Harmonotonic tuning

In mathematics, a monotonic sequence is one whose values do not both increase and decrease, so a monotonic tuning is one whose step sizes are monotonic.

• A diatonic tuning is not monotonic because it goes back and forth between whole and half steps.
• A segment of the harmonic series is monotonic because its steps always decrease in size (within the interval of repetition).
• An EDO tuning is monotonic because the steps are all the same size.

Categorization

Monotonic tunings can be categorized in two different ways: by shape, and by type.

Here are the three different shapes, according to their pitches sorted in ascending order:

1. decreasing step size (e.g. harmonic series)
2. equal step size (e.g. EDO)
3. increasing step size (e.g. subharmonic series)

And here are the three different types:

1. arithmetic & rational
2. arithmetic & irrational
3. non-arithmetic & irrational

Arithmetic tunings

An arithmetic tuning is one which has equal step sizes of any kind of quantity, whether that be pitch, frequency, or length (of the resonating entity producing the sound). All arithmetic tunings are monotonic tunings.

Basic examples of arithmetic tunings:

1. the harmonic series has equal steps of frequency (1, 2, 3, 4, etc.; adding 1 each step)
2. any EDO has equal steps of pitch (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)
3. the subharmonic series has equal steps of length (to play the first four steps of the subharmonic series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step)

Other arithmetic tunings can be found by changing the step size. For example, if you vary the harmonic series to have a step size of 3/4 instead of 1, then you get the tuning [math]\displaystyle{ 1, 1\frac 34, 2\frac 24, 3\frac14 }[/math], which is equivalent to [math]\displaystyle{ \frac 44, \frac 74, \frac{10}{4}, \frac{13}{4} }[/math], or in other words, a class iii isoharmonic tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: EFS, for equal frequency sequence. For example, if we wanted to move by steps of φ — 1, 1+φ, 1+2φ, 1+3φ, etc. — we could have the EFSφ.

OS and EFS are equivalent to taking a harmonic series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see Derivation of OS.

The same principles that were just described for frequency are also possible for length. By varying the subharmonic series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an equal length sequence (ELS). Analogously, by shifting the subharmonic series by a constant amount of string length, the step sizes remain equal in terms of length, but alter their relationship in pitch.

Non-arithmetic tunings

We've shown that new arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the harmonic series. But addition is not the only operation we could try applying to the frequencies of a basic monotonic harmonic series.

The next operation above addition is multiplication. This operation is not very interesting, however, because multiplying frequency is equivalent to adding pitch, which does not meaningfully change a tuning; this merely transposes it. The reason multiplying frequency is equivalent to adding pitch is because pitch is found by taking the logarithm of frequency, and taking the logarithm of something effectively gears it down one operation lower on the hierarchy of operations: addition, multiplication, exponentiation, tetration, etc.

The next operation above multiplication is exponentiation. Exponentiating frequency is equivalent to multiplying pitch. Multiplying all pitch values does give you meaningfully new tunings. However, it does not preserve the arithmetic quality of a tuning for frequency or for pitch. So, these are now non-arithmetic tunings.

For example, we could start with the harmonic series, then take the square root of all the frequencies. This results in something like the harmonic series, except you don't reach the 2nd harmonic until the 4th step, the 3rd harmonic until the 9th step, or the 4th harmonic until the 16th step, etc. Because the square root is the same as raising to the power of 1/2, this is equivalent to multiplying all pitches by 1/2 (i.e. dividing them by 2). We could call this the 1/2-powharmonic series.

All powharmonic tunings are monotonic, but non-arithmetic and ir-rational.

Gallery of monotonic tunings

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Derivation of OS

The tuning OS3/4 is the sequence [math]\displaystyle{ \frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}... }[/math] and so on. Any OS is equivalent to shifting the harmonic series by a constant amount of frequency. In the case of OS3/4, it is a shift by [math]\displaystyle{ \frac 13 }[/math]. Let's show how.

Begin with the harmonic series:

[math]\displaystyle{ 1, 2, 3, 4... }[/math]

Shift it by [math]\displaystyle{ \frac 13 }[/math]:

[math]\displaystyle{ 1\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\ }[/math]

Convert to improper fractions by first expanding the whole number:

[math]\displaystyle{ \frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\ }[/math]

...then consolidating numerators:

[math]\displaystyle{ \frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}... }[/math]

Resize to start at [math]\displaystyle{ \frac 11 }[/math] by multiplying every term by the reciprocal of the first term, [math]\displaystyle{ \frac 43 }[/math], which is [math]\displaystyle{ \frac 34 }[/math]:

[math]\displaystyle{ \frac 43 \cdot \frac 34, \frac 73 \cdot \frac 34, \frac{10}{3} \cdot \frac 34, \frac{13}{3} \cdot \frac 34... }[/math]

Cancel out:

[math]\displaystyle{ \frac{4}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{7}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{10}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{13}{\cancel{3}} \cdot \frac{\cancel{3}}{4}... }[/math]

And we've arrived:

[math]\displaystyle{ \frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}... }[/math]

So we can see that [math]\displaystyle{ \frac 13 }[/math] was the right amount to shift by because it is the delta from the starting position [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ \frac 43 }[/math], the latter of which is the reciprocal of the target step size [math]\displaystyle{ \frac 34 }[/math] and therefore the value that we need the starting position to equal in order to be sent back to [math]\displaystyle{ 1 }[/math] when we resize all steps from 1 to the target step size by multiplying everything by it.

A note on etymology

The "-tonic" root of "monotonic" does share etymology with the musical terms "tone" and "tonic". They both come from the Greek word "tonikos" which means "a stretching". This is also the explanation for "tonic" water, which supposedly relaxes you by stretching your muscles. So, the term "monotonic tuning" reunites these divergent applications of stretching — function values, and instrument strings — back into one place.