Pianotech

I consider myself to be a tuning theoretician. I need to ignore inharmonicity so that the mathematics of tuning operates in a consistent and predictable manner. The benefit of doing this is that it allows me to construct the perfect theoretical model of each equal temperament and each unequal temperament. The perfect theoretical model of each equal temperament and each unequal temperament can be set as the target that tuning practitioners should aim to achieve.


First, the mathematical design of each temperament needs to be established. The mathematical design of each temperament is almost always expressed in terms of the amount of a particular comma by which each of the 12 Fifths is tempered. The most common commas that are used are the Pythagorean Comma on its own and the Syntonic Comma in conjunction with the Schisma. Pythagorean Comma = Syntonic Comma + Schisma.

Second, a system of cents can be used to express the size of each of the 12 Fifths in cents. The cent is a logarithmic unit of measure that is used for measuring the size of intervals. The System of P8cents is the most popular system of cents in which 1 cent = 2^1/1200.

Third, the size of the Octave in cents can be calculated by using the following formula: Size of Stack of 12 Fifths in cents / 7 = Size of Octave in cents. This Fifths-led approach causes the size of the Octave to be determined by the size of the Stack of 12 Fifths. This method is particularly useful for temperaments such as Pure 12th Equal Temperament which contain tempered Fifths and tempered Octaves.

Fourth, the size of each of the 12 Fourths can be calculated via the following formula: Size of Fourth in cents = Size of Octave in cents − Size of Fifth in cents.

Fifth, once the size in cents of each of the 12 Fifths, each of the 12 Fourths and the Octave has been established, the size in cents of every other interval can be calculated for the purpose of constructing the entirety of a temperament in cents. The following sequence needs to be used to construct the entirety of a temperament in cents: C-G (Fifth), D-G (Fourth), D-A (Fifth), E-A (Fourth), E-B (Fifth), F#-B (Fourth), C#-F# (Fourth), C#-G# (Fifth), D#-G# (Fourth), D#-A# (Fifth), F-A# (Fourth) and F-C (Fifth).

Sixth, by using A4 = 440 Hz as the standard, the system of cents can be used to calculate the frequencies of each of the 88 notes from A0 to C8.

Seventh, the partials of each interval can be used to calculate the beat rate of each interval.

Eighth, the beat rate of each interval can be used to calculate the beat ratio of two or more intervals.

Note: the eight steps above provide a universal method which is applicable to equal temperaments as well as unequal temperaments.


The common ground between tuning theory and tuning practice consists of beat rates and beat ratios. Tuning practitioners can set the theoretical beat rates and the theoretical beat ratios of the intervals in their tuning sequence as the targets that they should achieve. Technology can be used to facilitate the achievement of target beat rates and target beat ratios.

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

Original Message:
Sent: 6/5/2020 1:58:00 PM
From: Roshan Kakiya
Subject: The common ground: Bridging the gap between tuning theory and tuning practice

I consider myself to be a tuning theoretician. I need to ignore inharmonicity so that the mathematics of tuning operates in a consistent and predictable manner. The benefit of doing this is that it allows me to construct the perfect theoretical model of each equal temperament and each unequal temperament. The perfect theoretical model of each equal temperament and each unequal temperament can be set as the target that tuning practitioners should aim to achieve.


First, the mathematical design of each temperament needs to be established. The mathematical design of each temperament is almost always expressed in terms of the amount of a particular comma by which each of the 12 Fifths is tempered. The most common commas that are used are the Pythagorean Comma on its own and the Syntonic Comma in conjunction with the Schisma. Pythagorean Comma = Syntonic Comma + Schisma.

Second, a system of cents can be used to express the size of each of the 12 Fifths in cents. The cent is a logarithmic unit of measure that is used for measuring the size of intervals. The System of P8cents is the most popular system of cents in which 1 cent = 2^1/1200.

Third, the size of the Octave in cents can be calculated by using the following formula: Size of Stack of 12 Fifths in cents / 7 = Size of Octave in cents. This Fifths-led approach causes the size of the Octave to be determined by the size of the Stack of 12 Fifths. This method is particularly useful for temperaments such as Pure 12th Equal Temperament which contain tempered Fifths and tempered Octaves.

Fourth, the size of each of the 12 Fourths can be calculated via the following formula: Size of Fourth in cents = Size of Octave in cents − Size of Fifth in cents.

Fifth, once the size in cents of each of the 12 Fifths, each of the 12 Fourths and the Octave has been established, the size in cents of every other interval can be calculated for the purpose of constructing the entirety of a temperament in cents. The following sequence needs to be used to construct the entirety of a temperament in cents: C-G (Fifth), D-G (Fourth), D-A (Fifth), E-A (Fourth), E-B (Fifth), F#-B (Fourth), C#-F# (Fourth), C#-G# (Fifth), D#-G# (Fourth), D#-A# (Fifth), F-A# (Fourth) and F-C (Fifth).

Sixth, by using A4 = 440 Hz as the standard, the system of cents can be used to calculate the frequencies of each of the 88 notes from A0 to C8.

Seventh, the partials of each interval can be used to calculate the beat rate of each interval.

Eighth, the beat rate of each interval can be used to calculate the beat ratio of two or more intervals.

Note: the eight steps above provide a universal method which is applicable to equal temperaments as well as unequal temperaments.


The common ground between tuning theory and tuning practice consists of beat rates and beat ratios. Tuning practitioners can set the theoretical beat rates and the theoretical beat ratios of the intervals in their tuning sequence as the targets that they should achieve. Technology can be used to facilitate the achievement of target beat rates and target beat ratios.

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

Ignoring inharmonicity can be a useful trick for some approximations, but like the spherical cow, it won't get you very far in calculating actual piano tuning frequencies. Perhaps the most distinctive characteristic in piano tone vs. the tone of other stringed instruments, is its high inharmonicity.

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

Original Message:
Sent: 06-18-2020 14:06
From: David Pinnegar
Subject: The common ground: Bridging the gap between tuning theory and tuning practice

The problem is that the departure between fundamentals and harmonics throws a spanner in the works which means that agreement in theory becomes a spectra of agreements and not in practice.

No one formula can always apply to all pianos as I found when tuning a Steinway C with iron or steel rather than copper bound bass strings.

A tuning is only successful with aural checking.

The piano is its own analoge of the equations and its own solution.

Best wishes

David P

--

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

Original Message:
Sent: 6/5/2020 1:58:00 PM
From: Roshan Kakiya
Subject: The common ground: Bridging the gap between tuning theory and tuning practice

I consider myself to be a tuning theoretician. I need to ignore inharmonicity so that the mathematics of tuning operates in a consistent and predictable manner. The benefit of doing this is that it allows me to construct the perfect theoretical model of each equal temperament and each unequal temperament. The perfect theoretical model of each equal temperament and each unequal temperament can be set as the target that tuning practitioners should aim to achieve.


First, the mathematical design of each temperament needs to be established. The mathematical design of each temperament is almost always expressed in terms of the amount of a particular comma by which each of the 12 Fifths is tempered. The most common commas that are used are the Pythagorean Comma on its own and the Syntonic Comma in conjunction with the Schisma. Pythagorean Comma = Syntonic Comma + Schisma.

Second, a system of cents can be used to express the size of each of the 12 Fifths in cents. The cent is a logarithmic unit of measure that is used for measuring the size of intervals. The System of P8cents is the most popular system of cents in which 1 cent = 2^1/1200.

Third, the size of the Octave in cents can be calculated by using the following formula: Size of Stack of 12 Fifths in cents / 7 = Size of Octave in cents. This Fifths-led approach causes the size of the Octave to be determined by the size of the Stack of 12 Fifths. This method is particularly useful for temperaments such as Pure 12th Equal Temperament which contain tempered Fifths and tempered Octaves.

Fourth, the size of each of the 12 Fourths can be calculated via the following formula: Size of Fourth in cents = Size of Octave in cents − Size of Fifth in cents.

Fifth, once the size in cents of each of the 12 Fifths, each of the 12 Fourths and the Octave has been established, the size in cents of every other interval can be calculated for the purpose of constructing the entirety of a temperament in cents. The following sequence needs to be used to construct the entirety of a temperament in cents: C-G (Fifth), D-G (Fourth), D-A (Fifth), E-A (Fourth), E-B (Fifth), F#-B (Fourth), C#-F# (Fourth), C#-G# (Fifth), D#-G# (Fourth), D#-A# (Fifth), F-A# (Fourth) and F-C (Fifth).

Sixth, by using A4 = 440 Hz as the standard, the system of cents can be used to calculate the frequencies of each of the 88 notes from A0 to C8.

Seventh, the partials of each interval can be used to calculate the beat rate of each interval.

Eighth, the beat rate of each interval can be used to calculate the beat ratio of two or more intervals.

Note: the eight steps above provide a universal method which is applicable to equal temperaments as well as unequal temperaments.


The common ground between tuning theory and tuning practice consists of beat rates and beat ratios. Tuning practitioners can set the theoretical beat rates and the theoretical beat ratios of the intervals in their tuning sequence as the targets that they should achieve. Technology can be used to facilitate the achievement of target beat rates and target beat ratios.

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

Original Message:
Sent: 6/18/2020 2:07:00 PM
From: David Pinnegar
Subject: RE: The common ground: Bridging the gap between tuning theory and tuning practice

The problem is that the departure between fundamentals and harmonics throws a spanner in the works which means that agreement in theory becomes a spectra of agreements and not in practice.

No one formula can always apply to all pianos as I found when tuning a Steinway C with iron or steel rather than copper bound bass strings.

A tuning is only successful with aural checking.

The piano is its own analoge of the equations and its own solution.

Best wishes

David P

--

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

Original Message:
Sent: 6/5/2020 1:58:00 PM
From: Roshan Kakiya
Subject: The common ground: Bridging the gap between tuning theory and tuning practice

I consider myself to be a tuning theoretician. I need to ignore inharmonicity so that the mathematics of tuning operates in a consistent and predictable manner. The benefit of doing this is that it allows me to construct the perfect theoretical model of each equal temperament and each unequal temperament. The perfect theoretical model of each equal temperament and each unequal temperament can be set as the target that tuning practitioners should aim to achieve.


First, the mathematical design of each temperament needs to be established. The mathematical design of each temperament is almost always expressed in terms of the amount of a particular comma by which each of the 12 Fifths is tempered. The most common commas that are used are the Pythagorean Comma on its own and the Syntonic Comma in conjunction with the Schisma. Pythagorean Comma = Syntonic Comma + Schisma.

Second, a system of cents can be used to express the size of each of the 12 Fifths in cents. The cent is a logarithmic unit of measure that is used for measuring the size of intervals. The System of P8cents is the most popular system of cents in which 1 cent = 2^1/1200.

Third, the size of the Octave in cents can be calculated by using the following formula: Size of Stack of 12 Fifths in cents / 7 = Size of Octave in cents. This Fifths-led approach causes the size of the Octave to be determined by the size of the Stack of 12 Fifths. This method is particularly useful for temperaments such as Pure 12th Equal Temperament which contain tempered Fifths and tempered Octaves.

Fourth, the size of each of the 12 Fourths can be calculated via the following formula: Size of Fourth in cents = Size of Octave in cents − Size of Fifth in cents.

Fifth, once the size in cents of each of the 12 Fifths, each of the 12 Fourths and the Octave has been established, the size in cents of every other interval can be calculated for the purpose of constructing the entirety of a temperament in cents. The following sequence needs to be used to construct the entirety of a temperament in cents: C-G (Fifth), D-G (Fourth), D-A (Fifth), E-A (Fourth), E-B (Fifth), F#-B (Fourth), C#-F# (Fourth), C#-G# (Fifth), D#-G# (Fourth), D#-A# (Fifth), F-A# (Fourth) and F-C (Fifth).

Sixth, by using A4 = 440 Hz as the standard, the system of cents can be used to calculate the frequencies of each of the 88 notes from A0 to C8.

Seventh, the partials of each interval can be used to calculate the beat rate of each interval.

Eighth, the beat rate of each interval can be used to calculate the beat ratio of two or more intervals.

Note: the eight steps above provide a universal method which is applicable to equal temperaments as well as unequal temperaments.


The common ground between tuning theory and tuning practice consists of beat rates and beat ratios. Tuning practitioners can set the theoretical beat rates and the theoretical beat ratios of the intervals in their tuning sequence as the targets that they should achieve. Technology can be used to facilitate the achievement of target beat rates and target beat ratios.

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