recreational tuning exercise
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Ozan Yarman
Feb 6, 2012, 7:09:52 PM2/6/12
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Albrecht Durer's magix square, you know or heard of, no doubt.
Here it is:
| 16 | 3 | 2 | 13 |
| 5 | 10 | 11 | 8 |
| 9 | 6 | 7 | 12 |
| 4 | 15 | 14 | 1 |
Let's make a recreational scale from it. I suggest every number in the box can be the numerator in contrast to every next number in any direction being the denominatior. The only rule is, the numerator must be greater than the denominator. The constraint of the box also applies, so that you cannot pair a number with a number outside the border, as though the square wrapped around.
So, one can easily discern 3:2, 13:11, 7:6, etc...
What is the list of musical ratios that can thus be extracted from Durer's square?
Oz.
✩ ✩ ✩
Ozan Yarman
Feb 7, 2012, 12:42:10 PM2/7/12
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If you haven't figured it out already, here is the scale:
|
0: 1/1 0.000
1: 15/14 119.443
2: 12/11 150.637
3: 11/10 165.004
4: 10/9 182.404
5: 7/6 266.871 minimal 3rd
6: 6/5 315.641 minor 3rd
7: 11/8 551.318
8: 10/7 617.488
9: 3/2 701.955 perfect 5th
10: 11/7 782.492
11: 8/5 813.686 minor 6th
12: 13/8 840.528
13: 5/3 884.359 major 6th
14: 12/7 933.129 septimal major 6th
15: 9/5 1017.596 minor 7th
16: 2/1 1200.000 1 octave
17: 15/7 1319.443
18: 9/4 1403.910 perfect 9th
19: 7/3 1466.871 minimal 10th
20: 5/2 1586.314 major 10th
21: 16/5 2013.686 minor 13th
22: 10/3 2084.359 major 13th
23: 11/3 2249.363
24: 15/4 2288.269 classic major 14th
25: 5/1 2786.314 major 17th
26: 16/3 2898.045 perfect 18th
27: 11/2 2951.318
28: 13/2 3240.528
29: 7/1 3368.826 harmonic 21st
30: 12/1 4301.955 perfect 26th
31: 14/1 4568.826 harmonic 28th
Octave normalized:
0: 1/1 0.000
1: 15/14 119.443
2: 12/11 150.637
3: 11/10 165.004
4: 10/9 182.404
5: 9/8 203.910 perfect 2nd
6: 7/6 266.871 minimal 3rd
7: 6/5 315.641 minor 3rd
8: 5/4 386.314 major 3rd
9: 4/3 498.045 perfect 4th
10: 11/8 551.318
11: 10/7 617.488
12: 3/2 701.955 perfect 5th
13: 11/7 782.492
14: 8/5 813.686 minor 6th
15: 13/8 840.528
16: 5/3 884.359 major 6th
17: 12/7 933.129 septimal major 6th
18: 7/4 968.826 harmonic 7th
19: 9/5 1017.596 minor 7th
20: 11/6 1049.363
21: 15/8 1088.269 classic major 7th
22: 2/1 1200.000 1 octave
__________________________
If we were to consider Durer's square wrapping around itself, we could expand the scale with new pitches:
0: 1/1 0.000
1: 16/15 111.731
2: 15/14 119.443
3: 14/13 128.298
4: 12/11 150.637
5: 11/10 165.004
6: 10/9 182.404
7: 9/8 203.910 perfect 2nd
8: 8/7 231.174 harmonic 2nd
9: 7/6 266.871 minimal 3rd
10: 6/5 315.641 minor 3rd
11: 16/13 359.472
12: 4/3 498.045 perfect 4th
13: 11/8 551.318
14: 10/7 617.488
15: 3/2 701.955 perfect 5th
16: 11/7 782.492
17: 8/5 813.686 minor 6th
18: 13/8 840.528
19: 5/3 884.359 major 6th
20: 12/7 933.129 septimal major 6th
21: 9/5 1017.596 minor 7th
22: 2/1 1200.000 1 octave
23: 15/7 1319.443
24: 9/4 1403.910 perfect 9th
25: 7/3 1466.871 minimal 10th
26: 12/5 1515.641 minor 10th
27: 5/2 1586.314 major 10th
28: 13/5 1654.214
29: 3/1 1901.955 perfect 12th
30: 16/5 2013.686 minor 13th
31: 13/4 2040.528
32: 10/3 2084.359 major 13th
33: 11/3 2249.363
34: 15/4 2288.269 classic major 14th
35: 4/1 2400.000 2 octaves
36: 14/3 2666.871 minimal 17th
37: 5/1 2786.314 major 17th
38: 16/3 2898.045 perfect 18th
39: 11/2 2951.318
40: 13/2 3240.528
41: 7/1 3368.826 harmonic 21st
42: 15/2 3488.269 classic major 21st
43: 9/1 3803.910 perfect 23rd
44: 12/1 4301.955 perfect 26th
45: 13/1 4440.528
46: 14/1 4568.826 harmonic 28th
47: 16/1 4800.000 4 octaves
Octave normalized:
0: 1/1 0.000
1: 16/15 111.731
2: 15/14 119.443
3: 14/13 128.298
4: 12/11 150.637
5: 11/10 165.004
6: 10/9 182.404
7: 9/8 203.910 perfect 2nd
8: 8/7 231.174 harmonic 2nd
9: 7/6 266.871 minimal 3rd
10: 6/5 315.641 minor 3rd
11: 16/13 359.472
12: 5/4 386.314 major 3rd
13: 13/10 454.214
14: 4/3 498.045 perfect 4th
15: 11/8 551.318
16: 10/7 617.488
17: 3/2 701.955 perfect 5th
18: 11/7 782.492
19: 8/5 813.686 minor 6th
20: 13/8 840.528
21: 5/3 884.359 major 6th
22: 12/7 933.129 septimal major 6th
23: 7/4 968.826 harmonic 7th
24: 9/5 1017.596 minor 7th
25: 11/6 1049.363
26: 15/8 1088.269 classic major 7th
27: 2/1 1200.000 1 octave
Looks neat.
Oz.
✩ ✩ ✩
Begin forwarded message:
Aaron Andrew Hunt
Feb 7, 2012, 1:46:32 PM2/7/12
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It's fun. But why this whimsical jargon "minimal", "classic", "harmonic" etc. to describe intervals?
<http://www.h-pi.com/wordpress/?p=4>
<http://www.h-pi.com/theory/huntintervalcalc.html>
<http://www.h-pi.com/wordpress/?p=4>
<http://www.h-pi.com/theory/huntintervalcalc.html>
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Ozan Yarman
Feb 7, 2012, 2:21:06 PM2/7/12
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