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Appendix A: COLLECTION OF TUNING ANALYSIS TABLES

In most web browsers you can listen to the intervals and tunings described here.  Wherever there is slight or distinct colour in a table you can click on the cell to hear a note or notes.  Right click (two finger tap on a MacBook, touch and hold on Android, etc.) for an alternative.  For intervals the alternate click plays a chord whilst the primary click plays a note sequence.  For single notes the alternate click may render the note in a different octave and for note sequences the sequence is reversed.  Try it!

This feature is known to work in Firefox and Chrome on the PC, MacBooks, and Android on a Samsung Galaxy.  Unfortunately, due to a lesser degree of audio support, it does not work in Internet Explorer, Safari on a PC or on older iPads.  You will need good quality headphones or speakers; all of the examples use pure tones and some use low frequencies so you may not hear it all on laptop speakers or inner ear headphones.

Advanced readers may like to know that the reference pitch for each page can be changed.  By default the reference pitch is A = 440 Hz but this can be changed by appending to the end of the URL in the address bar, for example ?A=432 or ?C=256.  Where a note other than A is used the offset from A will be calculated using equal temperament.  You need to do this for each page.

Technical specialists may wish to turn on the browser debug log to see the actual frequencies of notes being synthesized.


T 2.7.3 Intervals of Pythagorean Tuning - Ancient Greek Phrygian Mode (Descending)
pivot D C B A G F E D
D 1 / 1 9 / 16 16 / 27 2 / 3 3 / 4 27 / 32 8 / 9 1 / 2
C 8 / 9 1 / 1 128 / 243 16 / 27 2 / 3 3 / 4 64 / 81 8 / 9
B 27 / 32 243 / 256 1 / 1 9 / 16 81 / 128 729 / 1024 3 / 4 27 / 32
A 3 / 4 27 / 32 8 / 9 1 / 1 9 / 16 81 / 128 2 / 3 3 / 4
G 2 / 3 3 / 4 64 / 81 8 / 9 1 / 1 9 / 16 16 / 27 2 / 3
F 16 / 27 2 / 3 512 / 729 64 / 81 8 / 9 1 / 1 128 / 243 16 / 27
E 9 / 16 81 / 128 2 / 3 3 / 4 27 / 32 243 / 256 1 / 1 9 / 16
D 1 / 2 9 / 16 16 / 27 2 / 3 3 / 4 27 / 32 8 / 9 1 / 1

T 2.9.2 Intervals of the Pentatonic Scale
pivot D E G A C
D 1 / 1 16 / 9 3 / 2 4 / 3 9 / 8
E 9 / 8 1 / 1 27 / 16 3 / 2 81 / 64
G 4 / 3 32 / 27 1 / 1 16 / 9 3 / 2
A 3 / 2 4 / 3 9 / 8 1 / 1 27 / 16
C 16 / 9 128 / 81 4 / 3 32 / 27 1 / 1

T 3.6.2 Analysis of Pythagorean Tuning - Modern Ionian (major) mode (ascending)
pivot C D E F G A B C
C 1 / 1 16 / 9 128 / 81 3 / 2 4 / 3 32 / 27 256 / 243 2 / 1
D 9 / 8 1 / 1 16 / 9 27 / 16 3 / 2 4 / 3 32 / 27 9 / 8
E 81 / 64 9 / 8 1 / 1 243 / 128 27 / 16 3 / 2 4 / 3 81 / 64
F 4 / 3 32 / 27 256 / 243 1 / 1 16 / 9 128 / 81 1024 / 729 4 / 3
G 3 / 2 4 / 3 32 / 27 9 / 8 1 / 1 16 / 9 128 / 81 3 / 2
A 27 / 16 3 / 2 4 / 3 81 / 64 9 / 8 1 / 1 16 / 9 27 / 16
B 243 / 128 27 / 16 3 / 2 729 / 512 81 / 64 9 / 8 1 / 1 243 / 128
C 2 / 1 16 / 9 128 / 81 3 / 2 4 / 3 32 / 27 256 / 243 1 / 1

T 4.2.3 Analysis of Early Pythagorean Chromatic Tuning (Eb x G#)
pivot C C# D Eb E F F# G G# A Bb B C
C 1 / 1 4096 / 2187 16 / 9 27 / 16 128 / 81 3 / 2 1024 / 729 4 / 3 8192 / 6561 32 / 27 9 / 8 256 / 243 2 / 1
C# 2187 / 2048 1 / 1 243 / 128 59049 / 32768 27 / 16 6561 / 4096 3 / 2 729 / 512 4 / 3 81 / 64 19683 / 16384 9 / 8 2187 / 2048
D 9 / 8 256 / 243 1 / 1 243 / 128 16 / 9 27 / 16 128 / 81 3 / 2 1024 / 729 4 / 3 81 / 64 32 / 27 9 / 8
Eb 32 / 27 65536 / 59049 256 / 243 1 / 1 4096 / 2187 16 / 9 32768 / 19683 128 / 81 262144 / 177147 1024 / 729 4 / 3 8192 / 6561 32 / 27
E 81 / 64 32 / 27 9 / 8 2187 / 2048 1 / 1 243 / 128 16 / 9 27 / 16 128 / 81 3 / 2 729 / 512 4 / 3 81 / 64
F 4 / 3 8192 / 6561 32 / 27 9 / 8 256 / 243 1 / 1 4096 / 2187 16 / 9 32768 / 19683 128 / 81 3 / 2 1024 / 729 4 / 3
F# 729 / 512 4 / 3 81 / 64 19683 / 16384 9 / 8 2187 / 2048 1 / 1 243 / 128 16 / 9 27 / 16 6561 / 4096 3 / 2 729 / 512
G 3 / 2 1024 / 729 4 / 3 81 / 64 32 / 27 9 / 8 256 / 243 1 / 1 4096 / 2187 16 / 9 27 / 16 128 / 81 3 / 2
G# 6561 / 4096 3 / 2 729 / 512 177147 / 131072 81 / 64 19683 / 16384 9 / 8 2187 / 2048 1 / 1 243 / 128 59049 / 32768 27 / 16 6561 / 4096
A 27 / 16 128 / 81 3 / 2 729 / 512 4 / 3 81 / 64 32 / 27 9 / 8 256 / 243 1 / 1 243 / 128 16 / 9 27 / 16
Bb 16 / 9 32768 / 19683 128 / 81 3 / 2 1024 / 729 4 / 3 8192 / 6561 32 / 27 65536 / 59049 256 / 243 1 / 1 4096 / 2187 16 / 9
B 243 / 128 16 / 9 27 / 16 6561 / 4096 3 / 2 729 / 512 4 / 3 81 / 64 32 / 27 9 / 8 2187 / 2048 1 / 1 243 / 128
C 2 / 1 4096 / 2187 16 / 9 27 / 16 128 / 81 3 / 2 1024 / 729 4 / 3 8192 / 6561 32 / 27 9 / 8 256 / 243 1 / 1

T 4.2.5 Analysis of Late Pythagorean Chromatic Tuning (F# x B)
pivot C Db D Eb E F Gb G Ab A Bb B C
C 1 / 1 243 / 256 16 / 9 27 / 16 128 / 81 3 / 2 729 / 512 4 / 3 81 / 64 32 / 27 9 / 8 256 / 243 2 / 1
Db 256 / 243 1 / 1 4096 / 2187 16 / 9 32768 / 19683 128 / 81 3 / 2 729 / 512 4 / 3 8192 / 6561 32 / 27 65536 / 59049 256 / 243
D 9 / 8 2187 / 2048 1 / 1 243 / 128 16 / 9 27 / 16 6561 / 4096 3 / 2 729 / 512 4 / 3 81 / 64 32 / 27 9 / 8
Eb 32 / 27 9 / 8 256 / 243 1 / 1 4096 / 2187 16 / 9 27 / 16 128 / 81 3 / 2 1024 / 729 4 / 3 8192 / 6561 32 / 27
E 81 / 64 19683 / 16384 9 / 8 2187 / 2048 1 / 1 243 / 128 59049 / 32768 27 / 16 6561 / 4096 3 / 2 729 / 512 4 / 3 81 / 64
F 4 / 3 81 / 64 32 / 27 9 / 8 256 / 243 1 / 1 243 / 128 16 / 9 27 / 16 128 / 81 3 / 2 1024 / 729 4 / 3
Gb 1024 / 729 4 / 3 8192 / 6561 32 / 27 65536 / 59049 256 / 243 1 / 1 4096 / 2187 16 / 9 32768 / 19683 128 / 81 262144 / 177147 1024 / 729
G 3 / 2 729 / 512 4 / 3 81 / 64 32 / 27 9 / 8 2187 / 2048 1 / 1 243 / 128 16 / 9 27 / 16 128 / 81 3 / 2
Ab 128 / 81 3 / 2 1024 / 729 4 / 3 8192 / 6561 32 / 27 9 / 8 256 / 243 1 / 1 4096 / 2187 16 / 9 32768 / 19683 128 / 81
A 27 / 16 6561 / 4096 3 / 2 729 / 512 4 / 3 81 / 64 19683 / 16384 9 / 8 2187 / 2048 1 / 1 243 / 128 16 / 9 27 / 16
Bb 16 / 9 27 / 16 128 / 81 3 / 2 1024 / 729 4 / 3 81 / 64 32 / 27 9 / 8 256 / 243 1 / 1 4096 / 2187 16 / 9
B 243 / 128 59049 / 32768 27 / 16 6561 / 4096 3 / 2 729 / 512 177147 / 131072 81 / 64 19683 / 16384 9 / 8 2187 / 2048 1 / 1 243 / 128
C 2 / 1 243 / 128 16 / 9 27 / 16 128 / 81 3 / 2 729 / 512 4 / 3 81 / 64 32 / 27 9 / 8 256 / 243 1 / 1

T 4.3.2 Analysis of Just Diatonic Tuning
pivot C D E F G A B C
C 1 / 1 16 / 9 8 / 5 3 / 2 4 / 3 6 / 5 16 / 15 1 / 1
D 9 / 8 1 / 1 9 / 5 27 / 16 3 / 2 27 / 20 6 / 5 9 / 8
E 5 / 4 10 / 9 1 / 1 15 / 8 5 / 3 3 / 2 4 / 3 5 / 4
F 4 / 3 32 / 27 16 / 15 1 / 1 16 / 9 8 / 5 64 / 45 4 / 3
G 3 / 2 4 / 3 6 / 5 9 / 8 1 / 1 9 / 5 8 / 5 3 / 2
A 5 / 3 40 / 27 4 / 3 5 / 4 10 / 9 1 / 1 16 / 9 5 / 3
B 15 / 8 5 / 3 3 / 2 45 / 32 5 / 4 9 / 8 1 / 1 15 / 8
C 2 / 1 16 / 9 8 / 5 3 / 2 4 / 3 6 / 5 16 / 15 1 / 1

T 4.4.2 Analysis of Just Chromatic Tuning
pivot C Db D Eb E F F# G Ab A Bb B C
C 1 / 1 15 / 8 16 / 9 5 / 3 8 / 5 3 / 2 64 / 45 4 / 3 5 / 4 6 / 5 10 / 9 16 / 15 1 / 1
Db 16 / 15 1 / 1 256 / 135 16 / 9 128 / 75 8 / 5 1024 / 675 64 / 45 4 / 3 32 / 25 32 / 27 256 / 225 16 / 15
D 9 / 8 135 / 128 1 / 1 15 / 8 9 / 5 27 / 16 8 / 5 3 / 2 45 / 32 27 / 20 5 / 4 6 / 5 9 / 8
Eb 6 / 5 9 / 8 16 / 15 1 / 1 48 / 25 9 / 5 128 / 75 8 / 5 3 / 2 36 / 25 4 / 3 32 / 25 6 / 5
E 5 / 4 75 / 64 10 / 9 25 / 24 1 / 1 15 / 8 16 / 9 5 / 3 25 / 16 3 / 2 25 / 18 4 / 3 5 / 4
F 4 / 3 5 / 4 32 / 27 10 / 9 16 / 15 1 / 1 256 / 135 16 / 9 5 / 3 8 / 5 40 / 27 64 / 45 4 / 3
F# 45 / 32 675 / 512 5 / 4 75 / 64 9 / 8 135 / 128 1 / 1 15 / 8 225 / 128 27 / 16 25 / 16 3 / 2 45 / 32
G 3 / 2 45 / 32 4 / 3 5 / 4 6 / 5 9 / 8 16 / 15 1 / 1 15 / 8 9 / 5 5 / 3 8 / 5 3 / 2
Ab 8 / 5 3 / 2 64 / 45 4 / 3 32 / 25 6 / 5 256 / 225 16 / 15 1 / 1 48 / 25 16 / 9 128 / 75 8 / 5
A 5 / 3 25 / 16 40 / 27 25 / 18 4 / 3 5 / 4 32 / 27 10 / 9 25 / 24 1 / 1 50 / 27 16 / 9 5 / 3
Bb 9 / 5 27 / 16 8 / 5 3 / 2 36 / 25 27 / 20 32 / 25 6 / 5 9 / 8 27 / 25 1 / 1 48 / 25 9 / 5
B 15 / 8 225 / 128 5 / 3 25 / 16 3 / 2 45 / 32 4 / 3 5 / 4 75 / 64 9 / 8 25 / 24 1 / 1 15 / 8
C 2 / 1 15 / 8 16 / 9 5 / 3 8 / 5 3 / 2 64 / 45 4 / 3 5 / 4 6 / 5 10 / 9 16 / 15 1 / 1

T 5.5.1 Analysis of Quarter Comma Mean Tone Temperament
pivot C Db D Eb E F F# G G# A A# B C
C 0 1083 1007 890 814 697 621 503 427 310 234 117 1200
Db 117 0 1124 1007 931 814 738 620 544 427 351 234 117
D 193 76 0 1083 1007 890 814 696 620 503 427 310 193
Eb 310 193 117 0 1124 1007 931 813 737 620 544 427 310
E 386 269 193 76 0 1083 1007 889 813 696 620 503 386
F 503 386 310 193 117 0 1124 1006 930 813 737 620 503
F# 579 462 386 269 193 76 0 1082 1006 889 813 696 579
G 697 580 504 387 311 194 118 0 1124 1007 931 814 697
G# 773 656 580 463 387 270 194 76 0 1083 1007 890 773
A 890 773 697 580 504 387 311 193 117 0 1124 1007 890
A# 966 849 773 656 580 463 387 269 193 76 0 1083 966
B 1083 966 890 773 697 580 504 386 310 193 117 0 1083
C 1200 1083 1007 890 814 697 621 503 427 310 234 117 0

T 5.8.2 Analysis of Andreas Werckmeister Temperament III
pivot C Db D Eb E F Gb G Ab A Bb B C
C 0 1110 1008 906 810 702 612 504 408 312 204 108 0
Db 90 0 1098 996 900 792 702 594 498 402 294 198 90
D 192 102 0 1098 1002 894 804 696 600 504 396 300 192
Eb 294 204 102 0 1104 996 906 798 702 606 498 402 294
E 390 300 198 96 0 1092 1002 894 798 702 594 498 390
F 498 408 306 204 108 0 1110 1002 906 810 702 606 498
Gb 588 498 396 294 198 90 0 1092 996 900 792 696 588
G 696 606 504 402 306 198 108 0 1104 1008 900 804 696
Ab 792 702 600 498 402 294 204 96 0 1104 996 900 792
A 888 798 696 594 498 390 300 192 96 0 1092 996 888
Bb 996 906 804 702 606 498 408 300 204 108 0 1104 996
B 1092 1002 900 798 702 594 504 396 300 204 96 0 1092
C 1200 1110 1008 906 810 702 612 504 408 312 204 108 0

T 5.9.2 Analysis of Equal Temperament
pivot C Db D Eb E F Gb G Ab A Bb B C
C 0 1100 1000 900 800 700 600 500 400 300 200 100 0
Db 100 0 1100 1000 900 800 700 600 500 400 300 200 100
D 200 100 0 1100 1000 900 800 700 600 500 400 300 200
Eb 300 200 100 0 1100 1000 900 800 700 600 500 400 300
E 400 300 200 100 0 1100 1000 900 800 700 600 500 400
F 500 400 300 200 100 0 1100 1000 900 800 700 600 500
Gb 600 500 400 300 200 100 0 1100 1000 900 800 700 600
G 700 600 500 400 300 200 100 0 1100 1000 900 800 700
Ab 800 700 600 500 400 300 200 100 0 1100 1000 900 800
A 900 800 700 600 500 400 300 200 100 0 1100 1000 900
Bb 1000 900 800 700 600 500 400 300 200 100 0 1100 1000
B 1100 1000 900 800 700 600 500 400 300 200 100 0 1100
C 1200 1100 1000 900 800 700 600 500 400 300 200 100 0