Just Me Too

A few extra thoughts on musical scale systems.

If you want to play a melody, you need a scale with notes that are more-or-less evenly spaced along the exponential curve of subjective pitch.

For example, here's a scale where the frequency of each note is one and one eighth of the previous:

Note 1: 100Hz
Note 2: 112.5Hz
Note 3: 126.563Hz
Note 4: 142.383Hz
Note 5: 160.181Hz
Note 6: 180.202Hz
Note 7: 202.729Hz

If you want to play harmonies and chords, you need a scale whose notes have simple mathematical relationships.

For example, here's one where all the notes are fractional fifths, quarters and thirds above the base frequency:

Note 1: 100Hz = 100*1/1 [unison]
Note 2: 120Hz = 100*6/5 [one and one fifth]
Note 3: 125Hz = 100*5/4 [one and a quarter]
Note 4: 140Hz = 100*7/5 [one and two fifths]
Note 5: 150Hz = 100*3/2 [one and a half]
Note 6: 160Hz = 100*8/5 [one and three fifths]
Note 7: 175Hz = 100*7/4 [one and three quarters]
Note 8: 180Hz = 100*9/5 [one and four fifths]
Note 9: 200Hz = 100*2/1 [double]

If you want to play both chords and melody, you need to find a compromise - a system of simple ratios that approximates a linear scale closely enough to enable melodies, or conversely a linear scale that approximates a ratio-based one.

The ratio-based scale above produces some very pleasant chords, but it's almost impossible to play a tune on it. There's a big jump between the 8th and 9th notes, but almost no jump between the 7th and 8th, or the 1st and 2nd. The spaces are fairly even in the middle of the scale, but are badly "squashed" at either end.

Conversely, the linear scale I give barely contains any notes that sound pleasant together, and doesn't even have an exact octave relation.

The "Equal Temperament" linear system used over most of the western world is a pretty good compromise, but it is still a compromise. If we're playing in C, then in the tonic chord C-E-G, the E and G are both slightly sharp, the G and G# are squeezed together, and the gap between F and G is oddly wide. Also it contains no approximation of the 7/4 ratio.

So, I've been thinking. Given that western music relies more on harmony, counterpoint and chords than on melody, might it be better to use a ratio-based system that approximates a linear one, rather than the other way around?

And also, could there be a different ratio-based system specifically for making chord-centric ambient music, where there may not even be a melody line?

For the former, I've come up with the table below.

The figures in square brackets are the note measured in cents - 100ths of a semitone, as defined in Equal Temperament - and the deviation of the note from the ET equivalent. These numbers can be used in retuning synthesisers that have "microtuning" capabilities.

Note 1: 1/1 = 1 [0,0]
Note 2: 17/16 = 1.0625 [104.995, -4.995]
Note 3: 9/8 = 1.125 [203.91, -3.91]
Note 4: 6/5 = 1.2 [315.641, -15.641]
Note 5: 5/4 = 1.25 [386.314, +13.686]
Note 6: 4/3 = 1.33333 [498.045, +1.955]
Note 7: 7/5 = 1.4 [582.512, +17.488]
Note 8: 3/2 = 1.5 [701.955, -1.955]
Note 9: 8/5 = 1.6 [813.686, +13.686]
Note 10: 5/3 = 1.66667 [884.359, +15.641]
Note 11: 9/5 = 1.8 [1017.596, -17.596]
Note 12: 15/8 = 1.875 [1088.269, +11.731]
Note 13: 2/1 = 2 [1200, 0]

As you can see, ET notes deviate by up to 17.6 cents from the "ideal" - that's nearly a sixth of a semitone. That may not sound like much, but it's enough to give a harsh edge to chords - which is there is probably all the music you hear, including the most relaxing.

Here's the numbers for the "ambient scale", which is designed purely for chords, and doesn't conform to the familiar layout of intervals:

Note 1: 1/1 = 1 [0, 0]
Note 2: 7/6 = 1.16667 [266.871, -166.871]
Note 3: 6/5 = 1.2 [315.641, -115.641]
Note 4: 5/4 = 1.25 [386.314, -86.314]
Note 5: 4/3 = 1.33333 [498.045, -98.045]
Note 6: 7/5 = 1.4 [582.512, -82.512]
Note 7: 3/2 = 1.5 [701.955, -101.955]
Note 8: 8/5 = 1.6 [813.868, -113.868]
Note 9: 5/3 = 1.66667 [884.359, -84.359]
Note 10: 7/4 = 1.75 [986.826, -68.826]
Note 11: 9/5 = 1.8 [1017.596, -17.596]
Note 11: 11/6 = 1.83333 [1049.362, +50.638]
Note 13: 2/1 = 2 [1200, 0]

I may revise this one to make it a little less "strange".