Pianotech

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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

Roshan: Do you imagine all piano octaves stretch the same (3.35 cents) or only the temperament?
Original Message:
Sent: 01-19-2019 16:16
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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

The inharmonicity of a piano's strings increases as we move up the keyboard:

http://daffy.uah.edu/piano/page4/page8/index.html

Therefore, all octaves should not be stretched by the same amount.

The alternative version of 12-TET only contains octaves of equal size in order to theoretically fulfil the definition of equal temperament (every semitone must be equal in terms of size).

I have designed the alternative version of 12-TET to account for the practice of stretching octaves which means it can be used as a framework for controlling inharmonicity. The size of each octave can be adjusted in accordance with the specific inharmonicity of a piano.
------------------------------
Roshan Kakiya
------------------------------

Original Message:
Sent: 01-20-2019 12:57
From: Paul Klaus
Subject: 12-Tone Equal Temperament: An Alternative Approach

Roshan: Do you imagine all piano octaves stretch the same (3.35 cents) or only the temperament?
Original Message:
Sent: 01-19-2019 16:16
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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

A framework, yes. I understand you depict one average generic piano octave not all, and maybe not even any in particular. Thanks for your interesting post and explanation.
Original Message:
Sent: 01-20-2019 13:24
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

The inharmonicity of a piano's strings increases as we move up the keyboard:

http://daffy.uah.edu/piano/page4/page8/index.html

Therefore, all octaves should not be stretched by the same amount.

The alternative version of 12-TET only contains octaves of equal size in order to theoretically fulfil the definition of equal temperament (every semitone must be equal in terms of size).

I have designed the alternative version of 12-TET to account for the practice of stretching octaves which means it can be used as a framework for controlling inharmonicity. The size of each octave can be adjusted in accordance with the specific inharmonicity of a piano.
------------------------------
Roshan Kakiya

Original Message:
Sent: 01-20-2019 12:57
From: Paul Klaus
Subject: 12-Tone Equal Temperament: An Alternative Approach

Roshan: Do you imagine all piano octaves stretch the same (3.35 cents) or only the temperament?
Original Message:
Sent: 01-19-2019 16:16
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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

This might be good in the digital realm but a piano is tuned with the specific partial makeup of each note, pianos are not tuned the same because each has differing partial strengths (inharmonicity). I can't see forcing this on a piano would have an acceptable outcome.

------------------------------
Regards,

Jon Page
mailto:jonpage@pianocapecod.com
http://www.pianocapecod.com
------------------------------

Original Message:
Sent: 01-19-2019 16:16
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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

The Circle of Fifths of Standard 12-Tone Equal Temperament and Alternative 12-Tone Equal Temperament.

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

Original Message:
Sent: 01-19-2019 16:16
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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

Alternatives to ET are available, effective, musical and provide coherence to piano sound.

Best wishes

David P

------------------------------
David Pinnegar BSc ARCS
Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
antespam@gmail.com

Call for papers - Seminar 6th May 2019 - "Restoring emotion to classical music through tuning."
------------------------------

Original Message:
Sent: 01-20-2019 18:12
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

The Circle of Fifths of Standard 12-Tone Equal Temperament and Alternative 12-Tone Equal Temperament.

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

Original Message:
Sent: 01-19-2019 16:16
From: Roshan Kakiya
Subject: 12-Tone Equal Temperament: An Alternative Approach

Standard 12-Tone Equal Temperament

Semitone = 100 cents.
12 Narrowed Fifths = 8400 cents.
7 Pure Octaves = 8400 cents.

Note	Cents
C	0.00
C#	100.00
D	200.00
D#	300.00
E	400.00
F	500.00
F#	600.00
G	700.00
G#	800.00
A	900.00
A#	1000.00
B	1100.00
C	1200.00

Alternative 12-Tone Equal Temperament

Revised Semitone = 100.28 cents.
12 Pure Fifths = 8423.46 cents.
7 Widened Octaves = 8423.46 cents.

Note	Cents
C	0.00
C#	100.28
D	200.56
D#	300.84
E	401.12
F	501.40
F#	601.68
G	701.96
G#	802.23
A	902.51
A#	1002.79
B	1103.07
C	1203.35

The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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