Harmonotonic tunings

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A harmonotonic tuning, or step-monotonic tuning, is one whose step sizes are monotonic: they do not both increase and decrease.

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

Diatonic scale not monotonic.svgOvertone series segment monotonic.svgEDO monotonic.svg

Essentially, "harmonotonic" references the mathematical concept of monotonicity to form an umbrella term for tunings which are closely related to the harmonic series.

Categorization

Harmonotonic 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. overtone series)
  2. equal step size (e.g. EDO)
  3. increasing step size (e.g. undertone series)

And here are the three different types:

  1. arithmetic & rational (e.g. overtone or undertone series)
  2. arithmetic & irrational (e.g. EDO)
  3. non-arithmetic & irrational

Arithmetic tunings

See arithmetic tunings.

Non-arithmetic harmonotonic 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 harmonotonic 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:

relationship between effects on frequency and pitch of transformations
operation frequency pitch
shifting addition Gaussian logarithmic addition
transposition multiplication addition
stretching exponentiation multiplication
... tetration exponentiation

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

Table of harmonotonic tunings

Table of harmonotonic tunings
tuning type
arithmetic

rational

arithmetic

irrational

non-arithmetic

irrational

tuning

shape

decreasing

step size

basic overtone series, or harmonic series irrationally shifted overtone series (± frequency) (equivalent to AFS) stretched/compressed overtone series (exponentiated frequency, multiplied pitch) (equivalent to powharmonic series)
rationally shifted overtone series (± frequency) (equivalent to OS)
division overtone mode, or over-n scale (equivalent to n-ODO) n-EFDp: n equal frequency divisions of irrational interval p
n-ODp: n otonal divisions of rational interval p
sequence (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p (n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by irrational interval p
other open c-powharmonic series exponent c
b-logharmonic series base b
equal

step size

basic 1D JI lattice rank-1 temperament (equivalent to equal multiplications)
division n-EDp: n equal (pitch) divisions of interval p (e.g. 12-EDO) (equivalent to rank-1 temperament of p/n)
sequence (n-)ASp: (n pitches of an) ambitonal sequence adding by rational interval p (equivalent to 1D JI lattice of p) (n-)APSp: (n pitches of an) arithmetic pitch sequence adding by irrational interval p (equivalent to rank-1 temperament with generator p)
increasing

step size

basic undertone series, or subharmonic series irrationally shifted undertone series (± frequency) (equivalent to ALS) stretched/compressed undertone series (exponentiated frequency, multiplied pitch) (equivalent to subpowharmonic series)
rationally shifted undertone series (± frequency) (equivalent to US)
division undertone mode, or under-n scale (equivalent to n-UDO) n-ELDp: n equal length divisions of irrational interval p
n-UDp: n utonal divisions of rational interval p
sequence (n-)USp: (n pitches of a) utonal sequence adding by rational interval p (n-)ALSp: (n pitches of an) arithmetic length sequence adding by irrational interval p
other open c-subpowharmonic series exponent c
b-sublogharmonic series base b

Shaahin Mohajeri has previously developed some tunings which qualify as harmonotonic. 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 harmonotonic 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 harmonotonic 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 harmonotonic tunings which have equal steps in the given quantity kind.

comparison of example harmonotonic tunings
harmonotonic 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) 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) 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 (1) 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) 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) 0.21 0.39 0.55 0.69 0.90 1.08 1.24 1.39 (1) 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) 0.14 0.31 0.49 0.69 0.84 1.00 1.18 1.39 (1) 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) 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) 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) 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) 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) 0.88 1.42 1.82 2.13 2.38 2.60 2.78 2.95 (1) 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) 0.17 0.35 0.57 0.82 1.13 1.53 2.07 2.95 (1) 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) 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) 0.50 0.79 1.00 1.16 1.29 1.40 1.50 1.58 (1) 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) 0.08 0.18 0.29 0.42 0.58 0.79 1.08 1.58 (1) 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) 0.66 1.00 1.22 1.37 1.49 1.58 1.66 1.73 (1) 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) 0.07 0.15 0.24 0.36 0.52 0.73 1.07 1.73 (1) 0.95 0.90 0.85 0.78 0.70 0.60 0.48 0.30

Notes on etymology

The "-tonic" root of "harmonotonic" 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 "harmonotonic tuning" reunites these divergent applications of stretching — function values, and instrument strings — back into one place.

There is no relation between the "mono" Greek root meaning "one" and the "mon" in "harmony". It is just a fun coincidence that we can combine the two words this way.

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 is also used in another sense in xenharmonics, however, for scales, which are not order-agnostic. In these cases, the monotonicity does apply to the pitches, not the steps. See: https://en.xen.wiki/index.php?search=monotone&title=Special%3ASearch&go=Go

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. In this sense, all tunings are strictly monotonic. But only some tunings are strictly step-monotonic. Of the tunings classified here, only those with increasing or decreasing step size are strictly step-monotonic.

The sequence of step-sizes is the "first difference" of the sequence of pitches. If you were to list the differences between the sizes of successive steps, that would be the second difference. A monotonic sequence, all of whose differences are monotonic, is called "absolutely monotonic". All tunings categorized here are absolutely monotonic.

History

Harmonotonic tunings are not at all new concepts. However, the classifications and manners of specification for harmonotonic 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, Dave Keenan, and Mike Battaglia. Discussion mostly occurred on Facebook on this comment thread. Other discussion occurred over email and private Facebook messages.

Dave and Mike independently suggested the term "step-monotonic" and prefer it to "harmonotonic" because it is more descriptive and mathematically accurate. Douglas does not disagree with those motivations or facts, but nonetheless prefers the name "harmonotonic" because it is friendly for non-mathematicians: it references the harmonic series, which is the core of this umbrella naming effort (with monotonicity merely happening-to-be-the-case, providing a convenient etymological root to reference).