Appendix C: VISUALIZATION OF TUNING INTERVALS AS LISSAJOUS FIGURES

This appendix provides visualizations of 4 well known tunings as Lissajous figures.  These figures are produced from two perpendicular harmonic motions.  By representing a tuning ratio such as 3 / 2, a perfect fifth, with a figure having the relative frequencies 3 and 2 a figure depicting the simplicity or complexity of the ratio is obtained.  Purer intervals, those with lower numbers in their ratios, result in more simple curves.

In the case of the tempered tunings the figures move, illustrating the instability of these non ratiometric tunings.  For the purposes of illustration and comparison one of the relative frequencies is taken as the denominator of one of the ratiometric tunings and the other relative frequency is the tempered tuning multiplied by that factor.

As with other pages in this essay you can listen to the intervals.  Left click to hear the intervals as a sequence of notes and right click for a chord (you need decent speakers or headphones).  All the tunings are relative to A = 440 Hz (unless you changed it as described in Appendix B) and A is the major 6th degree of these tunings.  Consequently other notes, including the key note, may differ slightly amongst the tunings.

Tunings to show:

Pythagorean Chromatic (F# x B)

Just Chromatic

Quarter Comma Mean Tone Temperament

Equal Temperament

Ratiometric basis for tempered tunings

Pythagorean
Just

Stop animations to conserve power

T C 1 Visualization of Pythagorean (F# x B) tuning as Lissajous Figures

Pythagorean (F# x B)

1 / 1 Unison

256 / 243 Minor second

9 / 8 Major second

32 / 27 Minor third

81 / 64 Major third

4 / 3 Perfect fourth

1024 / 729 Tritone

3 / 2 Perfect fifth

128 / 81 Minor sixth

27 / 16 Major sixth

16 / 9 Minor seventh

243 / 128 Major seventh

2 / 1 Octave

T C 2 Visualization of Just tuning as Lissajous Figures

Just

1 / 1 Unison

16 / 15 Minor second

9 / 8 Major second

6 / 5 Minor third

5 / 4 Major third

4 / 3 Perfect fourth

45 / 32 Tritone

3 / 2 Perfect fifth

8 / 5 Minor sixth

5 / 3 Major sixth

9 / 5 Minor seventh

15 / 8 Major seventh

2 / 1 Octave

T C 3 Visualization of Quarter Comma Mean Tone tuning as Lissajous Figures

Quarter Comma Mean Tone

1 Unison

8 / 55/4 Minor second

52/4 / 2 Major second

4 / 53/4 Minor third

54/4 / 4 Major third

2 / 51/4 Perfect fourth

56/4 / 8 Tritone

51/4 Perfect fifth

58/4 / 16 Minor sixth

53/4 / 2 Major sixth

510/4 / 32 Minor seventh

55/4 / 4 Major seventh

2 Octave

T C 4 Visualization of Equal Temperament tuning as Lissajous Figures

Equal Temperament

0 cents Unison

100 cents Minor second

200 cents Major second

300 cents Minor third

400 cents Major third

500 cents Perfect fourth

600 cents Tritone

700 cents Perfect fifth

800 cents Minor sixth

900 cents Major sixth

1000 cents Minor seventh

1100 cents Major seventh

1200 cents Octave

In the ratiometric tunings (Pythagorean and Just) the interval of a tone (major second) is the same in both tunings, a relatively pure interval.  The Pythagorean thirds, of less importance in ancient Greek music, are less satisfactory.  The relative purity of the Just major third is striking.

In the tempered tunings (Mean tone and Equal tempered) the stability of the Mean tone major third is striking.  The slower moving figures for the fourths and fifths of these tunings illustrates that they are quite close to the ratiometric intervals, Equal Temperament more so than the Quarter Comma Mean Tone.