Pianotech

Roshan Kakiya's Idealised Young I



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


Cents

C	0.00
C#	94.13
D	196.09
D#	298.04
E	392.18
F	500.00
F#	592.18
G	698.04
G#	796.09
A	894.13
A#	1000.00
B	1092.18
C	1200.00

Offsets in Cents from Pure Octave Equal Temperament

C	0.00
C#	−5.87
D	−3.91
D#	−1.96
E	−7.82
F	0.00
F#	−7.82
G	−1.96
G#	−3.91
A	−5.87
A#	0.00
B	−7.82
C	0.00

Frequencies in Hz

A0	27.50
A#0	29.23
B0	30.83
C1	32.81
C#1	34.65
D1	36.75
D#1	38.98
E1	41.16
F1	43.80
F#1	46.20
G1	49.11
G#1	51.97
A1	55.00
A#1	58.47
B1	61.67
C2	65.63
C#2	69.30
D2	73.50
D#2	77.96
E2	82.31
F2	87.60
F#2	92.39
G2	98.22
G#2	103.94
A2	110.00
A#2	116.94
B2	123.33
C3	131.26
C#3	138.59
D3	147.00
D#3	155.92
E3	164.63
F3	175.21
F#3	184.79
G3	196.44
G#3	207.89
A3	220.00
A#3	233.87
B3	246.66
C4	262.51
C#4	277.18
D4	294.00
D#4	311.83
E4	329.26
F4	350.41
F#4	369.58
G4	392.88
G#4	415.77
A4	440.00
A#4	467.75
B4	493.33
C5	525.03
C#5	554.37
D5	587.99
D#5	623.66
E5	658.51
F5	700.83
F#5	739.15
G5	785.76
G#5	831.55
A5	880.00
A#5	935.49
B5	986.65
C6	1050.05
C#6	1108.73
D6	1175.99
D#6	1247.32
E6	1317.02
F6	1401.65
F#6	1478.31
G6	1571.53
G#6	1663.10
A6	1760.00
A#6	1870.98
B6	1973.30
C7	2100.11
C#7	2217.46
D7	2351.97
D#7	2494.64
E7	2634.04
F7	2803.31
F#7	2956.61
G7	3143.05
G#7	3326.19
A7	3520.00
A#7	3741.97
B7	3946.61
C8	4200.21

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.


a = 2b.

c = 0.


4a + 4b + 4c = −1.

4a + 4b + 4 × 0 = −1.

4a + 4b = −1.

4 × 2b + 4b = −1.

8b + 4b = −1.

12b = −1.

b = − 1/12.


a = 2 × (− 1/12).

a = − 1/6.


Features

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

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

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


Roshan Kakiya's Idealised Young I preserves the symmetry of the Major Thirds of Young I.

8 of the Fifths of Roshan Kakiya's Idealised Young I (C-G-D-A-E and F#-C#-G#-D#-A#) are the same as those of Young II.

Therefore, Roshan Kakiya's Idealised Young I combines the features of Young I and Young II.

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

Original Message------

Roshan Kakiya's Idealised Young I

Cents

C	0.00
C#	94.13
D	196.09
D#	298.04
E	392.18
F	500.00
F#	592.18
G	698.04
G#	796.09
A	894.13
A#	1000.00
B	1092.18
C	1200.00

Offsets in Cents from Pure Octave Equal Temperament

C	0.00
C#	−5.87
D	−3.91
D#	−1.96
E	−7.82
F	0.00
F#	−7.82
G	−1.96
G#	−3.91
A	−5.87
A#	0.00
B	−7.82
C	0.00

Frequencies in Hz

A0	27.50
A#0	29.23
B0	30.83
C1	32.81
C#1	34.65
D1	36.75
D#1	38.98
E1	41.16
F1	43.80
F#1	46.20
G1	49.11
G#1	51.97
A1	55.00
A#1	58.47
B1	61.67
C2	65.63
C#2	69.30
D2	73.50
D#2	77.96
E2	82.31
F2	87.60
F#2	92.39
G2	98.22
G#2	103.94
A2	110.00
A#2	116.94
B2	123.33
C3	131.26
C#3	138.59
D3	147.00
D#3	155.92
E3	164.63
F3	175.21
F#3	184.79
G3	196.44
G#3	207.89
A3	220.00
A#3	233.87
B3	246.66
C4	262.51
C#4	277.18
D4	294.00
D#4	311.83
E4	329.26
F4	350.41
F#4	369.58
G4	392.88
G#4	415.77
A4	440.00
A#4	467.75
B4	493.33
C5	525.03
C#5	554.37
D5	587.99
D#5	623.66
E5	658.51
F5	700.83
F#5	739.15
G5	785.76
G#5	831.55
A5	880.00
A#5	935.49
B5	986.65
C6	1050.05
C#6	1108.73
D6	1175.99
D#6	1247.32
E6	1317.02
F6	1401.65
F#6	1478.31
G6	1571.53
G#6	1663.10
A6	1760.00
A#6	1870.98
B6	1973.30
C7	2100.11
C#7	2217.46
D7	2351.97
D#7	2494.64
E7	2634.04
F7	2803.31
F#7	2956.61
G7	3143.05
G#7	3326.19
A7	3520.00
A#7	3741.97
B7	3946.61
C8	4200.21

Features

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

4 Pure Fifths reduced by 1/12 Pythagorean comma: E-B, B-F#, A#-F and F-C.

4 Pure Fifths reduced by 1/6 Pythagorean comma: C-G, G-D, D-A and A-E.


Roshan Kakiya's Idealised Young I preserves the symmetry of the Major Thirds of Young I.

8 of the Fifths of Roshan Kakiya's Idealised Young I (C-G-D-A-E and F#-C#-G#-D#-A#) are the same as those of Young II.

Therefore, Roshan Kakiya's Idealised Young I combines the features of Young I and Young II.

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

My Idealised Young I is my favourite Unequal Temperament.

It is the best out of all the Unequal Temperaments that I have made.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Original Message------

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Look at the offsets. 

Every offset is a fraction of the Pythagorean comma:

1/12 Pythagorean comma = 23.46 cents / 12 = 1.96 cents.

1/6 Pythagorean comma = 23.46 cents / 6 = 3.91 cents.

1/4 Pythagorean comma = 23.46 cents / 4 = 5.87 cents.

1/3 Pythagorean comma = 23.46 cents / 3 = 7.82 cents.


Look at the difference between two consecutive Major Thirds (red bars on the bar chart).

The difference between two consecutive Major Thirds (red bars on the bar chart) is either 1.96 cents (1/12 Pythagorean comma) or 3.91 cents (1/6 Pythagorean comma).


Look at the symmetry of the Major Thirds.

The Major Thirds are symmetrical on either side of F#. C-E is closest to Just and F#-A# is furthest from Just.

C-E is wider than Just by 1/4 Pythagorean comma. C-E = 386.31 cents + 5.87 cents = 392.18 cents.

F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.31 cents + 21.51 cents = 407.82 cents.


Look at the two types of Tempered Fifth (blue bars on the bar chart).

Pure Fifth narrowed by 1.96 cents (1/12 Pythagorean comma).

Pure Fifth narrowed by 3.91 cents (1/6 Pythagorean comma).


The difference between two consecutive Major Thirds (red bars on the bar chart) is either 1.96 cents (1/12 Pythagorean comma) or 3.91 cents (1/6 Pythagorean comma).

Of the two types of Tempered Fifth (blue bars on the bar chart), one is a Pure Fifth narrowed by 1.96 cents (1/12 Pythagorean comma) and the other is a Pure Fifth narrowed by 3.91 cents (1/6 Pythagorean comma).

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

Original Message:
Sent: 09-16-2019 09:52
From: David Pinnegar
Subject: Roshan Kakiya's Idealised Young I

I'm wondering what your definition of best is and what your criterion for favourite might be.

In comparison with Equal and Kellner

Proportion of same frequencies
Equal	38%
Kellner	43%
Roshan Young	41%
Around 1 beat
Equal	15%
Kellner	9%
Roshan Young	11%
1 to 5 beats
Equal	33%
Kellner	28%
Roshan Young	31%
2 to 5 beats
Equal	16%
Kellner	17%
Roshan Young	18%

So for resonance and ability to create stillness, Young might not be so good as Kellner

Best wishes

David P

--

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

Original Message------

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Roshan,

The temperament you describe here is undeniably most clever in its mathematical relationships. But this is music, not numerology.

In well temperaments, the triads associated with harsher M3rds tend to have pure 5ths, but the pure fifths do not ameliorate the harsh M3rds to my ears. Conversely, the triads associated with the mild M3rds tend to have busy P5ths, and to my ears those busy P5ths do greatly detract from the “calmness” that is claimed to be associated with those slow M3rds.

In other words, you can’t win. It seems to me that temperament and pieces of music need to be carefully matched for best musical effect, and there can be no universal temperament, no “best” temperament.

Sent from my iPad


Original Message------

Look at the offsets. 

Every offset is a fraction of the Pythagorean comma:

1/12 Pythagorean comma = 23.46 cents / 12 = 1.96 cents.

1/6 Pythagorean comma = 23.46 cents / 6 = 3.91 cents.

1/4 Pythagorean comma = 23.46 cents / 4 = 5.87 cents.

1/3 Pythagorean comma = 23.46 cents / 3 = 7.82 cents.


Look at the difference between two consecutive Major Thirds (red bars on the bar chart).

The difference between two consecutive Major Thirds (red bars on the bar chart) is either 1.96 cents (1/12 Pythagorean comma) or 3.91 cents (1/6 Pythagorean comma).


Look at the symmetry of the Major Thirds.

The Major Thirds are symmetrical on either side of F#. C-E is closest to Just and F#-A# is furthest from Just.

C-E is wider than Just by 1/4 Pythagorean comma. C-E = 386.31 cents + 5.87 cents = 392.18 cents.

F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.31 cents + 21.51 cents = 407.82 cents.


Look at the two types of Tempered Fifth (blue bars on the bar chart).

Pure Fifth reduced by 1.96 cents (1/12 Pythagorean comma).

Pure Fifth reduced by 3.91 cents (1/6 Pythagorean comma).


The difference between two consecutive Major Thirds (red bars on the bar chart) is either 1.96 cents (1/12 Pythagorean comma) or 3.91 cents (1/6 Pythagorean comma).

Of the two types of Tempered Fifth (blue bars on the bar chart), one is a Pure Fifth reduced by 1.96 cents (1/12 Pythagorean comma) and the other is a Pure Fifth reduced by 3.91 cents (1/6 Pythagorean comma).

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

Original Message:
Sent: 09-16-2019 09:52
From: David Pinnegar
Subject: Roshan Kakiya's Idealised Young I

I'm wondering what your definition of best is and what your criterion for favourite might be.

In comparison with Equal and Kellner

Proportion of same frequencies
Equal	38%
Kellner	43%
Roshan Young	41%
Around 1 beat
Equal	15%
Kellner	9%
Roshan Young	11%
1 to 5 beats
Equal	33%
Kellner	28%
Roshan Young	31%
2 to 5 beats
Equal	16%
Kellner	17%
Roshan Young	18%

So for resonance and ability to create stillness, Young might not be so good as Kellner

Best wishes

David P

--

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

Original Message------

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Kent,

Unequal Temperaments are designed in the way that you have described because there is a mathematical relationship between 4 Fifths and a Major Third.

4 Fifths, C-G, G-D, D-A and A-E, are required to get from C to E around the Circle of Fifths.

If each of these 4 Fifths is pure, C-E will become a Pythagorean Major Third (407.82 cents). This Major Third is 7.82 cents wider than a Major Third in Pure Octave Equal Temperament (400.00 cents).

On the other hand, if each of these 4 Fifths is narrowed by 1/4 Syntonic comma, C-E will become a Pure Major Third (386.31 cents). This Major Third is 13.69 cents narrower than a Major Third in Pure Octave Equal Temperament (400.00 cents).

Therefore, it is only possible to have calm Major Thirds if there are busy Fifths and it is only possible to have calm Fifths if there are busy Major Thirds.

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

Original Message:
Sent: 09-16-2019 13:54
From: Kent Swafford
Subject: Roshan Kakiya's Idealised Young I

Roshan,

The temperament you describe here is undeniably most clever in its mathematical relationships. But this is music, not numerology.

In well temperaments, the triads associated with harsher M3rds tend to have pure 5ths, but the pure fifths do not ameliorate the harsh M3rds to my ears. Conversely, the triads associated with the mild M3rds tend to have busy P5ths, and to my ears those busy P5ths do greatly detract from the "calmness" that is claimed to be associated with those slow M3rds.

In other words, you can't win. It seems to me that temperament and pieces of music need to be carefully matched for best musical effect, and there can be no universal temperament, no "best" temperament.

Sent from my iPad


Original Message------

Look at the offsets.

Every offset is a fraction of the Pythagorean comma:

1/12 Pythagorean comma = 23.46 cents / 12 = 1.96 cents.

1/6 Pythagorean comma = 23.46 cents / 6 = 3.91 cents.

1/4 Pythagorean comma = 23.46 cents / 4 = 5.87 cents.

1/3 Pythagorean comma = 23.46 cents / 3 = 7.82 cents.


Look at the difference between two consecutive Major Thirds (red bars on the bar chart).

The difference between two consecutive Major Thirds (red bars on the bar chart) is either 1.96 cents (1/12 Pythagorean comma) or 3.91 cents (1/6 Pythagorean comma).


Look at the symmetry of the Major Thirds.

The Major Thirds are symmetrical on either side of F#. C-E is closest to Just and F#-A# is furthest from Just.

C-E is wider than Just by 1/4 Pythagorean comma. C-E = 386.31 cents + 5.87 cents = 392.18 cents.

F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.31 cents + 21.51 cents = 407.82 cents.


Look at the two types of Tempered Fifth (blue bars on the bar chart).

Pure Fifth reduced by 1.96 cents (1/12 Pythagorean comma).

Pure Fifth reduced by 3.91 cents (1/6 Pythagorean comma).


The difference between two consecutive Major Thirds (red bars on the bar chart) is either 1.96 cents (1/12 Pythagorean comma) or 3.91 cents (1/6 Pythagorean comma).

Of the two types of Tempered Fifth (blue bars on the bar chart), one is a Pure Fifth reduced by 1.96 cents (1/12 Pythagorean comma) and the other is a Pure Fifth reduced by 3.91 cents (1/6 Pythagorean comma).

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

Original Message:
Sent: 09-16-2019 09:52
From: David Pinnegar
Subject: Roshan Kakiya's Idealised Young I

I'm wondering what your definition of best is and what your criterion for favourite might be.

In comparison with Equal and Kellner

Proportion of same frequencies
Equal	38%
Kellner	43%
Roshan Young	41%
Around 1 beat
Equal	15%
Kellner	9%
Roshan Young	11%
1 to 5 beats
Equal	33%
Kellner	28%
Roshan Young	31%
2 to 5 beats
Equal	16%
Kellner	17%
Roshan Young	18%

So for resonance and ability to create stillness, Young might not be so good as Kellner

Best wishes

David P

--

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

Original Message------

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Roshan wrote "It is my favourite temperament."

I inquire: Why is it your favorite temperament?
              Have you ever heard it on a musical instrument?
              Have you used it to play a commonly known piece of music?
              Have you composed and played one of your own pieces of music in this temperament?
              Are you just fascinated with numbers and charts?
              Do you hope to make a great discovery and become famous?





------------------------------
Ed Sutton
ed440@me.com
(980) 254-7413
------------------------------

Original Message:
Sent: 09-16-2019 09:19
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

My Idealised Young I is my favourite Unequal Temperament and the best out of all the Unequal Temperaments that I have made because of its underlying mathematical relationships. I have illustrated some of its mathematical relationships on my previous post.

I admire the original version of Young I because of the symmetry of its Major Thirds on either side of F#. However, the original version is slightly flawed in terms of design. My idealised version corrects the flaw in the design of the original version.



The Connection Between 4 Fifths and a Major Third

The size of a Major Third is related to a sequence of 4 Fifths.

4 Fifths are required to get from C to E. Therefore, the size of C-E would be affected by modifying the Fifths between C and E around the Circle of Fifths.

This mathematical relationship is a crucial part of the design of Original Young I and my Idealised Young I.



Original Young I

An image of Original Young I:

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


C-E is wider than Just by 1/4 Syntonic comma. C-E = 386.314 cents + 21.506 cents / 4 = 386.314 cents + 5.377 cents = 391.69 cents.

This can be achieved as follows:

Syntonic comma × 3/4 = 21.506 cents × 3/4 = 16.130 cents.

(Syntonic comma × 3/4) / 4 = 16.130 cents / 4 = 4.033 cents.

Pure Fifth − (Syntonic comma × 3/4) / 4 = 701.955 cents − 4.033 cents = 697.922 cents.

C-E = 697.922 cents − (1200.000 cents − 697.922 cents) + 697.922 cents − (1200.000 cents − 697.922 cents) = 391.69 cents.


F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.314 cents + 21.506 cents = 407.82 cents.

This can be achieved as follows:

Pure Fifth = 701.955 cents.

F#-A# = 701.955 cents − (1200.000 cents − 701.955 cents) + 701.955 cents − (1200.000 cents − 701.955 cents) = 407.82 cents.


The remainder of the Pythagorean comma must be tempered out.

This can be achieved as follows:

Remainder = 23.460 cents − 16.130 cents = 7.330 cents.

Remainder / 4 = 7.330 cents / 4 = 1.833 cents.

Pure Fifth − Remainder / 4 = 701.955 cents − 1.833 cents = 700.122 cents.


The Fifths of Original Young I are unevenly spaced: 

701.955 cents − 700.122 cents = 1.833 cents.

700.122 cents − 697.922 cents = 2.200 cents.

This is the flaw in the design of Original Young I.



Roshan Kakiya's Idealised Young I

An image of Roshan Kakiya's Idealised Young I:

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


C-E is wider than Just by 1/4 Pythagorean comma. C-E = 386.314 cents + 23.460 cents / 4 = 386.314 cents + 5.865 cents = 392.18 cents.

This can be achieved as follows:

Pythagorean comma × 2/3 = 23.460 cents × 2/3 = 15.640 cents.

(Pythagorean comma × 2/3) / 4 = 15.640 cents / 4 = 3.910 cents.

Pure Fifth − (Pythagorean comma × 2/3) / 4 = 701.955 cents − 3.910 cents = 698.045 cents.

C-E = 698.045 cents − (1200.000 cents − 698.045 cents) + 698.045 cents − (1200.000 cents − 698.045 cents) = 392.18 cents.


F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.314 cents + 21.506 cents = 407.82 cents.

This can be achieved as follows:

Pure Fifth = 701.955 cents.

F#-A# = 701.955 cents − (1200.000 cents − 701.955 cents) + 701.955 cents − (1200.000 cents − 701.955 cents) = 407.82 cents.


The remainder of the Pythagorean comma must be tempered out.

This can be achieved as follows:

Remainder = 23.460 cents − 15.640 cents = 7.820 cents.

Remainder / 4 = 7.820 cents / 4 = 1.955 cents.

Pure Fifth − Remainder / 4 = 701.955 cents − 1.955 cents = 700.000 cents.


The Fifths of Roshan Kakiya's Idealised Young I are evenly spaced:

701.955 cents − 700.000 cents = 1.955 cents.

700.000 cents − 698.045 cents = 1.955 cents.

Therefore, Roshan Kakiya's Idealised Young I corrects the flaw in the design of Original Young I.

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

Original Message:
Sent: 09-16-2019 17:22
From: Ed Sutton
Subject: Roshan Kakiya's Idealised Young I

Roshan wrote "It is my favourite temperament."

I inquire: Why is it your favorite temperament?
              Have you ever heard it on a musical instrument?
              Have you used it to play a commonly known piece of music?
              Have you composed and played one of your own pieces of music in this temperament?
              Are you just fascinated with numbers and charts?
              Do you hope to make a great discovery and become famous?





------------------------------
Ed Sutton
ed440@me.com
(980) 254-7413

Original Message:
Sent: 09-16-2019 09:19
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Roshan-
Consider the offsets of a Perfect Harmony Temperament:
C=0
C#=-100cts.
D=-200cts.
D#=-300cts.
etc.
Such that 
All fifths = 0.000 cents
All thirds = 0.000 cents
All intervals are perfectly pure!
Some adjustments will be needed for inharmonicity, but don't worry, the piano tuners will deal with that!

------------------------------
Ed Sutton
ed440@me.com
(980) 254-7413
------------------------------

Original Message:
Sent: 09-16-2019 22:43
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

My Idealised Young I is my favourite temperament and the best out of all the temperaments that I have made because of its underlying mathematical relationships. I have illustrated some of its mathematical relationships on my previous post.

I admire the original version of Young I because of the symmetry of its Major Thirds on either side of F#. However, the original version is slightly flawed in terms of design. My idealised version corrects the flaw in the design of the original version.



The Connection Between 4 Fifths and a Major Third

The size of a Major Third is related to a sequence of 4 Fifths.

4 Fifths are required to get from C to E. Therefore, the size of C-E would be affected by modifying the Fifths between C and E around the Circle of Fifths.

This mathematical relationship is a crucial part of the design of Original Young I and my Idealised Young I.



Original Young I

An image of Original Young I:

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


C-E is wider than Just by 1/4 Syntonic comma. C-E = 386.314 cents + 21.506 cents / 4 = 386.314 cents + 5.377 cents = 391.69 cents.

This can be achieved as follows:

Syntonic comma × 3/4 = 21.506 cents × 3/4 = 16.130 cents.

(Syntonic comma × 3/4) / 4 = 16.130 cents / 4 = 4.033 cents.

Pure Fifth − (Syntonic comma × 3/4) / 4 = 701.955 cents − 4.033 cents = 697.922 cents.

C-E = 697.922 cents − (1200.000 cents − 697.922 cents) + 697.922 cents − (1200.000 cents − 697.922 cents) = 391.69 cents.


F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.314 cents + 21.506 cents = 407.82 cents.

This can be achieved as follows:

Pure Fifth = 701.955 cents.

F#-A# = 701.955 cents − (1200.000 cents − 701.955 cents) + 701.955 cents − (1200.000 cents − 701.955 cents) = 407.82 cents.


The remainder of the Pythagorean comma must be tempered out.

This can be achieved as follows:

Remainder = 23.460 cents − 16.130 cents = 7.330 cents.

Remainder / 4 = 7.330 cents / 4 = 1.833 cents.

Pure Fifth − Remainder / 4 = 701.955 cents − 1.833 cents = 700.122 cents.


The Fifths of Original Young I are unevenly spaced: 

701.955 cents − 700.122 cents = 1.833 cents.

700.122 cents − 697.922 cents = 2.200 cents.

This is the flaw in the design of Original Young I.



Roshan Kakiya's Idealised Young I

C-E is wider than Just by 1/4 Pythagorean comma. C-E = 386.314 cents + 23.460 cents / 4 = 386.314 cents + 5.865 cents = 392.18 cents.

This can be achieved as follows:

Pythagorean comma × 2/3 = 23.460 cents × 2/3 = 15.640 cents.

(Pythagorean comma × 2/3) / 4 = 15.640 cents / 4 = 3.910 cents.

Pure Fifth − (Pythagorean comma × 2/3) / 4 = 701.955 cents − 3.910 cents = 698.045 cents.

C-E = 698.045 cents − (1200.000 cents − 698.045 cents) + 698.045 cents − (1200.000 cents − 698.045 cents) = 392.18 cents.


F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.314 cents + 21.506 cents = 407.82 cents.

This can be achieved as follows:

Pure Fifth = 701.955 cents.

F#-A# = 701.955 cents − (1200.000 cents − 701.955 cents) + 701.955 cents − (1200.000 cents − 701.955 cents) = 407.82 cents.


The remainder of the Pythagorean comma must be tempered out.

This can be achieved as follows:

Remainder = 23.460 cents − 15.640 cents = 7.820 cents.

Remainder / 4 = 7.820 cents / 4 = 1.955 cents.

Pure Fifth − Remainder / 4 = 701.955 cents − 1.955 cents = 700.000 cents.


The Fifths of Roshan Kakiya's Idealised Young I are evenly spaced:

701.955 cents − 700.000 cents = 1.955 cents.

700.000 cents − 698.045 cents = 1.955 cents.

Therefore, Roshan Kakiya's Idealised Young I corrects the flaw in the design of Original Young I.

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

Original Message:
Sent: 09-16-2019 17:22
From: Ed Sutton
Subject: Roshan Kakiya's Idealised Young I

Roshan wrote "It is my favourite temperament."

I inquire: Why is it your favorite temperament?
              Have you ever heard it on a musical instrument?
              Have you used it to play a commonly known piece of music?
              Have you composed and played one of your own pieces of music in this temperament?
              Are you just fascinated with numbers and charts?
              Do you hope to make a great discovery and become famous?





------------------------------
Ed Sutton
ed440@me.com
(980) 254-7413

Original Message:
Sent: 09-16-2019 09:19
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

Ed,

1-limit just intonation is perfectly harmonious:

https://my.ptg.org/communities/community-home/digestviewer/viewthread?MessageKey=9ee06d55-9d72-4afd-94a3-ec9c05bf3614&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

1-limit just intonation does not have any tempered intervals so it is not a temperament!

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

Original Message:
Sent: 09-17-2019 08:58
From: Ed Sutton
Subject: Roshan Kakiya's Idealised Young I

Roshan-
Consider the offsets of a Perfect Harmony Temperament:
C=0
C#=-100cts.
D=-200cts.
D#=-300cts.
etc.
Such that
All fifths = 0.000 cents
All thirds = 0.000 cents
All intervals are perfectly pure!
Some adjustments will be needed for inharmonicity, but don't worry, the piano tuners will deal with that!

------------------------------
Ed Sutton
ed440@me.com
(980) 254-7413

Original Message:
Sent: 09-16-2019 22:43
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

My Idealised Young I is my favourite temperament and the best out of all the temperaments that I have made because of its underlying mathematical relationships. I have illustrated some of its mathematical relationships on my previous post.

I admire the original version of Young I because of the symmetry of its Major Thirds on either side of F#. However, the original version is slightly flawed in terms of design. My idealised version corrects the flaw in the design of the original version.



The Connection Between 4 Fifths and a Major Third

The size of a Major Third is related to a sequence of 4 Fifths.

4 Fifths are required to get from C to E. Therefore, the size of C-E would be affected by modifying the Fifths between C and E around the Circle of Fifths.

This mathematical relationship is a crucial part of the design of Original Young I and my Idealised Young I.



Original Young I

An image of Original Young I:

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


C-E is wider than Just by 1/4 Syntonic comma. C-E = 386.314 cents + 21.506 cents / 4 = 386.314 cents + 5.377 cents = 391.69 cents.

This can be achieved as follows:

Syntonic comma × 3/4 = 21.506 cents × 3/4 = 16.130 cents.

(Syntonic comma × 3/4) / 4 = 16.130 cents / 4 = 4.033 cents.

Pure Fifth − (Syntonic comma × 3/4) / 4 = 701.955 cents − 4.033 cents = 697.922 cents.

C-E = 697.922 cents − (1200.000 cents − 697.922 cents) + 697.922 cents − (1200.000 cents − 697.922 cents) = 391.69 cents.


F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.314 cents + 21.506 cents = 407.82 cents.

This can be achieved as follows:

Pure Fifth = 701.955 cents.

F#-A# = 701.955 cents − (1200.000 cents − 701.955 cents) + 701.955 cents − (1200.000 cents − 701.955 cents) = 407.82 cents.


The remainder of the Pythagorean comma must be tempered out.

This can be achieved as follows:

Remainder = 23.460 cents − 16.130 cents = 7.330 cents.

Remainder / 4 = 7.330 cents / 4 = 1.833 cents.

Pure Fifth − Remainder / 4 = 701.955 cents − 1.833 cents = 700.122 cents.


The Fifths of Original Young I are unevenly spaced: 

701.955 cents − 700.122 cents = 1.833 cents.

700.122 cents − 697.922 cents = 2.200 cents.

This is the flaw in the design of Original Young I.



Roshan Kakiya's Idealised Young I

C-E is wider than Just by 1/4 Pythagorean comma. C-E = 386.314 cents + 23.460 cents / 4 = 386.314 cents + 5.865 cents = 392.18 cents.

This can be achieved as follows:

Pythagorean comma × 2/3 = 23.460 cents × 2/3 = 15.640 cents.

(Pythagorean comma × 2/3) / 4 = 15.640 cents / 4 = 3.910 cents.

Pure Fifth − (Pythagorean comma × 2/3) / 4 = 701.955 cents − 3.910 cents = 698.045 cents.

C-E = 698.045 cents − (1200.000 cents − 698.045 cents) + 698.045 cents − (1200.000 cents − 698.045 cents) = 392.18 cents.


F#-A# is wider than Just by a Syntonic comma. F#-A# = 386.314 cents + 21.506 cents = 407.82 cents.

This can be achieved as follows:

Pure Fifth = 701.955 cents.

F#-A# = 701.955 cents − (1200.000 cents − 701.955 cents) + 701.955 cents − (1200.000 cents − 701.955 cents) = 407.82 cents.


The remainder of the Pythagorean comma must be tempered out.

This can be achieved as follows:

Remainder = 23.460 cents − 15.640 cents = 7.820 cents.

Remainder / 4 = 7.820 cents / 4 = 1.955 cents.

Pure Fifth − Remainder / 4 = 701.955 cents − 1.955 cents = 700.000 cents.


The Fifths of Roshan Kakiya's Idealised Young I are evenly spaced:

701.955 cents − 700.000 cents = 1.955 cents.

700.000 cents − 698.045 cents = 1.955 cents.

Therefore, Roshan Kakiya's Idealised Young I corrects the flaw in the design of Original Young I.

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

Original Message:
Sent: 09-16-2019 17:22
From: Ed Sutton
Subject: Roshan Kakiya's Idealised Young I

Roshan wrote "It is my favourite temperament."

I inquire: Why is it your favorite temperament?
              Have you ever heard it on a musical instrument?
              Have you used it to play a commonly known piece of music?
              Have you composed and played one of your own pieces of music in this temperament?
              Are you just fascinated with numbers and charts?
              Do you hope to make a great discovery and become famous?





------------------------------
Ed Sutton
ed440@me.com
(980) 254-7413

Original Message:
Sent: 09-16-2019 09:19
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I

In my opinion, my Idealised Young I is the perfect compromise.

It is my favourite temperament.

It is the best out of all the temperaments that I have made.

It is the conclusion of my research into the Equal Temperament tuning system, the Unequal Temperament tuning system and the Just/Rational Intonation tuning system.

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

Original Message:
Sent: 09-15-2019 15:32
From: Roshan Kakiya
Subject: Roshan Kakiya's Idealised Young I