshan Kakiya

Original Message:
Sent: 10-30-2019 02:02
From: Paul Klaus
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)
Original Message:
Sent: 10-30-2019 00:42
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-28-2019 10:13
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

The solution in cents is as follows:

Stack of 7 Octaves = Stack of 12 Fifths.

Octave = Stack of 12 Fifths / 7.

The total value of a stack of 12 Fifths can be calculated by adding the value of each Fifth.

The total value of a stack of 12 Fifths must be divided by 7 to calculate the value of 1 Octave.


Roshan Kakiya's Stretched Young I:

Pythagorean Comma = 23.460 cents.


Pure Fifth narrowed by 2/19 Pythagorean Comma = 699.486 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.720 cents.

Pure Fifth = 701.955 cents.


Stack of 12 Fifths = 4 × 699.486 cents + 4 × 700.720 cents + 4 × 701.955 cents = 8408.644 cents.

Octave = Stack of 12 Fifths / 7 = 8408.644 cents / 7 = 1201.235 cents.


Conclusion:

A Circle of 12 Fifths is all that is required to calculate the value of the Octave.


------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Original Message------

The solution in cents is as follows:

Stack of 7 Octaves = Stack of 12 Fifths.

Octave = Stack of 12 Fifths / 7.

The total value of a stack of 12 Fifths can be calculated by adding the value of each Fifth.

The total value of a stack of 12 Fifths must be divided by 7 to calculate the value of 1 Octave.


Roshan Kakiya's Stretched Young I:

Pythagorean Comma = 23.460 cents.


Pure Fifth narrowed by 2/19 Pythagorean Comma = 699.486 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.720 cents.

Pure Fifth = 701.955 cents.


Stack of 12 Fifths = 4 × 699.486 cents + 4 × 700.720 cents + 4 × 701.955 cents = 8408.644 cents.

Octave = Stack of 12 Fifths / 7 = 8408.644 cents / 7 = 1201.235 cents.


Conclusion:

A Circle of 12 Fifths is all that is required to calculate the value of the Octave.


------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Thomas Young's First Temperament has symmetrical Major Thirds on either side of F# around the Circle of Fifths. This temperament balances the tempering of the Major Thirds and the Fifths.

Pure 12th Equal Temperament causes the Fifth and the Octave to beat at the same rate. The Fifth and the Octave are tempered by the same amount in opposite directions. This temperament balances the tempering of the Fifths and the Octaves.

Roshan Kakiya's Stretched Young I preserves the symmetrical Major Thirds of Thomas Young's First Temperament and preserves the beat symmetry of the Fifth and the Octave in Pure 12th Equal Temperament. This temperament balances the tempering of the Major Thirds, the Fifths and the Octaves.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 11-02-2019 20:30
From: David Pinnegar
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

: come through an exciting and exhausting week tuning for the Nice International Piano Competition. The Conservatoire wasn't available so we had to juggle between venues and this led me to different instruments, a Yamaha C7 for most of the week and today a Fazioli 280.

Whilst being with Roshan in the encouragement of exploration of unequal temperaments I'm less than enthusiastic about stretch and it was for that reason that I asked what musical consequence was the aim to which this mathematical work led.

Today's performers in Nice played concertos with full orchestra.

Not only did the piano sound different, more harmonious, and to the extent that a recording engineer heard the difference and wants me to tune for an upcoming disc, but time and time again the orchestra and the piano at both extremes came in and interwove at the same pitch. A viola player in particular noticed and bothered to ask me questions.

Last night I was asked to tune an upright outside of anything to do with the competition. To my chagri it hadn't been tuned for 20 years and was at 425. I took it up to 440 using a procedure based on Michael Gamble's techniques and think the result will be stable. It was four hours of battle and I charged double. The result was an instrument that sounded like a Steinway.

Effectively I'm tuning the piano as an organ. So the other day in the "Beginners'" class - although at the age of one winner 8 and an 11 year old, both achieving joint first award these children far outpassed anything I've heard before . . . an 8 year old lad played Bach. It was so superb, so musical that I knew he was up to the experiment. So I took him back in to shock the Jury.

In my opinion most fashionable playing of Bach on the piano is atrocious and unmusical. Because of the resonance my tuning achieves and because I'm reducing the total modes of vibration in which the whole instrument can work the sustaining pedal can be held down for lengths undreamed of in modern times, but as specified by both Chopin and by Beethoven. So I took the lad back in upon the stage and got him to play his Bach as if the instrument were the organ in a cathedral. The experiment had worked and had taught the Jury a bit about music, sound and composers. There had been some dispute on the Jury as to who to place in which position and my experiment with the lad affirmed those who had vouched for him in their deliberation. When all returned back into the room he performed and everyone was dumbfounded.

The result was a revelation. The President of the Jury on the day with the boy playing Bach hadn't voted for him, wanting another in his place but was outvoted. After the organ in the cathedral experiment, her face lit up and she came over and kissed me, as is the French manner.

I will be putting recordings onto YouTube in due course, although with many hours recorded it will take some time.

So my objectives in my tunings are:

1. To restore differences in sound between keys, that we can actually hear, even measure, to increase the purity of home keys and not cause unpleasantness in wide thirds

2. To improve the sound of the instrument  

- a. to bring it resonance, aweetness and 

-  b. to increase its dynamic power and therefore dynamic range.

3. to reduce available total numbers of modes of vibration

- a. to reduce confusion of sound

- b. to re-enable original specified pedalling by Chopin and Beethoven.

4. To enable greater compatibility with instrumentalists and even full orchestra

and the results of the Nice International Concours de Piano have found all four objectives to have been achieved.

 

So what objectives do you, and anyone else, expect to achieve with your chosen stretching of the octaves?

Best wishes

David P

 - - - - - - - - - - - - - - - 

David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594

Original Message------

The solution in cents is as follows:

Stack of 7 Octaves = Stack of 12 Fifths.

Octave = Stack of 12 Fifths / 7.

The total value of a stack of 12 Fifths can be calculated by adding the value of each Fifth.

The total value of a stack of 12 Fifths must be divided by 7 to calculate the value of 1 Octave.


Roshan Kakiya's Stretched Young I:

Pythagorean Comma = 23.460 cents.


Pure Fifth narrowed by 2/19 Pythagorean Comma = 699.486 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.720 cents.

Pure Fifth = 701.955 cents.


Stack of 12 Fifths = 4 × 699.486 cents + 4 × 700.720 cents + 4 × 701.955 cents = 8408.644 cents.

Octave = Stack of 12 Fifths / 7 = 8408.644 cents / 7 = 1201.235 cents.


Conclusion:

A Circle of 12 Fifths is all that is required to calculate the value of the Octave.


------------------------------
Roshan Kakiya

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Original Message------

Thomas Young's First Temperament has symmetrical Major Thirds on either side of F# around the Circle of Fifths. This temperament balances the tempering of the Major Thirds and the Fifths.

Pure 12th Equal Temperament causes the Fifth and the Octave to beat at the same rate. The Fifth and the Octave are tempered by the same amount in opposite directions. This temperament balances the tempering of the Fifths and the Octaves.

Roshan Kakiya's Stretched Young I preserves the symmetrical Major Thirds of Thomas Young's First Temperament and preserves the beat symmetry of the Fifth and the Octave in Pure 12th Equal Temperament. This temperament balances the tempering of the Major Thirds, the Fifths and the Octaves.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 11-02-2019 20:30
From: David Pinnegar
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions




Original Message------

The solution in cents is as follows:

Stack of 7 Octaves = Stack of 12 Fifths.

Octave = Stack of 12 Fifths / 7.

The total value of a stack of 12 Fifths can be calculated by adding the value of each Fifth.

The total value of a stack of 12 Fifths must be divided by 7 to calculate the value of 1 Octave.


Roshan Kakiya's Stretched Young I:

Pythagorean Comma = 23.460 cents.


Pure Fifth narrowed by 2/19 Pythagorean Comma = 699.486 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.720 cents.

Pure Fifth = 701.955 cents.


Stack of 12 Fifths = 4 × 699.486 cents + 4 × 700.720 cents + 4 × 701.955 cents = 8408.644 cents.

Octave = Stack of 12 Fifths / 7 = 8408.644 cents / 7 = 1201.235 cents.


Conclusion:

A Circle of 12 Fifths is all that is required to calculate the value of the Octave.


------------------------------
Roshan Kakiya

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions




Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

In Pure Octave Equal Temperament, 1 Pythagorean Comma is equally spread across 12 Fifths and 7 Octaves are Pure:

1 Pythagorean Comma = 23.460 cents.

1/12 Pythagorean Comma = 1/12 × 23.460 cents = 1.955 cents.

Pure Fifth narrowed by 1/12 Pythagorean Comma = 701.955 cents − 1.955 cents = 700.000 cents.


Pure Octave = 1200.000 cents.



In Pure 12th Equal Temperament, 12/19 Pythagorean Comma is equally spread across 12 Fifths and 7/19 Pythagorean Comma is equally spread across 7 Octaves:

12/19 Pythagorean Comma = 12/19 × 23.460 cents = 14.817 cents.

1/12 × 12/19 Pythagorean Comma = 1/12 × 14.817 cents = 1.235 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 701.955 cents − 1.235 cents = 700.720 cents.


7/19 Pythagorean Comma = 7/19 × 23.460 cents = 8.643 cents.

1/7 × 7/19 Pythagorean Comma = 1/7 × 8.643 cents = 1.235 cents.

Pure Octave widened by 1/19 Pythagorean Comma = 1200.000 cents + 1.235 cents = 1201.235 cents.



In Roshan Kakiya's Stretched Young I, 8/19 Pythagorean Comma is equally spread across 4 Fifths, 4/19 Pythagorean Comma is equally spread across 4 Fifths, 4 Fifths are Pure and 7/19 Pythagorean Comma is equally spread across 7 Octaves:

8/19 Pythagorean Comma = 8/19 × 23.460 cents = 9.878 cents.

1/4 × 8/19 Pythagorean Comma = 1/4 × 9.878 cents = 2.470 cents.

Pure Fifth narrowed by 2/19 Pythagorean Comma = 701.955 cents − 2.470 cents = 699.485 cents.


4/19 Pythagorean Comma = 4/19 × 23.460 cents = 4.939 cents.

1/4 × 4/19 Pythagorean Comma = 1/4 × 4.939 cents = 1.235 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 701.955 cents − 1.235 cents = 700.720 cents.


Pure Fifth = 701.955 cents.


7/19 Pythagorean Comma = 7/19 × 23.460 cents = 8.643 cents.

1/7 × 7/19 Pythagorean Comma = 1/7 × 8.643 cents = 1.235 cents.

Pure Octave widened by 1/19 Pythagorean Comma = 1200.000 cents + 1.235 cents = 1201.235 cents.


------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Well done. By laying it out for us with the portions of the comma converted to cents, it all becomes much easier to follow. Thanks for this.

I said I would comment upon my first reactions to the pure12-Young unequal temperament. I liked it.

It is interesting to me the different ways that different people hear unequal temperaments. Some are sensitive to the varying speeds of the Major 3rds, even to the point of saying that the various beat speeds of the M3rds can enhance the emotional content of music containing them.

But I don’t much hear it that way.

However, for better or worse, I actually hear the balance of fifths and octaves that occurs when one uniformly stretches equal temperament to form pure12 ET. (As has been mentioned many times before, the M3rd beat speeds of pure12 ET are similar to that of pure8 ET if transposed up one half step. In terms of M3rd beat speeds, this is a minor difference between pure8 ET and pure12 ET and not at all significant, at least to my ears. Your ears are your ears and may very well tell you something very different. That is OK.)

All of this, I think, explains why I may have liked pure12-Young UET. With the exception of that F#-A#, which when exposed did seem a bit fast to my ears when first hearing pure12-Young, I remained fairly unaffected by the varying beat speeds of the M3rds.

However, what I did hear in pure12-Young was relative balance between the fifths and octaves. The octaves are all 1.24 cents expanded. The fifths are all pure, contracted by 1.24 cents, or contracted by 2.47 cents. A contraction of the fifth by 2.47 cents as a practical matter is just barely more that that of pure8 equal temperament, especially when compared to the 4 cent contraction of the 5th in pure8 UETs, and for one reason or another did not seem to detract too much from the balance of 5ths and octaves. I strongly suspect that the beat-masking effect of pure12 ET is still at work in this UET.

For a while I didn’t much understand how I could be so unaffected by the beat rates of the M3rds, but at the same time so affected by the beat rates of the fifths and octaves. The only thing I can think of is to point out that the beat rates of the fifths and octaves tend to be formed by lower partials than that of the fast-beating intervals, and perhaps sound more intrinsic to my ears.

At any “rate”, pure12-Young is a welcome contribution to tuning, at least, to my ears.


Original Message------

In Pure Octave Equal Temperament, 1 Pythagorean Comma is equally spread across 12 Fifths and 7 Octaves are Pure:

1 Pythagorean Comma = 23.460 cents.

1/12 Pythagorean Comma = 1/12 × 23.460 cents = 1.955 cents.

Pure Fifth narrowed by 1/12 Pythagorean Comma = 701.955 cents − 1.955 cents = 700.000 cents.


Pure Octave = 1200.000 cents.



In Pure 12th Equal Temperament, 12/19 Pythagorean Comma is equally spread across 12 Fifths and 7/19 Pythagorean Comma is equally spread across 7 Octaves:

12/19 Pythagorean Comma = 12/19 × 23.460 cents = 14.817 cents.

1/12 × 12/19 Pythagorean Comma = 1/12 × 14.817 cents = 1.235 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 701.955 cents − 1.235 cents = 700.720 cents.


7/19 Pythagorean Comma = 7/19 × 23.460 cents = 8.643 cents.

1/7 × 7/19 Pythagorean Comma = 1/7 × 8.643 cents = 1.235 cents.

Pure Octave widened by 1/19 Pythagorean Comma = 1200.000 cents + 1.235 cents = 1201.235 cents.



In Roshan Kakiya's Stretched Young I, 8/19 Pythagorean Comma is equally spread across 4 Fifths, 4/19 Pythagorean Comma is equally spread across 4 Fifths, 4 Fifths are Pure and 7/19 Pythagorean Comma is equally spread across 7 Octaves:

8/19 Pythagorean Comma = 8/19 × 23.460 cents = 9.878 cents.

1/4 × 8/19 Pythagorean Comma = 1/4 × 9.878 cents = 2.470 cents.

Pure Fifth narrowed by 2/19 Pythagorean Comma = 701.955 cents − 2.470 cents = 699.485 cents.


4/19 Pythagorean Comma = 4/19 × 23.460 cents = 4.939 cents.

1/4 × 4/19 Pythagorean Comma = 1/4 × 4.939 cents = 1.235 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 701.955 cents − 1.235 cents = 700.720 cents.


Pure Fifth = 701.955 cents.


7/19 Pythagorean Comma = 7/19 × 23.460 cents = 8.643 cents.

1/7 × 7/19 Pythagorean Comma = 1/7 × 8.643 cents = 1.235 cents.

Pure Octave widened by 1/19 Pythagorean Comma = 1200.000 cents + 1.235 cents = 1201.235 cents.


------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Kent - how did you tune the RK stretched Young I?

------------------------------
Jason Kanter
Lynnwood WA
425-830-1561
------------------------------

Original Message:
Sent: 11-02-2019 08:01
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Well done. By laying it out for us with the portions of the comma converted to cents, it all becomes much easier to follow. Thanks for this.

I said I would comment upon my first reactions to the pure12-Young unequal temperament. I liked it.

It is interesting to me the different ways that different people hear unequal temperaments. Some are sensitive to the varying speeds of the Major 3rds, even to the point of saying that the various beat speeds of the M3rds can enhance the emotional content of music containing them.

But I don't much hear it that way.

However, for better or worse, I actually hear the balance of fifths and octaves that occurs when one uniformly stretches equal temperament to form pure12 ET. (As has been mentioned many times before, the M3rd beat speeds of pure12 ET are similar to that of pure8 ET if transposed up one half step. In terms of M3rd beat speeds, this is a minor difference between pure8 ET and pure12 ET and not at all significant, at least to my ears. Your ears are your ears and may very well tell you something very different. That is OK.)

All of this, I think, explains why I may have liked pure12-Young UET. With the exception of that F#-A#, which when exposed did seem a bit fast to my ears when first hearing pure12-Young, I remained fairly unaffected by the varying beat speeds of the M3rds.

However, what I did hear in pure12-Young was relative balance between the fifths and octaves. The octaves are all 1.24 cents expanded. The fifths are all pure, contracted by 1.24 cents, or contracted by 2.47 cents. A contraction of the fifth by 2.47 cents as a practical matter is just barely more that that of pure8 equal temperament, especially when compared to the 4 cent contraction of the 5th in pure8 UETs, and for one reason or another did not seem to detract too much from the balance of 5ths and octaves. I strongly suspect that the beat-masking effect of pure12 ET is still at work in this UET.

For a while I didn't much understand how I could be so unaffected by the beat rates of the M3rds, but at the same time so affected by the beat rates of the fifths and octaves. The only thing I can think of is to point out that the beat rates of the fifths and octaves tend to be formed by lower partials than that of the fast-beating intervals, and perhaps sound more intrinsic to my ears.

At any "rate", pure12-Young is a welcome contribution to tuning, at least, to my ears.


Original Message------

In Pure Octave Equal Temperament, 1 Pythagorean Comma is equally spread across 12 Fifths and 7 Octaves are Pure:

1 Pythagorean Comma = 23.460 cents.

1/12 Pythagorean Comma = 1/12 × 23.460 cents = 1.955 cents.

Pure Fifth narrowed by 1/12 Pythagorean Comma = 701.955 cents − 1.955 cents = 700.000 cents.


Pure Octave = 1200.000 cents.



In Pure 12th Equal Temperament, 12/19 Pythagorean Comma is equally spread across 12 Fifths and 7/19 Pythagorean Comma is equally spread across 7 Octaves:

12/19 Pythagorean Comma = 12/19 × 23.460 cents = 14.817 cents.

1/12 × 12/19 Pythagorean Comma = 1/12 × 14.817 cents = 1.235 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 701.955 cents − 1.235 cents = 700.720 cents.


7/19 Pythagorean Comma = 7/19 × 23.460 cents = 8.643 cents.

1/7 × 7/19 Pythagorean Comma = 1/7 × 8.643 cents = 1.235 cents.

Pure Octave widened by 1/19 Pythagorean Comma = 1200.000 cents + 1.235 cents = 1201.235 cents.



In Roshan Kakiya's Stretched Young I, 8/19 Pythagorean Comma is equally spread across 4 Fifths, 4/19 Pythagorean Comma is equally spread across 4 Fifths, 4 Fifths are Pure and 7/19 Pythagorean Comma is equally spread across 7 Octaves:

8/19 Pythagorean Comma = 8/19 × 23.460 cents = 9.878 cents.

1/4 × 8/19 Pythagorean Comma = 1/4 × 9.878 cents = 2.470 cents.

Pure Fifth narrowed by 2/19 Pythagorean Comma = 701.955 cents − 2.470 cents = 699.485 cents.


4/19 Pythagorean Comma = 4/19 × 23.460 cents = 4.939 cents.

1/4 × 4/19 Pythagorean Comma = 1/4 × 4.939 cents = 1.235 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 701.955 cents − 1.235 cents = 700.720 cents.


Pure Fifth = 701.955 cents.


7/19 Pythagorean Comma = 7/19 × 23.460 cents = 8.643 cents.

1/7 × 7/19 Pythagorean Comma = 1/7 × 8.643 cents = 1.235 cents.

Pure Octave widened by 1/19 Pythagorean Comma = 1200.000 cents + 1.235 cents = 1201.235 cents.


------------------------------
Roshan Kakiya

Original Message:
Sent: 10-31-2019 22:08
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Re: "Don't think in terms of one octave"

OK, I don't have a problem with widening the octave and narrowing all the fifths slightly less as you described. But I still don't see how the unequal temperament proposed above with 12 fifths narrowed by 1/19 or 2/19 P.C. would work. The Pythagorean Comma of 23.46 cents needs to fit somehow inside an octave of 12 notes, not 19 notes. If we want to define a new "octave" of 19 notes and divide a comma into an unequal temperament, then that comma would need to be bigger (37.15 cents?), because we're not going around the circle of fifths a whole number of times in the new "octave". I think. I don't know. It's confusing. 

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-31-2019 16:58
From: Kent Swafford
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

"I think I'm getting lost here. The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)"

 

Don't think in terms of "one octave".

 

It is as if the _difference_ between the "height" of a stack of 7 pure octaves and the "height" of 12 pure fifths is 23.46 cents, the Pythagorean comma.

 

To resolve the difference in "height" between the two stacks one traditionally shortens the stack of 12 fifths to match the "height" of the octave stack.

 

23.46 cents divided by 12 is 1.955 cents, the traditional contraction of the fifth in pure octave ET. Contract each 5th by 1.955 cents and the two stacks of intervals are equalized.

 

But there are other possible solutions. One could both shorten the stack of fifths _and_ make the octave stack higher. And it is entirely possible to contract each 5th and expand each octave by the same amount to accomplish this equalization of stacks.

 

There are 19 intervals in the two stacks, 7 octaves and 12 5ths.

 

23.48 cents / 19 = 1.2358 cents

 

So, if the stack of 7 octaves was made higher by expanding each octave 1.24 cents, and the stack of 12 5ths is made shorter by contracting each 5th by the same 1.24 cents, then the stacks are equalized. This is the solution used by pure 12th ET.

 

Unequal temperaments will have various "solutions", but in any event the two stacks must be equalized, one way or another.

 

 

 

 

Original Message------

The P8 example made twelve 5ths. P12 similarly makes nineteen 5ths. Two additional 5ths narrowed by 2/19 Pythagorean comma and three 5ths narrowed by 1/19 of the comma makes the difference. ( with two more "pure" 5ths)

Anthony,

Pure Fifth narrowed by 2/19 Pythagorean Comma = 699.486 cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.720 cents.

Pure Fifth = 701.955 cents.


12 Fifths = 4 × 699.486 cents + 4 × 700.720 cents + 4 × 701.955 cents = 8408.644 cents.

Roshan Kakiya's Stretched Young I Octave = 8408.644 cents / 7 = 1201.235 cents.

Pure 12th Equal Temperament Octave = 1201.235 cents.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-30-2019 00:42
From: Anthony Willey
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com

Original Message:
Sent: 10-28-2019 10:13
From: Roshan Kakiya
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

Circle of Fifths Conversion Formulas: P8fractions and P12fractions


Conversion Formula: P8fraction to P12fraction

P12fraction = 12/19 × P8fraction


Conversion Formula: P12fraction to P8fraction

P8fraction = 19/12 × P12fraction



Example

Roshan Kakiya's Idealised Young I (P8fractions)

Pure Fifth narrowed by 1/6 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/12 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


Roshan Kakiya's Stretched Young I (P12fractions)

P12fraction = 12/19 × 1/6 Pythagorean Comma = 2/19 Pythagorean Comma.

P12fraction = 12/19 × 1/12 Pythagorean Comma = 1/19 Pythagorean Comma.


Pure Fifth narrowed by 2/19 Pythagorean Comma: C-G, G-D, D-A and A-E.

Pure Fifth narrowed by 1/19 Pythagorean Comma: E-B, B-F#, A#-F and F-C.

Pure Fifth: F#-C#, C#-G#, G#-D# and D#-A#.


------------------------------
Roshan Kakiya
------------------------------

Original Message------

I think I'm getting lost here. 
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes? 
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

Circle of Fifths Conversion Formulas

The Circle of Fifths Conversion Formulas will produce the correct answer regardless of which System of Cents is being used (P8cent = 2^1/1200 or P12cent = 3^1/1900).



Roshan Kakiya's Stretched Young I

P8cent = 2^1/1200

Pythagorean Comma = 23.460 P8cents.


Pure Fifth narrowed by 2/19 Pythagorean Comma = 699.486 P8cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.720 P8cents.

Pure Fifth = 701.955 P8cents.


12 Fifths = 4 × 699.486 P8cents + 4 × 700.720 P8cents + 4 × 701.955 P8cents = 8408.644 P8cents.

Roshan Kakiya's Stretched Young I Octave = 8408.644 P8cents / 7 = 1201.235 P8cents.

Pure 12th Equal Temperament Octave = 1201.235 P8cents.


P12cent = 3^1/1900

Pythagorean Comma = 23.436 P12cents.


Pure Fifth narrowed by 2/19 Pythagorean Comma = 698.767 P12cents.

Pure Fifth narrowed by 1/19 Pythagorean Comma = 700.000 P12cents.

Pure Fifth = 701.233 P12cents.


12 Fifths = 4 × 698.767 P12cents + 4 × 700.000 P12cents + 4 × 701.233 P12cents = 8400.000 P12cents.

Roshan Kakiya's Stretched Young I Octave = 8400.000 P12cents / 7 = 1200.000 P12cents.

Pure 12th Equal Temperament Octave = 1200.000 P12cents.



Successful Conversion of the Circle of Fifths

In Roshan Kakiya's Stretched Young I, the value of the Octave is 1201.235 P8cents and 1200.000 P12cents.

In Pure 12th Equal Temperament, the value of the Octave is 1201.235 P8cents and 1200.000 P12cents.

These results confirm the successful conversion of the Circle of Fifths.


------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-30-2019 18:43
From: Jason Kanter
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

I need to think this through again, but I think we have two different sizes of "cents", one of which applies to the 2:1 octave and a different one applies to the 3:1 twelfth. The PythComma is 23.46 of the octave-related cents, and is determined by the diff between twelve pure-fifths and seven pure 2:1 octaves. So that distance obviously still exists in the P12 universe, but as the P12-cents size is different, the number of P12-cents that comprise the PythComma must also be different. Likewise the Syntonic Comma. I'll give it some more thought.

| || ||| || ||| || ||| || ||| || ||| || ||| || |||

jasonkanter@gmail.com

jason's cell 425 830 1561

bellevuepianotuning.com

Original Message------

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

Original Message------

I think I'm getting lost here. 
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes? 
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

We cannot remain within the traditional theoretical world of tuning forever. We must try new things to make progress in the theoretical world of tuning.

The Circle of Fifths Conversion Formulas can be used to fit more Unequal Temperaments within the Pure 12th Equal Temperament Octave.

Each width of Equal Temperament can be modified to create more Unequal Temperaments.

There are numerous ​widths of Equal Temperament. Therefore, there are also numerous widths of Unequal Temperament. They exist and are just waiting to be discovered.

I look forward to seeing further progress in the theoretical world of tuning in the future.

The future of the theoretical world of tuning is bright.

------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 10-30-2019 22:42
From: David Pinnegar
Subject: Circle of Fifths Conversion Formulas: P8fractions and P12fractions

The mathematics are interesting juggling but I ask Roshan why? To what musical effect? 

It's important to the tuner to ask where the music is coming from, where the harmony is coming from and where the harmonic support arises. This consideration is important.

Around London in England there is a circular motorway known as the M25. The mathematics here are like going around the motorway without knowing which junction goes where until one runs out of petrol. There has to be a purpose of getting on the motorway so that one knows the destination so as to be able to choose the right junction. I don't see the purpose.

Is it to tune the inharmonics together? Or to tune the fundamentals better to a better spread of fundamental frequency accordances? So much depends on the individual piano and what we decide to listen for.

Best wishes

David P

--

- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594

Original Message------

I think I'm getting lost here.
The Pythagorean Comma is 23.46 cents, and that has to be spread throughout the octave somehow. Yes?
Perfect 12th temperament is something like 1.24 cents wider than pure octave temperament, so there's a little less comma to spread around. (23.46 - 1.24 = 22.22 cents)
In your temperament you have 4 fifths narrowed by 1/19 P.C. and 4 fifths narrowed by 2/19 P.C. That accounts for 14.82 cents of the comma. (23.46 * (4/19 + 8/19) = 14.82)
That leaves a difference of 22.22 - 14.82 = 7.4 cents that are unaccounted for.
What happens to those 7.4 cents?

------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
------------------------------

This post was removed