SIMULATION OF 12-EDO and 19-edo with 640-edl

hi all

Here is simulation of 12-edo in a string with a length of 640 mm (with
accuracy of 1 mm) which belongs to 640-EDL :

0: 1/1 0.000 unison, perfect prime

1: 100.228 cents 100.228

2: 200.532 cents 200.532

3: 300.559 cents 300.559

4: 399.892 cents 399.892

5: 501.655 cents 501.655

6: 598.273 cents 598.273

7: 700.603 cents 700.603

8: 800.750 cents 800.750

9: 897.937 cents 897.937

10: 1000.906 cents 1000.906

11: 1100.144 cents 1100.144

12: 1200.000 cents 1200.000

|640-EDL ( ratioes are simplified) :

0: 1/1 0.000 unison, perfect prime

1: 160/151 100.228

2: 64/57 200.532

3: 320/269 300.559

4: 160/127 399.892

5: 640/479 501.655

6: 640/453 598.273

7: 640/427 700.603

8: 640/403 800.750

9: 640/381 897.937

10: 640/359 1000.906

11: 640/339 1100.144

12: 2/1 1200.000 octave

The maximum and minimum error are 2.062 and - 1.655 cent.this scale has
these fifths :

0: 0.000 cents

7: 700.603 cents

2: 699.929 cents

9: 697.405 cents

4: 701.955 cents

11: 700.252 cents

6: 698.129 cents

1: 701.955 cents

8: 700.522 cents

3: 699.808 cents

10: 700.347 cents

5: 700.750 cents

12: 698.345 cents

And these thirds :

0: 0.000 cents

4: 399.892 cents

8: 400.858 cents

11: 399.541 cents

3: 400.415 cents

7: 400.044 cents

12: 399.250 cents

So for a 640 mm string ( for example in setar) , 12-edo is simulated by
degrees of 640-edl with tiny errors for melodic sense.

This is anexample and you can test it for other edos . for example
19-edo :

0: 1/1 0.000 unison, perfect prime

1: 640/617 63.362

2: 128/119 126.219

3: 320/287 188.425

4: 640/553 252.951

5: 640/533 316.724

6: 320/257 379.564

7: 40/31 441.278

8: 320/239 505.274

9: 640/461 567.966

10: 160/111 633.015

11: 160/107 696.553

12: 640/413 758.316

13: 320/199 822.364

14: 5/3 884.359 major sixth, BP sixth

15: 64/37 948.656 37th subharmonic

16: 640/357 1010.577

17: 80/43 1074.796

18: 160/83 1136.266

19: 2/1 1200.000 octave

And differences with 19-edo are :

1: 63.362: 1: 63.1579 cents, diff. -0.003226 steps, -0.2038 cents

2: 126.219: 2: 126.3158 cents, diff. 0.001537 steps, 0.0971
cents

3: 188.425: 3: 189.4737 cents, diff. 0.016597 steps, 1.0483
cents

4: 252.951: 4: 252.6316 cents, diff. -0.005056 steps, -0.3193
cents

5: 316.724: 5: 315.7895 cents, diff. -0.014791 steps, -0.9342
cents

6: 379.564: 6: 378.9474 cents, diff. -0.009767 steps, -0.6169
cents

7: 441.278: 7: 442.1053 cents, diff. 0.013096 steps, 0.8271
cents

8: 505.274: 8: 505.2632 cents, diff. -0.000164 steps, -0.0104
cents

9: 567.966: 9: 568.4211 cents, diff. 0.007202 steps, 0.4549
cents

10: 633.015: 10: 631.5789 cents, diff. -0.022732 steps, -1.4357
cents

11: 696.553: 11: 694.7368 cents, diff. -0.028761 steps, -1.8165
cents

12: 758.316: 12: 757.8947 cents, diff. -0.006672 steps, -0.4214
cents

13: 822.364: 13: 821.0526 cents, diff. -0.020766 steps, -1.3115
cents

14: 884.359: 14: 884.2105 cents, diff. -0.002346 steps, -0.1482
cents

15: 948.656: 15: 947.3684 cents, diff. -0.020386 steps, -1.2875
cents

16: 1010.577: 16: 1010.5263 cents, diff. -0.000808 steps, -0.0511
cents

17: 1074.796: 17: 1073.6842 cents, diff. -0.017603 steps, -1.1118
cents

18: 1136.266: 18: 1136.8421 cents, diff. 0.009115 steps, 0.5757
cents

19: 1200.000: 19: 1200.0000 cents, diff. -0.000000 steps, -0.0000
cents

The maximum and minimum error are 1.0483 cents and -1.8165 cents

It is interesting that you cant find any relation between fifth and
fourth as :

fifth + fourth = 1200 ,

you see in (640-edl version of 12-edo) :

501.655 + 700.603 = 1202.258

And this is nature of persian music and all music played with string
instruments with or without fret.

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html

www.harmonytalk.com/archives/000288.html

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm

Dear brother, can you send me the SCL files for easy scrutiny?

Cordially,
Oz.
----- Original Message -----
From: Mohajeri Shahin
To: tuning@yahoogroups.com
Sent: 01 Mayıs 2006 Pazartesi 15:53
Subject: [tuning] SIMULATION OF 12-EDO and 19-edo with 640-edl

hi all

Here is simulation of 12-edo in a string with a length of 640 mm (with accuracy of 1 mm) which belongs to 640-EDL :

0: 1/1 0.000 unison, perfect prime

1: 100.228 cents 100.228

2: 200.532 cents 200.532

3: 300.559 cents 300.559

4: 399.892 cents 399.892

5: 501.655 cents 501.655

6: 598.273 cents 598.273

7: 700.603 cents 700.603

8: 800.750 cents 800.750

9: 897.937 cents 897.937

10: 1000.906 cents 1000.906

11: 1100.144 cents 1100.144

12: 1200.000 cents 1200.000

|640-EDL ( ratioes are simplified) :

0: 1/1 0.000 unison, perfect prime

1: 160/151 100.228

2: 64/57 200.532

3: 320/269 300.559

4: 160/127 399.892

5: 640/479 501.655

6: 640/453 598.273

7: 640/427 700.603

8: 640/403 800.750

9: 640/381 897.937

10: 640/359 1000.906

11: 640/339 1100.144

12: 2/1 1200.000 octave

The maximum and minimum error are 2.062 and – 1.655 cent.this scale has these fifths :

0: 0.000 cents

7: 700.603 cents

2: 699.929 cents

9: 697.405 cents

4: 701.955 cents

11: 700.252 cents

6: 698.129 cents

1: 701.955 cents

8: 700.522 cents

3: 699.808 cents

10: 700.347 cents

5: 700.750 cents

12: 698.345 cents

And these thirds :

0: 0.000 cents

4: 399.892 cents

8: 400.858 cents

11: 399.541 cents

3: 400.415 cents

7: 400.044 cents

12: 399.250 cents

So for a 640 mm string ( for example in setar) , 12-edo is simulated by degrees of 640-edl with tiny errors for melodic sense.

This is anexample and you can test it for other edos . for example 19-edo :

0: 1/1 0.000 unison, perfect prime

1: 640/617 63.362

2: 128/119 126.219

3: 320/287 188.425

4: 640/553 252.951

5: 640/533 316.724

6: 320/257 379.564

7: 40/31 441.278

8: 320/239 505.274

9: 640/461 567.966

10: 160/111 633.015

11: 160/107 696.553

12: 640/413 758.316

13: 320/199 822.364

14: 5/3 884.359 major sixth, BP sixth

15: 64/37 948.656 37th subharmonic

16: 640/357 1010.577

17: 80/43 1074.796

18: 160/83 1136.266

19: 2/1 1200.000 octave

And differences with 19-edo are :

1: 63.362: 1: 63.1579 cents, diff. -0.003226 steps, -0.2038 cents

2: 126.219: 2: 126.3158 cents, diff. 0.001537 steps, 0.0971 cents

3: 188.425: 3: 189.4737 cents, diff. 0.016597 steps, 1.0483 cents

4: 252.951: 4: 252.6316 cents, diff. -0.005056 steps, -0.3193 cents

5: 316.724: 5: 315.7895 cents, diff. -0.014791 steps, -0.9342 cents

6: 379.564: 6: 378.9474 cents, diff. -0.009767 steps, -0.6169 cents

7: 441.278: 7: 442.1053 cents, diff. 0.013096 steps, 0.8271 cents

8: 505.274: 8: 505.2632 cents, diff. -0.000164 steps, -0.0104 cents

9: 567.966: 9: 568.4211 cents, diff. 0.007202 steps, 0.4549 cents

10: 633.015: 10: 631.5789 cents, diff. -0.022732 steps, -1.4357 cents

11: 696.553: 11: 694.7368 cents, diff. -0.028761 steps, -1.8165 cents

12: 758.316: 12: 757.8947 cents, diff. -0.006672 steps, -0.4214 cents

13: 822.364: 13: 821.0526 cents, diff. -0.020766 steps, -1.3115 cents

14: 884.359: 14: 884.2105 cents, diff. -0.002346 steps, -0.1482 cents

15: 948.656: 15: 947.3684 cents, diff. -0.020386 steps, -1.2875 cents

16: 1010.577: 16: 1010.5263 cents, diff. -0.000808 steps, -0.0511 cents

17: 1074.796: 17: 1073.6842 cents, diff. -0.017603 steps, -1.1118 cents

18: 1136.266: 18: 1136.8421 cents, diff. 0.009115 steps, 0.5757 cents

19: 1200.000: 19: 1200.0000 cents, diff. -0.000000 steps, -0.0000 cents

The maximum and minimum error are 1.0483 cents and -1.8165 cents

It is interesting that you cant find any relation between fifth and fourth as :

fifth + fourth = 1200 ,

you see in (640-edl version of 12-edo) :

501.655 + 700.603 = 1202.258

And this is nature of persian music and all music played with string instruments with or without fret.

Shaahin Mohaajeri