Tuning Methods

Harmonic Series Tunings


Arithmetic Series Plus a Constant Tunings

Anyone who has tried simple harmonic series tunings knows that their intervals tend to dwindle into unusability fairly quickly. One way around this is to use algorithms to skip certain members of a set. ASPluCTs were among my earlier attempts at a technique to achieve approximate intervalic evenness while tuning with harmonic fragments. They didn't behave the way I expected, but they did provide some unique harmonic frameworks.

ASPluCTs are generated by beginning an arithmetic series at any integer, rather than at zero, and using the corresponding harmonic frequency relationships. The number that is added to each member of the arithmetic series is called the base.

For example, beginning with a base of 7, find the first term of the series by adding 0 to that =7, then add 1 to that =8 +2=10 +3=13, and so on. This produces the true arithmetic series, which I call the first-order series, transposed by +7. The ASPluCT designation for the resulting tuning is 7,1.

Finding the 7,1 ASPluCT Series

Previous member +: 0 1  2  3  4  5  6  7  8  9 10...
Arithmetic Series: 0 1  3  6 10 15 21 28 36 45 55...
           + Base: 7
   ASPluCT Series: 7 8 10 13 17 22 28 35 43 52 62...

The resulting tuning would have frequency relationships corresponding to the harmonics 7, 8, 10, 13, and so on. The second-order series is simply double the first order series: 0, 2, 6, 12, 20.... The tuning generated by transposing the second-order series by, say, +31 would be designated 31,2.

Finding the 31,2 ASPluCT Series

Second-Order Series:  0  2  6 12 20 30 42 56  72 ...              + Base: 31
     ASPluCT Series: 31 33 37 43 51 61 73 87 103 ...

It is theoretically possible to use third-order, fourth-order, and so on. Of course, to keep your intervals from getting unusably large, you have to use higher bases.

The "negative" arithmetic series can be used to find subharmonic series by beginning with "negative" integers. They're not really negative--they're positive inversions. But it's more convenient, in spreadsheets, to do the math with negative integers than with fractions. For example, the subharmonic, first-order ASPluCT based on 1/12 would be called -12,-1.

Finding the -12,-1 ASPluCT Series

Negative Arthmtc Series:   0  -1  -3  -6 -10 -15 ...
                 + Base: -12
         ASPluCT Series: -12 -13 -15 -18 -22 -27 ...

A minus sign in this case designates a subharmonic, so a number so designated is placed in a denominator rather than a numerator. The -12,-1 series would contain the rational frequency relationships 1/12, 1/13, 1/15, 1/18, 1/22, 1/27 ...

It's theoretically possible to build a positive series starting on a negative integer, or vice versa. But there is a caveate with this. Since the distance from 1:1 up to 2:1 is an octave, and the distance from 1:1 down to 1:2 is also an octave, crossing from "negative" to "positive" territory or vice versa puts such large leaps in a tuning as to make it unusable. At best, only a handful of the tones will lie within human hearing range, and those will be separated at the extremes of the range, say, below 50 Hz and above 5 KHz. Typically, you would go from below the range of the lowest musical instruments to above the range of the highest ones in one or two steps! Such tunings might be playable on a large pipe organ, but the resulting sounds would tend toward incoherence, and would probably be merely annoying. So, when building +,- or -,+ style tunings, be sure to start far enough away from 1:1 to avoid reaching it in the number of tones you wish to use. Also, be aware that the character of these backwards ASPluCTs will be opposite to what ASPluCT was designed to for -- the intervals will tend to be very small at one end and very large at the other. Generally, the tuning will go from unusably small intervals to unusably large ones in a very few tones. It will be an extremely violent tuning rather than a relatively even one.

Finding the -81,1 ASPluCT Series

Arithmetic Series:   0   1   3   6  10  15  21 ...
           + Base: -81
   ASPluCT Series: -81 -80 -78 -75 -71 -66 -60 ...

I prefer to think of ASPluCTs simply as tunings, as distinguished from scales, because they are open-ended (non-repeating). Of course, it would also be possible to use subsets of their octave equivalents as octave-repeating scales, but the intent behind the design is to provide orderly series of true harmonics or subharmonics.

Many synthesizers tuning tables are inherently octave equivalent, assume a 12-tone per octave MIDI controller, and allow only +/-99c with reference to 12 EDO. If your equipment has those restrictions, it helps to put a strip of labeling tape along the keyboard and mark it up with a Sharpie. A much better solution for Windows users is to get Robert Walker's excellent Fractal Tune Smithy software. Then you can just enter your ratios and play chromatically, allowing up to 128 distinct tones.

Copyright 2005 by David J. Finnamore
All rights reserved.