Pianotech

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are “proven” and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD’s are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Traditionally, the ETD developers have always claimed that the cents deviations for the various unequal temperaments would be "close enough" when applied to stretched tunings, even without conversion.  Perhaps it would be worth the effort to calculate the cents deviation for each unequal temperament _and_ for each level of stretch. I don’t know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan’s numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Kent,

Thank you for your comments.

I have created my Stretched Young I by directly modifying Pure 12th Equal Temperament so that it benefits from the features of both Young I and Pure 12th Equal Temperament.

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically redesigned to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically redesigned version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Thanks for this. I appreciate it very much. I have edited my original post to try to eliminate the portions at issue. Thanks again.

Leading up to a discussion of conversion of unequal temperaments to other widths of ET besides pure8, could you please describe your pure12-Young temperament and which elements of pure12ET are retained (only the expanded octave?) and which elements of Young are retained in your temperament?

To the extent that, say, Vallotti-Young is described as a temperament with pure 5ths tuned down from C through the flat side, and 4 cent contracted 5ths tuned up from C through the sharp side, how would you capsulize your pure12-Young?

------------------------------
Kent Swafford
Lenexa KS
913-631-8227
------------------------------

Original Message:
Sent: 10-19-2019 08:27
From: Roshan Kakiya
Subject: Cents Conversion Formulas: P8cents and P12cents

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically rewritten to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically rewritten version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Roshan Kakiya's Stretched Young I

Arrangement of the Fifths:

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#.


Retained elements of Young I:

The Major Thirds are symmetrical on either side of F# around the Circle of Fifths.

The Major Third C-E is closest to Just and the Major Third F#-A# is furthest from Just.

The Fifths F#-C#, C#-G#, G#-D# and D#-A# are the same.


Retained elements of Pure 12th Equal Temperament:

The Fifth and the Octave will beat at the same rate within the Twelfths E-B, B-F#, A#-F and F-C.

The Fifths and the Twelfths E-B, B-F#, A#-F and F-C are the same. 

Every Octave is the same.

The Major Thirds A-C# and D#-G are the same.

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

Original Message:
Sent: 10-19-2019 13:04
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Thanks for this. I appreciate it very much. I have edited my original post to try to eliminate the portions at issue. Thanks again.

Leading up to a discussion of conversion of unequal temperaments to other widths of ET besides pure8, could you please describe your pure12-Young temperament and which elements of pure12ET are retained (only the expanded octave?) and which elements of Young are retained in your temperament?

To the extent that, say, Vallotti-Young is described as a temperament with pure 5ths tuned down from C through the flat side, and 4 cent contracted 5ths tuned up from C through the sharp side, how would you capsulize your pure12-Young?

------------------------------
Kent Swafford
Lenexa KS
913-631-8227

Original Message:
Sent: 10-19-2019 08:27
From: Roshan Kakiya
Subject: Cents Conversion Formulas: P8cents and P12cents

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically rewritten to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically rewritten version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Original Message------

Roshan Kakiya's Stretched Young I

Retained elements of Young I: The Major Thirds are symmetrical on either side of F# around the Circle of Fifths. The Major Third C-E is closest to Just and the Major Third F#-A# is furthest from Just.

Retained elements of Pure 12th ET: The Fifths and the Twelfths E-B, B-F#, A#-F and F-C are the same. Every Octave is the same. The Major Thirds A-C# and D#-G are the same.


2/19 Pythagorean comma Fifth: C-G, G-D, D-A and A-E.

1/19 Pythagorean comma Fifth: E-B, B-F#, A#-F and F-C.

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

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

Original Message:
Sent: 10-19-2019 13:04
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Thanks for this. I appreciate it very much. I have edited my original post to try to eliminate the portions at issue. Thanks again.

Leading up to a discussion of conversion of unequal temperaments to other widths of ET besides pure8, could you please describe your pure12-Young temperament and which elements of pure12ET are retained (only the expanded octave?) and which elements of Young are retained in your temperament?

To the extent that, say, Vallotti-Young is described as a temperament with pure 5ths tuned down from C through the flat side, and 4 cent contracted 5ths tuned up from C through the sharp side, how would you capsulize your pure12-Young?

------------------------------
Kent Swafford
Lenexa KS
913-631-8227

Original Message:
Sent: 10-19-2019 08:27
From: Roshan Kakiya
Subject: Cents Conversion Formulas: P8cents and P12cents

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically rewritten to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically rewritten version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

David,

Young I contains 4 Pure Fifths at F#-C#, C#-G#, G#-D# and D#-A#:

http://rollingball.com/images/Young.gif


Roshan Kakiya's Stretched Young I contains 4 Pure Fifths at F#-C#, C#-G#, G#-D# and D#-A#.


The remaining 8 Fifths have been tempered in a specific way to achieve symmetrical Major Thirds on either side of F# around the Circle of Fifths. This is illustrated by the red bars on the graphs.

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

Original Message:
Sent: 10-19-2019 15:32
From: David Pinnegar
Subject: Cents Conversion Formulas: P8cents and P12cents

What I really don't understand is why one would want to take a temperament with many pure fifths and then make them impure . . . ?

The only purpose would be to get the 5ths in line with the 3rd harmonics. There's only sense in this as far as I can see if the central octave does actually set out the temperament unstretched, as is, pure, in the central octave . . . or two. Then different portions of the instrument have different inharmonic stretches and it really has to be done by ear and each according to the instrument which therefor cannot be prescribed in advance.

Best wishes

David P

--

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

Original Message------

Roshan Kakiya's Stretched Young I

Retained elements of Young I: The Major Thirds are symmetrical on either side of F# around the Circle of Fifths. The Major Third C-E is closest to Just and the Major Third F#-A# is furthest from Just.

Retained elements of Pure 12th ET: The Fifths and the Twelfths E-B, B-F#, A#-F and F-C are the same. Every Octave is the same. The Major Thirds A-C# and D#-G are the same.


2/19 Pythagorean comma Fifth: C-G, G-D, D-A and A-E.

1/19 Pythagorean comma Fifth: E-B, B-F#, A#-F and F-C.

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

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

Original Message:
Sent: 10-19-2019 13:04
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Thanks for this. I appreciate it very much. I have edited my original post to try to eliminate the portions at issue. Thanks again.

Leading up to a discussion of conversion of unequal temperaments to other widths of ET besides pure8, could you please describe your pure12-Young temperament and which elements of pure12ET are retained (only the expanded octave?) and which elements of Young are retained in your temperament?

To the extent that, say, Vallotti-Young is described as a temperament with pure 5ths tuned down from C through the flat side, and 4 cent contracted 5ths tuned up from C through the sharp side, how would you capsulize your pure12-Young?

------------------------------
Kent Swafford
Lenexa KS
913-631-8227

Original Message:
Sent: 10-19-2019 08:27
From: Roshan Kakiya
Subject: Cents Conversion Formulas: P8cents and P12cents

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically rewritten to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically rewritten version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

Original Message------

David,

Young I contains 4 Pure Fifths at F#-C#, C#-G#, G#-D# and D#-A#:

http://rollingball.com/images/Young.gif


Roshan Kakiya's Stretched Young I contains 4 Pure Fifths at F#-C#, C#-G#, G#-D# and D#-A#.


The remaining 8 Fifths have been tempered in a specific way to achieve symmetrical Major Thirds on either side of F# around the Circle of Fifths. This is illustrated by the red bars on the graphs.

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

Original Message:
Sent: 10-19-2019 15:32
From: David Pinnegar
Subject: Cents Conversion Formulas: P8cents and P12cents

What I really don't understand is why one would want to take a temperament with many pure fifths and then make them impure . . . ?

The only purpose would be to get the 5ths in line with the 3rd harmonics. There's only sense in this as far as I can see if the central octave does actually set out the temperament unstretched, as is, pure, in the central octave . . . or two. Then different portions of the instrument have different inharmonic stretches and it really has to be done by ear and each according to the instrument which therefor cannot be prescribed in advance.

Best wishes

David P

--

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

Original Message------

Roshan Kakiya's Stretched Young I

Retained elements of Young I: The Major Thirds are symmetrical on either side of F# around the Circle of Fifths. The Major Third C-E is closest to Just and the Major Third F#-A# is furthest from Just.

Retained elements of Pure 12th ET: The Fifths and the Twelfths E-B, B-F#, A#-F and F-C are the same. Every Octave is the same. The Major Thirds A-C# and D#-G are the same.


2/19 Pythagorean comma Fifth: C-G, G-D, D-A and A-E.

1/19 Pythagorean comma Fifth: E-B, B-F#, A#-F and F-C.

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

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

Original Message:
Sent: 10-19-2019 13:04
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Thanks for this. I appreciate it very much. I have edited my original post to try to eliminate the portions at issue. Thanks again.

Leading up to a discussion of conversion of unequal temperaments to other widths of ET besides pure8, could you please describe your pure12-Young temperament and which elements of pure12ET are retained (only the expanded octave?) and which elements of Young are retained in your temperament?

To the extent that, say, Vallotti-Young is described as a temperament with pure 5ths tuned down from C through the flat side, and 4 cent contracted 5ths tuned up from C through the sharp side, how would you capsulize your pure12-Young?

------------------------------
Kent Swafford
Lenexa KS
913-631-8227

Original Message:
Sent: 10-19-2019 08:27
From: Roshan Kakiya
Subject: Cents Conversion Formulas: P8cents and P12cents

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically rewritten to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically rewritten version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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

David,

Roshan Kakiya's Stretched Young I preserves the overall structure of Young I. The Major Thirds are symmetrical on either side of F# around the Circle of Fifths.

Roshan Kakiya's Stretched Young I preserves the beat symmetry of Pure 12th Equal Temperament within the Twelfths E-B, B-F#, A#-F and F-C. The Fifth and the Octave will beat at the same rate within the Twelfths E-B, B-F#, A#-F and F-C.



Thomas Young's First Temperament (Young I) balances the tempering of the Fifths and the Major Thirds. This temperament is ingeniously designed. The Major Third C-E is closest to Just and the Major Third F#-A# is furthest from Just. The Major Thirds are symmetrical on either side of F# around the Circle of Fifths:

http://rollingball.com/images/Young.gif



I have shown that the Fifth and the Octave beat at the same rate in Pure 12th Equal Temperament by analysing beat rates:

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=d66c7c67-9643-4c50-9bf7-c1d5864b8f2d&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-20-2019 07:08
From: David Pinnegar
Subject: Cents Conversion Formulas: P8cents and P12cents

Hmmm. . . . Young is often presented as Vallotti with the circle of fifths rotated by one key so I wasn't aware it was only 4 rather than 6 perfect fifths.

Using my usual tuning of 7 perfect fifths a trio with flute and cello yesterday was particularly successful, the player saying that it enabled them to tune more orchestrally between them. The programme included Trois Aquarelles by Gaubert (1879 - 1941) and Debussy, Grand Trio in G major and demonstrated the benefit that purer intervals, and no stretching in the important harmonic regions, can bring.

Best wishes

David P

--

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

Original Message------

David,

Young I contains 4 Pure Fifths at F#-C#, C#-G#, G#-D# and D#-A#:

http://rollingball.com/images/Young.gif


Roshan Kakiya's Stretched Young I contains 4 Pure Fifths at F#-C#, C#-G#, G#-D# and D#-A#.


The remaining 8 Fifths have been tempered in a specific way to achieve symmetrical Major Thirds on either side of F# around the Circle of Fifths. This is illustrated by the red bars on the graphs.

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

Original Message:
Sent: 10-19-2019 15:32
From: David Pinnegar
Subject: Cents Conversion Formulas: P8cents and P12cents

What I really don't understand is why one would want to take a temperament with many pure fifths and then make them impure . . . ?

The only purpose would be to get the 5ths in line with the 3rd harmonics. There's only sense in this as far as I can see if the central octave does actually set out the temperament unstretched, as is, pure, in the central octave . . . or two. Then different portions of the instrument have different inharmonic stretches and it really has to be done by ear and each according to the instrument which therefor cannot be prescribed in advance.

Best wishes

David P

--

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

Original Message------

Roshan Kakiya's Stretched Young I

Retained elements of Young I: The Major Thirds are symmetrical on either side of F# around the Circle of Fifths. The Major Third C-E is closest to Just and the Major Third F#-A# is furthest from Just.

Retained elements of Pure 12th ET: The Fifths and the Twelfths E-B, B-F#, A#-F and F-C are the same. Every Octave is the same. The Major Thirds A-C# and D#-G are the same.


2/19 Pythagorean comma Fifth: C-G, G-D, D-A and A-E.

1/19 Pythagorean comma Fifth: E-B, B-F#, A#-F and F-C.

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

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

Original Message:
Sent: 10-19-2019 13:04
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Thanks for this. I appreciate it very much. I have edited my original post to try to eliminate the portions at issue. Thanks again.

Leading up to a discussion of conversion of unequal temperaments to other widths of ET besides pure8, could you please describe your pure12-Young temperament and which elements of pure12ET are retained (only the expanded octave?) and which elements of Young are retained in your temperament?

To the extent that, say, Vallotti-Young is described as a temperament with pure 5ths tuned down from C through the flat side, and 4 cent contracted 5ths tuned up from C through the sharp side, how would you capsulize your pure12-Young?

------------------------------
Kent Swafford
Lenexa KS
913-631-8227

Original Message:
Sent: 10-19-2019 08:27
From: Roshan Kakiya
Subject: Cents Conversion Formulas: P8cents and P12cents

Kent,

The mathematical relationship between the Fifth and the Fourth changes when the Pure Octave (Frequency Ratio: 2/1) is stretched.

The relationship between the Fifth and the Fourth in Pure Octave Equal Temperament is not the same as the relationship between the Fifth and the Fourth in Pure 12th Equal Temperament.


I will be using P8cents to illustrate this:

Fifth in Pure Octave Equal Temperament = 700.000 cents. Deviation from Just = 700.000 cents − 701.955 cents = −1.955 cents.

Fourth in Pure Octave Equal Temperament = 500.000 cents. Deviation from Just = 500.000 cents − 498.045 cents = +1.955 cents.


Fifth in Pure 12th Equal Temperament = 700.720 cents. Deviation from Just = 700.720 cents − 701.955 cents = −1.235 cents.

Fourth in Pure 12th Equal Temperament = 500.514 cents. Deviation from Just = 500.514 cents − 498.045 cents = +2.469 cents.


Deviation Ratios (Fourth / Fifth):

1.955 cents / 1.955 cents = 1.000 / 1.000 = 1/1.

2.469 cents / 1.235 cents = 1.999 / 1.000 = 2/1.


In Pure Octave Equal Temperament, the Fourth is as tempered as the Fifth.

In Pure 12th Equal Temperament, the Fourth is twice as tempered as the Fifth.

Therefore, the traditional theoretical models of Unequal Temperament (that are based on the P8octave) need to be mathematically rewritten to fit within the P12octave.


Roshan Kakiya's Stretched Young I is a mathematically rewritten version of Young I that fits within the P12octave. I have achieved this by directly modifying Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I (Cent = 3^1/1900):

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=fca9f5d4-f379-44d7-9f36-06a67b759524&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 19:46
From: Kent Swafford
Subject: Cents Conversion Formulas: P8cents and P12cents

Roshan,

I appreciate your command of math. Really. And I rather like your pure12 Young. But...

In tuning pure12 ET with an ETD setup for a pure8 world, it is necessary to be able to do these conversions in a practical way.

1200 pure12 cents = 1201.23 pure8 cents
100 pure12 cents = 100.103 pure8 cents

So if the ETD cents deviation of A4 is 0.0 assuming no inharmonicity, then the cents deviation for A#4 in pure 12th would be 0.10, and the cents deviation for G#4 would be -0.10. The cents deviation for A3 would be -1.23, and the cents deviation for A5 would be 1.23. And so on.

I have zero inharmonicity tuning files in RCT and Verituner formats for pure12, pure5, pure19, pure26, plus a few other widths of ET. The files are useful for harpsichords, toy pianos, electric pianos, and similar. These are "proven" and have been in use for years.

As for tuning pianos in pure12 unequal temperaments with RCT or Verituner, keep in mind that both ETD's are perfectly capable of tuning pure12 equal temperament and then simply using that calculated pure12 tuning as a starting point for conversion to an unequal temperament.

Now, if you converted the pure8 cents deviations for a given unequal temperament to pure12 cents using the conversion numbers above, making use of the custom temperament functions for the revised figures, then you might be able to produce a more accurate pure12 Young (for example). No? But for the most part the difference wrought by converting to pure12 cents from pure8 cents might be exceedingly small. Perhaps it would be worth the effort. I don't know, but then, as I said above, I rather liked the sound of the pure12 Young which I realized in a Pianoteq harpsichord with a Scala file based on Roshan's numbers.

This is a worthwhile discussion. I am salivating at the prospect that OnlyPure might someday be capable of tuning unequal temperaments, I might add.

Sent from my iPad


Original Message------

P8cents

2^P8cents/1200


P12cents

3^{P12cents/1900}


Conversion Formula: P8cents to P12cents

3^{P12cents/1900} = 2^P8cents/1200

P12cents/1900 × log₃(3) = P8cents/1200 × log₃(2)

P12cents/1900 = P8cents/1200 × log₃(2)

P12cents = 1900 × P8cents/1200 × log₃(2)


P12cents = 19/12 × log₃(2) × P8cents



Conversion Formula: P12cents to P8cents

2^P8cents/1200 = 3^{P12cents/1900}

P8cents/1200 × log₂(2) = P12cents/1900 × log₂(3)

P8cents/1200 = P12cents/1900 × log₂(3)

P8cents = 1200 × P12cents/1900 × log₂(3)


P8cents = 12/19 × log₂(3) × P12cents


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