Harmonotonic tuning
A monotonic tuning is one whose step sizes are monotonic: they do not both increase and decrease.
- A diatonic tuning is not monotonic because it goes back and forth between whole and half steps.
- A segment of the overtone 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:
- decreasing step size (e.g. overtone series)
- equal step size (e.g. EDO)
- increasing step size (e.g. undertone series)
And here are the three different types:
- arithmetic & rational (e.g. overtone or undertone series)
- arithmetic & irrational (e.g. EDO)
- non-arithmetic & irrational
Arithmetic tunings
See arithmetic tunings.
Non-arithmetic monotonic tunings
New arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the overtone series. But addition is not the only operation we could try applying to the frequencies of a basic monotonic overtone 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 overtone series, then take the square root of all the frequencies. This results in something like the overtone 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.
The next operation above exponentiation is tetration. Tetrating frequency is equivalent to exponentiating pitch. This operation and beyond will not be explored here.
Adding frequency is called shifting a tuning. Exponentiating frequency (or multiplying pitch) is called stretching (or compressing) a tuning.
Here is a table to illustrate:
| operation | frequency | pitch |
|---|---|---|
| shifting | addition | Gaussian logarithm |
| transposition | multiplication | addition |
| stretching | exponentiation | multiplication |
| ... | tetration | exponentiation |
All powharmonic tunings are monotonic, but non-arithmetic and ir-rational.
Table of monotonic tunings
Shaahin Mohajeri has previously developed some tunings which qualify as monotonic. His n-ADO is equivalent to n-ODO, and his n-EDL is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
Example monotonic tuning charts and graphs for comparison
Typically, the undertone series is displayed moving from 1/1, descending in pitch. To better illustrate the analogies between these different types of monotonic tunings, however, all tunings are shown starting on 1/1 but ascending in pitch. This means that all utonal tunings here were necessary to truncate, so that the lowest pitch would be known, so that the pitches' order could be reversed and the lowest pitch repositioned to 1/1 and all other pitches relative to that instead.
For a variety of perspectives, the frequency table values have been left unreduced, to illustrate the ideas behind the tuning better; the length values have been reduced.
The first several examples repeat after 4 steps. Their second repetition is italicized in the tables below.
In the charts, dots indicate places where values are rational with respect to the given quantity kind. Thicker lines indicate monotonic tunings which have equal steps in the given quantity kind.
| monotonic tuning | frequency (f) | pitch (log₂f) | length (1/f) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||
| 4-ODO = 4th overtone mode = 4-ADO | (4/4) | 5/4 | 6/4 | 7/4 | 8/4 | 5/2 | 3/1 | 7/2 | 4/1 | 0.00 | 0.32 | 0.58 | 0.81 | 1.00 | 1.32 | 1.58 | 1.81 | 2.00 | 1/1 | 4/5 | 2/3 | 4/7 | 1/2 | 2/5 | 1/3 | 2/7 | 1/4 | |||
| 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189 | (2⁰⸍⁴) | 2¹⸍⁴ | 2²⸍⁴ | 2³⸍⁴ | 2⁴⸍⁴ | 2.38 | 2.83 | 3.36 | 4 | 0.00 | 0.25 | 0.50 | 0.75 | 1.00 | 1.25 | 1.50 | 1.75 | 2.00 | 1.00 | 0.84 | 0.71 | 0.59 | 0.50 | 0.42 | 0.35 | 0.30 | 0.25 | |||
| 4-UDO = 4th undertone mode | (8/8) | 8/7 | 8/6 | 8/5 | 8/4 | 16/7 | 8/3 | 16/5 | 4/1 | 0.00 | 0.19 | 0.42 | 0.68 | 1.00 | 1.19 | 1.42 | 1.68 | 2.00 | 1/1 | 7/8 | 3/4 | 5/8 | 1/2 | 7/16 | 3/8 | 5/16 | 1/4 | |||
| 4-EFDφ | 1+(0/4)(φ-1) | 1+(1/4)(φ-1) | 1+(2/4)(φ-1) | 1+(3/4)(φ-1) | 1+(4/4)(φ-1) | 1.87 | 2.12 | 2.37 | 2.62 | 0.00 | 0.21 | 0.39 | 0.55 | 0.69 | 0.90 | 1.08 | 1.24 | 1.39 | 1.00 | 0.87 | 0.76 | 0.68 | 0.62 | 0.54 | 0.47 | 0.42 | 0.38 | |||
| 4-ELDφ | (1) | 1.11 | 1.24 | 1.40 | φ | 1.79 | 2.00 | 2.27 | 2.62 | 0.00 | 0.14 | 0.31 | 0.49 | 0.69 | 0.84 | 1.00 | 1.18 | 1.39 | 1.00 | 0.90 | 0.81 | 0.71 | 0.62 | 0.56 | 0.50 | 0.44 | 0.38 | |||
| overtone series segment = 8-OS = 8-OD9 | (1/1) | 2/1 | 3/1 | 4/1 | 5/1 | 6/1 | 7/1 | 8/1 | 9/1 | 0.00 | 1.00 | 1.58 | 2.00 | 2.32 | 2.58 | 2.81 | 3.00 | 3.17 | 1/1 | 1/2 | 1/3 | 1/4 | 1/5 | 1/6 | 1/7 | 1/8 | 1/9 | |||
| undertone series segment = 8-US = 8-UD9 | (9/9) | 9/8 | 9/7 | 9/6 | 9/5 | 9/4 | 9/3 | 9/2 | 9/1 | 0.00 | 0.17 | 0.36 | 0.58 | 0.85 | 1.17 | 1.58 | 2.17 | 3.17 | 1/1 | 8/9 | 7/9 | 2/3 | 5/9 | 4/9 | 1/3 | 2/9 | 1/9 | |||
| 8-OS(3/4) | (4/4) | 7/4 | 10/4 | 13/4 | 16/4 | 19/4 | 22/4 | 25/4 | 28/4 | 0.00 | 0.81 | 1.32 | 1.70 | 2.00 | 2.25 | 2.46 | 2.64 | 2.81 | 1/1 | 4/7 | 2/5 | 4/13 | 1/4 | 4/19 | 2/11 | 4/25 | 1/7 | |||
| 8-US(3/4) | (28/28) | 28/25 | 28/22 | 28/19 | 28/16 | 28/13 | 28/10 | 28/7 | 28/4 | 0.00 | 0.16 | 0.35 | 0.56 | 0.81 | 1.11 | 1.49 | 2.00 | 2.81 | 1/1 | 25/28 | 11/14 | 19/28 | 4/7 | 13/28 | 5/14 | 1/4 | 1/7 | |||
| (1/⁴√2)-shifted overtone series segment = 8-AFS(1/⁴√2) | (1) | 1.84 | 2.68 | 3.52 | 4.36 | 5.20 | 6.05 | 6.89 | 7.73 | 0.00 | 0.88 | 1.42 | 1.82 | 2.13 | 2.38 | 2.60 | 2.78 | 2.95 | 1.00 | 0.54 | 0.37 | 0.28 | 0.23 | 0.19 | 0.17 | 0.15 | 0.13 | |||
| (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2) | (1) | 1.12 | 1.28 | 1.48 | 1.77 | 2.19 | 2.88 | 4.20 | 7.73 | 0.00 | 0.17 | 0.35 | 0.57 | 0.82 | 1.13 | 1.53 | 2.07 | 2.95 | 1.00 | 0.89 | 0.78 | 0.67 | 0.56 | 0.46 | 0.35 | 0.24 | 0.13 | |||
| AS5/4 = 1D JI lattice of 5/4 = 5/4 chain | (5⁰/4⁰) | 5¹/4¹ | 5²/4² | 5³/4³ | 5⁴/4⁴ | 5⁵/4⁵ | 5⁶/4⁶ | 5⁷/4⁷ | 5⁸/4⁸ | 0.00 | 0.32 | 0.64 | 0.97 | 1.29 | 1.61 | 1.93 | 2.25 | 2.58 | 1/1 | 4/5 | 16/25 | 64/125 | 256/625 | 1024/3125 | 4096/15625 | 16384/78125 | 65536/390625 | |||
| 8 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2 | √1 | √2 | √3 | √4 | √5 | √6 | √7 | √8 | √9 | 0.00 | 0.50 | 0.79 | 1.00 | 1.16 | 1.29 | 1.40 | 1.50 | 1.58 | 1.00 | 0.71 | 0.58 | 0.50 | 0.45 | 0.41 | 0.38 | 0.35 | 0.33 | |||
| 8 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2 | (1) | 1.06 | 1.13 | 1.22 | 1.34 | 1.50 | 1.73 | 2.12 | 3 | 0.00 | 0.08 | 0.18 | 0.29 | 0.42 | 0.58 | 0.79 | 1.08 | 1.58 | 1.00 | 0.94 | 0.88 | 0.82 | 0.75 | 0.67 | 0.58 | 0.47 | 0.33 | |||
| 8 pitches of 2-logharmonic series | (1) | 1.58 | 2.00 | 2.32 | 2.58 | 2.81 | 3.00 | 3.17 | 3.32 | 0.00 | 0.66 | 1.00 | 1.22 | 1.37 | 1.49 | 1.58 | 1.66 | 1.73 | 1.00 | 0.63 | 0.50 | 0.43 | 0.39 | 0.36 | 0.33 | 0.32 | 0.30 | |||
| 8 pitches of 2-sublogharmonic series | (1) | 1.05 | 1.11 | 1.18 | 1.29 | 1.43 | 1.66 | 2.10 | 3.32 | 0.00 | 0.07 | 0.15 | 0.24 | 0.36 | 0.52 | 0.73 | 1.07 | 1.73 | 1.00 | 0.95 | 0.90 | 0.85 | 0.78 | 0.70 | 0.60 | 0.48 | 0.30 | |||
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Notes 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.
Notes on monotonicity
Pitches of any tuning could be sorted in order to be monotonic, so for monotonicity to be meaningful for tunings, it must be applied to their steps after sorting their pitches.
Monotonicity must also be restricted to within the interval of repetition to have much value. Otherwise, only scales with equal step sizes would qualify as monotonic, because any increase or decrease in step size would be countered with an opposite change once the scale repeated.
In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same if the values are increasing or decreasing. In this sense, all monotonic tunings classified here are strictly monotonic.
Sometimes also absolutely monotonic sequences are distinguished, whose derivatives are all also monotonic. This is true for all tunings classified here. So all tunings here are absolutely and strictly monotonic by step size.
History
Monotonic tunings are not at all new concepts. However, the classifications and manners of specification for monotonic and arithmetic tunings described here were developed by Douglas Blumeyer in March 2021, incorporating existing ideas and further significant input and guidance from Billy Stiltner, Shaahin Mohajeri, Paul Erlich, Joakim Bang Larsen, and Dave Keenan. Discussion occurred on Facebook on this comment thread.