Reflected sunset

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Appendix B: COLLECTION OF EXPLANATORY 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 1.1.1 A pure tone with a frequency of 110 cycles per second
A musical note
(click me)

T 1.1.2 Beating sound between two similar frequencies
Two similar notes
(right click)

T 1.2.1 A series of harmonically related tones
First harmonic F (Fundamental) Second harmonic 2F Third harmonic 3F Fourth harmonic 4F Fifth harmonic 5F Sixth harmonic 6F

T 1.3.1 Perception of pitch intervals is related to ratios between frequencies
16 cycles per second 32 cycles per second 64 cycles per second ... 1024 cycles per second 2048 cycles per second 4096 cycles per second

T 1.3.2 The interval of an octave
F 2F
octave
(right click for sequence, left click for chord)

T 1.4.1 Notes of the same pitch class
F 2F 4F 8F

T 1.5.1 Transposing up an octave
fundamental F octave 2F
interval 2 / 1

T 1.5.2 Transposing down an octave
octave 2F fundamental F
interval 1 / 2

T 1.7.1 Derivation of a fifth from the third harmonic
F 2F 3F
twelfth 3 / 1
octave 2 / 1
fifth 3 / 2

T 1.8.1 Subtracting a fifth from an octave to yield a fourth
octave 2 / 1
fifth 3 / 2 fourth 4 / 3

T 1.8.2 A fourth exists between the 3rd and 4th harmonics
3F 4F
fourth 4 / 3

T 1.9.1 A major third exists between the 4th and 5th harmonics
4F 5F
major third 5 / 4

T 1.9.2 A minor third exists between the 5th and 6th harmonics
5F 6F
minor third 6 / 5

T 1.9.3 A major third and a minor third constitute a fifth
major third 5 / 4 minor third 6 / 5
fifth 3 / 2

T 2.3.1 Interpretation of detail from School of Athens
TONOS
VI VIII VIIII XII
DIATESSARON DIATESSARON
DIAPENTE
DIAPENTE
DIAPASON

T 2.4.1 Greater Perfect System
TETRACHORD TETRACHORD
A G F E D C B A G F E D C B A
TETRACHORD TETRACHORD


T 2.5.1 Ancient Greek Modes
E D CB A G FE Dorian
D CB A G FE D Phrygian
CB A G FE D C Lydian
B A G FE D CB Mixolydian
A G FE D CB A Hypodorian
G FE D CB A G Hypophrygian
FE D CB A G F Hypolydian

T 2.6.1 Intervals of the Greater Perfect System
TETRACHORD TETRACHORD
A G F E D C B A G F E D C B A
16 12 9 8 6 4
TETRACHORD TETRACHORD

T 2.7.1 Descending and Ascending Fifths
Original fifth Transposed fifth
27 / 8 27 / 32
9 / 4 9 / 16
3 / 2 3 / 4
1 / 1 1 / 1
2 / 3 2 / 3
4 / 9 8 / 9
8 / 27 16 / 27
T 2.7.2 Sorted Descending and Ascending Fifths
Sorted fifth Note name Interval to next
1 / 1 D 8 / 9
8 / 9 C 243 / 256
27 / 32 B 8 / 9
3 / 4 A 8 / 9
2 / 3 G 8 / 9
16 / 27 F 243 / 256
9 / 16 E 8 / 9
1 / 2 D 8 / 9

T 2.9.1 Pentatonic Scale
Sorted fifth Step in sequence of fifths Note name Interval to next
1 / 1 0 D 9 / 8
9 / 8 2 E 32 / 27
4 / 3 -1 G 9 / 8
3 / 2 1 A 32 / 27
16 / 9 -2 C 9 / 8

T 3.2.1 Series of Fifths
Original fifth Octaves to drop Transposed fifth
1 / 1 0 1 / 1
3 / 2 0 3 / 2
9 / 4 1 9 / 8
27 / 8 1 27 / 16
81 / 16 2 81 / 64
243 / 32 2 243 / 128
729 / 64 3 729 / 512
T 3.2.2 Sorted Series of Fifths
Sorted fifth Note name Interval to next
1 / 1 F 9 / 8
9 / 8 G 9 / 8
81 / 64 A 9 / 8
729 / 512 B 256 / 243
3 / 2 C 9 / 8
27 / 16 D 9 / 8
243 / 128 E 256 / 243
2 / 1 F 9 / 8

T 3.3.1 Medieval Modes
Authentic Plagal
I Dorian D EF G A BC D II Hypodorian A BC D EF G A
III Phrygian EF G A BC D E IV Hypophrygian BC D EF G A B
V Lydian F G A BC D EF VI Hypolydian C D EF G A BC
VII Mixolydian G A BC D EF G VIII Hypomixolydian D EF G A BC D

T 3.5.1 Extended Modes
Authentic Plagal
IX Aeolian A BC D EF G A X Hypoaeolian EF G A BC D E
( Locrian ) ( BC D EF G A B ) ( Hypolochrian ) ( F G A BC D EF )
XI Ionian C D EF G A BC XII Hypoionian G A BC D EF G

T 3.6.1 Diatonic Major Scale - Pythagorean Tuning
Original fifth
(was called)		 Transposed fifth
(now called)		 Note name Interval to next step
3 / 2 1 / 1 C 9 / 8 tetrachord
27 / 16 9 / 8 D 9 / 8
243 / 128 81 / 64 E 256 / 243
1 / 1 4 / 3 F 9 / 8
9 / 8 3 / 2 G 9 / 8 tetrachord
81 / 64 27 / 16 A 9 / 8
729 / 512 243 / 128 B 256 / 243
2 / 1 C 9 / 8

T 4.2.1 Pythagorean Diatonic Tuning with Added Bb and F#
Transposed fifth Note name Interval to next step
1 / 1 C 9 / 8
9 / 8 D 9 / 8
81 / 64 E 256 / 243
4 / 3 F 2187 / 2048
729 / 512 F# 256 / 243
3 / 2 G 9 / 8
27 / 16 A 256 / 243
16 / 9 Bb 2187 / 2048
243 / 128 B 256 / 243
2 / 1 C 9 / 8

T 4.2.2 Early Pythagorean Chromatic Tuning (Eb x G#)
Transposed fifth Note name Interval to next step
1 / 1 C 2187 / 2048
2187 / 2048 C# 256 / 243
9 / 8 D 256/ 243
32 / 27 Eb (D#) 2187 / 2048
81 / 64 E 256 / 243
4 / 3 F 2187 / 2048
729 / 512 F# 256 / 243
3 / 2 G 2187 2048
6561 / 4096 G# 256 / 243
27 / 16 A 256 / 243
16 / 9 Bb 2187 / 2048
243 / 128 B 256 / 243
2 / 1 C 256 / 243

T 4.2.4 Late Pythagorean Chromatic Tuning (F# x B)
Transposed fifth Note name Interval to next step
1 / 1 C 256 / 243
256 / 243 Db 2187 / 2048
9 / 8 D 256/ 243
32 / 27 Eb 2187 / 2048
81 / 64 E 256 / 243
4 / 3 F 256 / 243
1024 / 729 Gb (F#) 2187 / 2048
3 / 2 G 256 / 243
128 / 81 Ab 2187 / 2048
27 / 16 A 256 / 243
16 / 9 Bb 2187 / 2048
243 / 128 B 256 / 243
2 / 1 C 256 / 243

T 4.3.1 Just Tuning - Zarlino
Interval from tonic Note name Interval to next step Zarlino's harmonic series
1 / 1 C 9 / 8 180
9 / 8 D 10 / 9 160
5 / 4 E 16 / 15 144
4 / 3 F 9 / 8 135
3 / 2 G 10 / 9 120
5 / 3 A 9 / 8 108
15 / 8 B 16 / 15 96
2 / 1 C 9 / 8 90

T 4.4.1 Chromatic Just Tuning
Interval from tonic Note name Interval to next step
1 / 1 C 16 / 15
16 / 15 Db 135 / 128
9 / 8 D 16 / 15
6 / 5 Eb 25 / 24
5 / 4 E 16 / 15
4 / 3 F 135 / 128
45 / 32 F# 16 / 15
3 / 2 G 16 / 15
8 / 5 Ab 25 / 24
5 / 3 A 27 / 25
9 / 5 Bb 25 / 24
15 / 8 B 16 / 15
2 / 1 C 16 / 15

T 4.6.1 Sestina
1 : 2 : 3 : 4 : 5 : 6

T 5.1.1 Extended Series of Fifths
Index Original fifth Octaves to drop Transposed fifth Approximate decimal value
1 1 / 1 0 1 / 1 1
2 3 / 2 0 3 / 2 1.5
3 9 / 4 1 9 / 8 1.125
4 27 / 8 1 27 / 16 1.6875
5 81 / 16 2 81 / 64 1.265625
6 243 / 32 2 243 / 128 1.898437
7 729 / 64 3 729 / 512 1.423828
8 2187 / 128 4 2187 / 2048 1.067871
9 6561 / 256 4 6561 / 4096 1.601806
10 19683 / 512 5 19683 / 16384 1.201354
11 59049 / 1024 5 59049 / 32768 1.802032
12 177147 / 2048 6 177147 / 131072 1.351524
13 531441 / 4096 7 531441 / 524288 1.013643

T 5.1.2 Sorted Series of Fifths
Index Transposed fifth Approximate decimal value Interval to next step Approximate decimal step size Note name
1 1 / 1 1 2187 / 2048 1.067871 F
8 2187 / 2048 1.067871 256 / 243 1.053498 F#
3 9 / 8 1.125 2187 / 2048 1.067871 G
10 19683 / 16384 1.201354 256 / 243 1.053498 G#
5 81 / 64 1.26562 2187 / 2048 1.067871 A
12 177147 / 131072 1.351524 256 / 243 1.053498 A#
7 729 / 512 1.423828 256 / 243 1.053498 B
2 3 / 2 1.5 2187 / 2048 1.067871 C
9 6561 / 4096 1.601806 256 / 243 1.053498 C#
4 27 / 16 1.6875 2187 / 2048 1.067871 D
11 59049 / 32768 1.802032 256 / 243 1.053498 D#
6 243 / 128 1.898437 256 / 243 1.053498 E

T 5.2.1 Quarter Comma Mean Tone Temperament
Note name Place in series Interval from tonic using quarter comma Interval from tonic using reduced fifth Approximate decimal value
C 0 1 / 1 1 / 1 1
D 2 9/8 x (80/81)2/4 52/4 / 2 1.118034
E 4 81/64 x (80/81)4/4 = 5/4 54/4 / 4 1.25
F -1 4/3 x (81/80)1/4 2 / 51/4 1.337481
G 1 3/2 x (80/81)1/4 51/4 1.495349
A 3 27/16 x (80/81)3/4 53/4 / 2 1.671851
B 5 243/128 x (80/81)5/4 55/4 / 4 1.869186
C - 2 / 1 2 / 1 2

T 5.2.2 Chromatic Quarter Comma Mean Tone Temperament
Note name Place in series Interval from tonic using quarter comma Interval from tonic using reduced fifth Approximate decimal value
C 0 1 / 1 1 / 1 1
C# 7 2187/2048 x (80/81)7/4 57/4 / 16 1.044907
Db -5 256/243 x (81/80)5/4 8 / 55/4 1.069984
D 2 9/8 x (80/81)2/4 52/4 / 2 1.118034
D# 9 19683/16384 x (80/81)9/4 59/4 / 32 1.168241
Eb -3 32/27 x (81/80)3/4 4 / 53/4 1.196279
E 4 81/64 x (80/81)4/4 = 5/4 54/4 / 4 1.25
F -1 4/3 x (81/80)1/4 2 / 51/4 1.337481
F# 6 729/512 x (80/81)6/4 56/4 / 8 1.397542
Gb -6 1024/729 x (81/80)6/4 16 / 56/4 1.431084
G 1 3/2 x (80/81)1/4 51/4 1.495349
G# 8 6561/4096 x (80/81)8/4 = 25/16 58/4 / 16 1.5625
Ab -4 128/81 x (81/80)4/4 8 / 54/4 1.6
A 3 27/16 x (80/81)3/4 53/4 / 2 1.671851
A# 10 59049/32768 x (80/81)10/4 510/4 / 32 1.746928
Bb -2 16/9 x (81/80)2/4 4 / 52/4 1.788854
B 5 243/128 x (80/81)5/4 55/4 / 4 1.869186
C - 2 / 1 2 / 1 2

T 5.4.1 Common Tuning Intervals
Interval Ratio Approximate decimal value Measure to nearest cent
Octave 2 / 1 2 1200
Harmonic fifth 3 / 2 1.5 702
Harmonic fourth 4 / 3 1.333333 498
Pythagorean major third 81 / 64 1.265625 408
Just major third 5 / 4 1.25 386
Pythagorean minor third 32 / 27 1.185185 294
Just minor third 6 / 5 1.2 316
Tone 9 / 8 1.125 204
Just minor tone 10 / 9 1.111111 182
Pythagorean chromatic semitone 2187 / 2048 1.067871 113
Just diatonic semitone 16 / 15 1.066667 112
Pythagorean diatonic semitone 256 / 243 1.053498 90
Pythagorean comma 531441 / 524288 1.013643 23
Syntonic comma 81 / 80 1.0125 22
Quarter Pythagorean comma (531441 / 524288)1/4 1.003394 6
Quarter syntonic comma (81/80)1/4 1.003110 5

T 5.6.1 Introduction of Sharps
Tonic Dominant Leading note
C G B
G D F#
D A C#
A E G#
E B D#
B F# A#

T 5.6.2 Introduction of Flats
Tonic Subdominant
C F
F Bb
Bb Eb
Eb Ab
Ab Db
Db Gb

T 5.7.1 Intervals and their Inversions
Interval Semitone steps Inversion
Unison 0 Octave
Minor second 1 Major seventh
Major second 2 Minor seventh
Minor third 3 Major sixth
Major third 4 Minor sixth
Perfect fourth 5 Perfect fifth
Tritone 6 Tritone
Perfect fifth 7 Perfect fourth
Minor sixth 8 Major third
Major sixth 9 Minor third
Minor seventh 10 Major second
Major seventh 11 Minor second
Octave 12 Unison

T 5.8.1 Andreas Werckmeister Temperament III
Note name Position in series Interval from tonic Approximate decimal value Measure to nearest cent
C 0 1 / 1 1 0
Db . 256 / 243 1.053498 90
D 2 9/8 x 1/P2/4 1.117403 192
Eb . 32 / 27 1.185185 294
E 4 81/64 x 1/P3/4 1.252827 390
F . 4 / 3 1.333333 498
Gb . 1024 / 729 1.404664 588
G 1 3/2 x 1/P1/4 1.494927 696
Ab . 128 / 81 1.580247 792
A 3 27/16 x 1/P3/4 1.670436 888
Bb . 16 / 9 1.777778 996
B 5 243/128 x 1/P3/4 1.879241 1092
C . 2 / 1 2 1200

T 5.9.1 Equal Temperament
Note name Interval from tonic Exact tuning in cents relative to first note
C 1 / 1 0
C# / Db 1 / 21/12 100
D 1 / 22/12 200
D# / Eb 1 / 23/12 300
E 1 / 24/12 400
F 1 / 25/12 500
F# / Gb 1 / 26/12 600
G 1 / 27/12 700
G# / Ab 1 / 28/12 800
A 1 / 29/12 900
A# / Bb 1 / 210/12 1000
B 1 / 211/12 1100
C 1 / 212/12 1200

T 5.10.1 Numerical Comparison of Tuning Systems
Note name Equal temperament Pythagorean (F# x B) Just
cents decimal ratio cents decimal ratio cents decimal
C 0 1.0 1 / 1 0 1.0 1 / 1 0 1.0
C# / Db 100 1.0594 256 / 243 90.22 1.0535 16 / 15 111.73 1.0667
D 200 1.1225 9 / 8 203.91 1.125 9 / 8 203.91 1.125
D# / Eb 300 1.1892 32 / 27 294.13 1.1852 6 / 5 315.64 1.2
E 400 1.2599 81 / 64 407.82 1.2656 5 / 4 386.31 1.25
F 500 1.3348 4 / 3 498.04 1.3333 4 / 3 498.04 1.3333
F# / Gb 600 1.4142 1024 / 729 588.27 1.4047 45 / 32 590.22 1.4063
G 700 1.4983 3 / 2 701.96 1.5 3 / 2 701.96 1.5
G# / Ab 800 1.5874 128 / 81 792.18 1.5802 8 / 5 813.69 1.6
A 900 1.6818 27 / 16 905.87 1.6875 5 / 3 884.36 1.6667
A# / Bb 1000 1.7818 16 / 9 996.09 1.7778 9 / 5 1017.60 1.8
B 1100 1.8877 243 / 128 1109.78 1.8984 15 / 8 1088.27 1.875
C 1200 2.0 2 / 1 1200.00 2.0 2 / 1 1200.00 2.0

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Contents
Tables
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