Pianotech

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



This image is available at: http://rollingball.com/images/Kakiya-StretchedYoung.png


System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is an Unequal Temperament that is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

12/19 Pythagorean comma = 12/19 × −1 = − 12/19.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


a = 2b.

c = 0.


4a + 4b + 4c = − 12/19.

4a + 4b + 4 × 0 = − 12/19.

4a + 4b = − 12/19.

4 × 2b + 4b = − 12/19.

8b + 4b = − 12/19.

12b = − 12/19.

b = − 12/19 × 1/12.

b = − 1/19.


a = 2 × (− 1/19).

a = − 2/19.


Features

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
------------------------------

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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

Roshan has kindly pointed out that the horizontal reference lines on my chart are still in normal ET mode, and they need to be adjusted for the stretch of a P12 tuning. Accordingly, here is the revision.
An interesting exercise, but I can't imagine how this could be useful to anyone, as the offsets do not correspond to ETD cents, and this tuning cannot be done "by ear" with anything like this degree of precision.
Anyway ...

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

Original Message:
Sent: 10-17-2019 21:32
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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

Jason,

Thank you for providing these graphs.

My Stretched Young I (Cent = 3^1/1900) will be usable if Electronic Tuning Devices are configured to use Cent = 3^1/1900.

This could be just the beginning of Unequal Temperaments that are based on Cent = 3^1/1900 rather than Cent = 2^1/1200.

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

Original Message:
Sent: 10-18-2019 13:00
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Roshan has kindly pointed out that the horizontal reference lines on my chart are still in normal ET mode, and they need to be adjusted for the stretch of a P12 tuning. Accordingly, here is the revision.
An interesting exercise, but I can't imagine how this could be useful to anyone, as the offsets do not correspond to ETD cents, and this tuning cannot be done "by ear" with anything like this degree of precision.
Anyway ...

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

Original Message:
Sent: 10-17-2019 21:32
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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

OnlyPure uses 3^(1/1900) cents since its introduction in 2007.

------------------------------
Bernhard Stopper
Klavierbaumeister
Tuebingen
------------------------------

Original Message:
Sent: 10-18-2019 13:29
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Jason,

Thank you for providing these graphs.

My Stretched Young I (Cent = 3^1/1900) will become usable if electronic tuning devices are configured to use Cent = 3^1/1900 in the future.

This could be just the beginning of unequal temperaments that are based on Cent = 3^1/1900 rather than Cent = 2^1/1200.

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

Original Message:
Sent: 10-18-2019 13:00
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Roshan has kindly pointed out that the horizontal reference lines on my chart are still in normal ET mode, and they need to be adjusted for the stretch of a P12 tuning. Accordingly, here is the revision.
An interesting exercise, but I can't imagine how this could be useful to anyone, as the offsets do not correspond to ETD cents, and this tuning cannot be done "by ear" with anything like this degree of precision.
Anyway ...

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

Original Message:
Sent: 10-17-2019 21:32
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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

So, Bernhard, how would one tune ANY unequal temperament in OnlyPure?

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

Original Message:
Sent: 10-18-2019 14:08
From: Bernhard Stopper
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

OnlyPure uses 3^(1/1900) cents since its introduction in 2007.

------------------------------
Bernhard Stopper
Klavierbaumeister
Tuebingen

Original Message:
Sent: 10-18-2019 13:29
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Jason,

Thank you for providing these graphs.

My Stretched Young I (Cent = 3^1/1900) will become usable if electronic tuning devices are configured to use Cent = 3^1/1900 in the future.

This could be just the beginning of unequal temperaments that are based on Cent = 3^1/1900 rather than Cent = 2^1/1200.

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

Original Message:
Sent: 10-18-2019 13:00
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Roshan has kindly pointed out that the horizontal reference lines on my chart are still in normal ET mode, and they need to be adjusted for the stretch of a P12 tuning. Accordingly, here is the revision.
An interesting exercise, but I can't imagine how this could be useful to anyone, as the offsets do not correspond to ETD cents, and this tuning cannot be done "by ear" with anything like this degree of precision.
Anyway ...

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

Original Message:
Sent: 10-17-2019 21:32
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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

"So, Bernhard, how would one tune ANY unequal temperament in OnlyPure?"

Jason,

a reasonable approximation on pianos with real word inharmonicity is to just read the standard 2^(1/1200) cents offsets of any unequal temperament from the zero cent target OnlyPure provides for any note.

For higher precision you need to translate any unequal temperament into 3^(1/1900) cent offsets with a spreadsheet, then read those offsets from the zero cents target indicated by OnlyPure. I could make a version which can read unequal temperaments with 2^(1/1200) cents format directly from Scala tables and translate them in OnlyPure, if there is enough demand for it.

------------------------------
Bernhard Stopper
Klavierbaumeister
Tuebingen
------------------------------

Original Message:
Sent: 10-18-2019 14:46
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

So, Bernhard, how would one tune ANY unequal temperament in OnlyPure?

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

Original Message:
Sent: 10-18-2019 14:08
From: Bernhard Stopper
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

OnlyPure uses 3^(1/1900) cents since its introduction in 2007.

------------------------------
Bernhard Stopper
Klavierbaumeister
Tuebingen

Original Message:
Sent: 10-18-2019 13:29
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Jason,

Thank you for providing these graphs.

My Stretched Young I (Cent = 3^1/1900) will become usable if electronic tuning devices are configured to use Cent = 3^1/1900 in the future.

This could be just the beginning of unequal temperaments that are based on Cent = 3^1/1900 rather than Cent = 2^1/1200.

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

Original Message:
Sent: 10-18-2019 13:00
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Roshan has kindly pointed out that the horizontal reference lines on my chart are still in normal ET mode, and they need to be adjusted for the stretch of a P12 tuning. Accordingly, here is the revision.
An interesting exercise, but I can't imagine how this could be useful to anyone, as the offsets do not correspond to ETD cents, and this tuning cannot be done "by ear" with anything like this degree of precision.
Anyway ...

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

Original Message:
Sent: 10-17-2019 21:32
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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

Jason and Bernhard,

The two Circle of Fifths Conversion Formulas that I have devised are useful for converting P8fractions to P12fractions and P12fractions to P8fractions:

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=a897453d-7fad-42fb-a2a2-844a38c8c721&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf


Additionally, the two Cents Conversion Formulas that I have devised are useful for converting P8cents to P12cents and P12cents to P8cents:

https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=875d159a-e924-4abc-b42d-bbba97bec2af&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

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

Original Message:
Sent: 10-18-2019 15:19
From: Bernhard Stopper
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

"So, Bernhard, how would one tune ANY unequal temperament in OnlyPure?"

Jason,

a reasonable approximation on pianos with real word inharmonicity is to just read the standard 2^(1/1200) cents offsets of any unequal temperament from the zero cent target OnlyPure provides for any note.

For higher precision you need to translate any unequal temperament into 3^(1/1900) cent offsets with a spreadsheet, then read those offsets from the zero cents target indicated by OnlyPure. I could make a version which can read unequal temperaments with 2^(1/1200) cents format directly from Scala tables and translate them in OnlyPure, if there is enough demand for it.

------------------------------
Bernhard Stopper
Klavierbaumeister
Tuebingen

Original Message:
Sent: 10-18-2019 14:46
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

So, Bernhard, how would one tune ANY unequal temperament in OnlyPure?

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

Original Message:
Sent: 10-18-2019 14:08
From: Bernhard Stopper
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

OnlyPure uses 3^(1/1900) cents since its introduction in 2007.

------------------------------
Bernhard Stopper
Klavierbaumeister
Tuebingen

Original Message:
Sent: 10-18-2019 13:29
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Jason,

Thank you for providing these graphs.

My Stretched Young I (Cent = 3^1/1900) will become usable if electronic tuning devices are configured to use Cent = 3^1/1900 in the future.

This could be just the beginning of unequal temperaments that are based on Cent = 3^1/1900 rather than Cent = 2^1/1200.

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

Original Message:
Sent: 10-18-2019 13:00
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

Roshan has kindly pointed out that the horizontal reference lines on my chart are still in normal ET mode, and they need to be adjusted for the stretch of a P12 tuning. Accordingly, here is the revision.
An interesting exercise, but I can't imagine how this could be useful to anyone, as the offsets do not correspond to ETD cents, and this tuning cannot be done "by ear" with anything like this degree of precision.
Anyway ...

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

Original Message:
Sent: 10-17-2019 21:32
From: Jason Kanter
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

See the attached graphic. Roshan has worked this out on P12ET, which has expanded octaves, expanded semitones, and expanded cents -- so the offset numbers do not represent normal cents and will not work if put into an ETD. However, IF one were to tune Young I on a piano using a P12 stretch, the beat rates of M3, m3, and 5ths in the C3-C4 octave would closely match what's shown in the chart. How would you accomplish that, though?

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

Original Message:
Sent: 10-17-2019 15:25
From: Roshan Kakiya
Subject: Roshan Kakiya's Stretched Young I [Cent = 3^(1/1900)]

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

System of Cents

Cent = 3^1/1900.

This system of cents is appropriate for Pure 12th Equal Temperament.

Roshan Kakiya's Stretched Young I is based on Young I and Pure 12th Equal Temperament.


Cents

C	0.00
C#	96.30
D	197.53
D#	298.77
E	395.07
F	500.00
F#	595.07
G	698.77
G#	797.53
A	896.30
A#	1000.00
B	1095.07
C	1200.00

Offsets in Cents from Pure 12th Equal Temperament

C	0.00
C#	−3.70
D	−2.47
D#	−1.23
E	−4.93
F	0.00
F#	−4.93
G	−1.23
G#	−2.47
A	−3.70
A#	0.00
B	−4.93
C	0.00

Frequencies in Hz

A0	27.42
A#0	29.12
B0	30.76
C1	32.69
C#1	34.56
D1	36.64
D#1	38.85
E1	41.07
F1	43.64
F#1	46.11
G1	48.96
G#1	51.84
A1	54.88
A#1	58.27
B1	61.57
C2	65.42
C#2	69.16
D2	73.33
D#2	77.75
E2	82.21
F2	87.35
F#2	92.28
G2	97.99
G#2	103.75
A2	109.84
A#2	116.63
B2	123.22
C3	130.93
C#3	138.43
D3	146.77
D#3	155.62
E3	164.53
F3	174.82
F#3	184.70
G3	196.11
G#3	207.64
A3	219.84
A#3	233.43
B3	246.62
C4	262.05
C#4	277.05
D4	293.75
D#4	311.46
E4	329.29
F4	349.89
F#4	369.66
G4	392.51
G#4	415.58
A4	440.00
A#4	467.19
B4	493.59
C5	524.47
C#5	554.50
D5	587.92
D#5	623.36
E5	659.06
F5	700.29
F#5	739.86
G5	785.58
G#5	831.75
A5	880.63
A#5	935.05
B5	987.88
C6	1049.68
C#6	1109.79
D6	1176.69
D#6	1247.62
E6	1319.06
F6	1401.57
F#6	1480.77
G6	1572.27
G#6	1664.68
A6	1762.51
A#6	1871.43
B6	1977.18
C7	2100.86
C#7	2221.15
D7	2355.05
D#7	2497.02
E7	2640.00
F7	2805.14
F#7	2963.65
G7	3146.79
G#7	3331.73
A7	3527.54
A#7	3745.53
B7	3957.18
C8	4204.71

Mathematical Structure

Pythagorean comma = −1.

a = The amount of the Pythagorean comma by which C-G, G-D, D-A and A-E are each narrower than Just.

b = The amount of the Pythagorean comma by which E-B, B-F#, A#-F and F-C are each narrower than Just.

c = The amount of the Pythagorean comma by which F#-C#, C#-G#, G#-D# and D#-A# are each narrower than Just.


c = 0.


b = − 1/19.


a = 2b.

a = 2 × (− 1/19).

a = − 2/19.


Features

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

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

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


Roshan Kakiya's Stretched Young I preserves the symmetry of the Major Thirds of Young I. C-E is the only Major Third that is closest to Just and F#-A# is the only Major Third that is furthest from Just.

4 of the Fifths (E-B, B-F#, A#-F and F-C), 4 of the Twelfths (E-B, B-F#, A#-F and F-C) and all of the Octaves of Roshan Kakiya's Stretched Young I are the same as those of Pure 12th Equal Temperament.

Therefore, Roshan Kakiya's Stretched Young I combines the features of Young I and Pure 12th Equal Temperament.

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Roshan Kakiya
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