Harmonotonic tuning: Difference between revisions

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A '''monotonic tuning''' is one whose ''step sizes'' are [https://en.wikipedia.org/wiki/Monotonic_function monotonic]: they do not both increase and decrease.

A '''harmonotonic tuning''', or '''step-monotonic tuning''', is one whose ''step sizes'' are [https://en.wikipedia.org/wiki/Monotonic_function 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 diatonic tuning is ''not'' harmonotonic 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).

* A segment of the harmonic series ''is'' harmonotonic 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.

* An EDO tuning ''is'' harmonotonic because the steps are all the same size.

{|

[[File:Diatonic scale not monotonic.svg|400px]][[File:Overtone series segment monotonic.svg|400px]][[File:EDO monotonic.svg|400px]]

|-

| [[File:Diatonic scale not monotonic.svg|~~thumb~~]] || [[File:Overtone series segment monotonic.svg|~~thumb~~]] || [[File:EDO monotonic.svg|~~thumb~~]]

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

|}

== Categorization ==

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

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:

# '''decreasing''' step size (e.g. ~~overtone~~ series)

# '''decreasing''' step size (e.g. harmonic series)

# '''equal''' step size (e.g. EDO)

# '''increasing''' step size (e.g. ~~undertone~~ series)

# '''increasing''' step size (e.g. subharmonic series)

And here are the three different '''types''':

# '''[[~~Monotonic~~ tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. ~~overtone~~ or ~~undertone~~ series)

# '''[[Harmonotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. harmonic or subharmonic series)

# '''arithmetic & irrational''' (e.g. EDO)

# '''non-arithmetic & irrational'''

Because different acronyms are used to distinguish rational (JI) tunings from general tunings which include irrational (non-JI) tunings, while the acronyms used for general tunings technically include the JI tunings, these general acronyms are more useful when reserved for non-JI tunings, and this is what is typically done. So when "irrational" is used on this page, it more accurately means "probably 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).~~

See [[arithmetic tunings]].

~~All arithmetic tunings are monotonic tunings.~~

~~Basic examples of arithmetic tunings:~~

~~# the '''overtone''' series has equal steps of '''frequency''' (1, 2, 3, 4, etc.; adding 1 each step)~~

~~# any '''EDO''' has equal steps of '''pitch''' (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)~~

# the '''undertone''' series has equal steps of '''length''' (to play the first four steps of the undertone 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)

~~=== Sequences ===~~

Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[~~isoharmonic_chords|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: AFS, for~~ arithmetic ~~frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ.~~

OS and AFS are equivalent to taking an overtone 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 the later section on the [[Monotonic tunings~~#Derivation of OS|derivation of OS~~]].

The same principles that were just described for frequency are also possible for length: by varying the undertone 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 arithmetic ~~length sequence (ALS). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.~~

== Non-arithmetic harmonotonic tunings ==

~~=== Divisions ===~~

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 harmonotonic harmonic series.

So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.

The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).

But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.

~~=== Comparing Arithmetic Tunings ===~~

~~We can state a few helpful analogies:~~

~~'''(n-)AQSp : n-EQDp'''~~

An arithmetic sequence of some kind of quantity Q is analogous to an equal division. Both require an interval '''p''' to be specified. The key difference is that a sequence is potentially open-ended — proceeding forever without repeating (such as the overtone series) — so its parameter '''n''', for the total number of pitches, is optional.

~~'''OS : AFS :: OD : EFD'''~~

~~'''AS : APS :: __ : EPD'''~~

~~'''US : ALS :: UD : ELD'''~~

~~Each of these rows has the form rational sequence : irrational division :: rational division : irrational division. The first row is for frequency, the second for pitch, and the third for length.~~

We haven't looked in detail at the middle row, for pitch. EPD, again, is long for simply ED. AS stands for ambitonal sequence; these are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. There is one blank space in the system of analogies for rational divisions of pitch; these are theoretically impossible.

~~Every rational arithmetic tuning is a subtype of its corresponding irrational arithmetic tuning:~~

* An OS is a specific (rational) type of AFS.

* An OD is a specific (rational) type of EFD.

* An AS is a specific (rational) type of APS.

* A US is a specific (rational) type of ALS.

* A UD is a specific (rational) type of ELD.

~~== Non-arithmetic monotonic tunings ==~~

~~We've shown that 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.

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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]].

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]].

The next operation above exponentiation is tetration. Tetrating frequency is equivalent to exponentiating pitch. This operation and beyond will not be explored here.

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!shifting

|addition

|[[wikipedia:Gaussian_logarithm|Gaussian ~~logarithm~~]]

|[[wikipedia:Gaussian_logarithm|Gaussian logarithmic addition]]

|-

!transposition

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|exponentiation

|}

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

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

== Table of ~~monotonic~~ tunings ==

== Table of harmonotonic tunings ==

{| class="wikitable"

|+Table of ~~monotonic~~ tunings

|+Table of harmonotonic tunings

! colspan="2" rowspan="2" style="background-color: white; border-left: 1px solid white; border-top: 1px solid white;" |

! colspan="3" rowspan="2" style="background-color: white; border-left: 1px solid white; border-top: 1px solid white;" |

! colspan="3" |tuning type

! colspan="5" |tuning type

|-

! arithmetic

rational

!

! arithmetic

irrational

!

! non-arithmetic

irrational

|-

! rowspan="15" |'''tuning'''

! rowspan="19" |'''tuning'''

'''shape'''

! rowspan="6" |'''decreasing'''

! rowspan="7" |'''decreasing'''

'''step size'''

|'''[[~~Overtone series|overtone series, or~~ harmonic series]]'''||[[AFS|'''shifted ~~overtone~~ series''' (± frequency) ''(equivalent to AFS)'']]

! rowspan="2" |basic

| [[Powharmonic series|'''stretched/compressed ~~overtone~~ series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'']]

|'''[[harmonic series]]'''

! rowspan="2" |

| rowspan="2" |[[AFS|'''irrationally shifted harmonic series''' (± frequency) ''(equivalent to AFS)'']]

! rowspan="2" |

| rowspan="2" | [[Powharmonic series|'''stretched/compressed harmonic series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'']]

|-

|[[OS|'''rationally shifted harmonic series''' (± frequency) ''(equivalent to OS)'']]

|-

|[[Overtone scale#Over-n ~~Scales~~|'''~~overtone~~ mode, or over-n scale''' ''(equivalent to n-ODO)'']]

! rowspan="2" |division

|

|[[Overtone scale#Over-n scales|'''harmonic mode, or over-n scale''' ''(equivalent to n-ODO)'']]

|

! rowspan="2" |

| rowspan="2" |[[EFD|'''n-EFDp:''' n equal frequency divisions of irrational interval p]]

! rowspan="2" |

| rowspan="2" |

|-

|[[OD|'''n-ODp:''' n otonal divisions of interval p]]

|[[OD|'''n-ODp:''' n otonal divisions of rational interval p]]

~~|[[EFD|'''n-EFDp:''' n equal frequency divisions of interval p]]~~

|

|-

|[[OS|'''(n-)OSp:''' (n pitches of an) otonal sequence adding by p]]

!sequence

|[[AFS|'''(n-)AFSp:''' (n pitches of an) arithmetic frequency sequence adding by p]]

|[[OS|'''(n-)OSp:''' (n pitches of an) otonal sequence adding by rational interval p]]

!

|[[AFS|'''(n-)AFSp:''' (n pitches of an) arithmetic frequency sequence adding by irrational interval p]]

!

|

|-

|

! rowspan="2" |other open

|

| rowspan="2" |

! rowspan="2" |

| rowspan="2" |

! rowspan="2" |

|'''[[Powharmonic series|c-powharmonic series]]''' exponent c

|-

|

|'''[[Logharmonic series|b-logharmonic series]]''' base b

|-

! rowspan="3" |'''equal'''

! colspan="7" |

'''step size'''

|-

! rowspan="3" |'''[[Equal-step tuning|equal]]'''

'''[[Equal-step tuning|step size]]'''

!basic

|'''1D [[Harmonic Lattice Diagram|JI lattice]]'''

| [[Tour of Regular Temperaments#Equal temperaments .28Rank-1 temperaments.29|'''rank-1 temperament''']] || rowspan="3" style="background-color: white; border-right: 1px solid white;" |

!

|[[Tour of Regular Temperaments#Equal temperaments .28Rank-1 temperaments.29|'''rank-1 temperament''']] [[Equal-step tuning#Equal multiplications|(equivalent to equal multiplications)]]

!

| rowspan="3" style="background-color: white; border-right: 1px solid white;" |

|-

!division

|

!

|[[EPD|'''n-EDp:''' n equal (pitch) divisions of interval p (e.g. 12-EDO) ''(equivalent to rank-1 temperament of p/n)'']]

!

|-

|[[AS|'''(n-)ASp:''' (n pitches of an) ambitonal sequence adding by p ''(equivalent to 1D JI lattice of p)'']]

!sequence

|[[APS|'''(n-)APSp:''' (n pitches of an) arithmetic pitch sequence adding by p ''(equivalent to rank-1 temperament with generator p)'']]

|[[AS|'''(n-)ASp:''' (n pitches of an) ambitonal sequence adding by rational interval p ''(equivalent to 1D JI lattice of p)'']]

!

|[[APS|'''(n-)APSp:''' (n pitches of an) arithmetic pitch sequence adding by irrational interval p ''(equivalent to rank-1 temperament with generator p)'']]

!

|-

! rowspan="6" |'''increasing'''

! colspan="7" |

|-

! rowspan="7" |'''increasing'''

'''step size'''

| '''[[~~wikipedia:Undertone_series|undertone series, or~~ subharmonic series]]''' || [[ALS|'''shifted ~~undertone~~ series''' (± frequency) ''(equivalent to ALS)'']] || [[Powharmonic series|'''stretched/compressed ~~undertone~~ series''' (exponentiated frequency, multiplied pitch) ''(equivalent to subpowharmonic series)'']]

! rowspan="2" |basic

|'''[[subharmonic series]]'''

! rowspan="2" |

| rowspan="2" |[[ALS|'''irrationally shifted subharmonic series''' (± frequency) ''(equivalent to ALS)'']]

! rowspan="2" |

|[[Powharmonic series|'''stretched/compressed subharmonic series''' (exponentiated frequency, multiplied pitch) ''(equivalent to subpowharmonic series)'']]

|-

|[[~~Overtone scale#Next Steps~~|'''~~undertone mode, or under-n scale~~''' ''(equivalent to ~~n-UDO~~)'']]

|[[US|'''rationally shifted subharmonic series''' (± frequency) ''(equivalent to US)'']]

|

|-

|[[UD|'''n-~~UDp~~:''' n u~~tonal~~ divisions of interval p]]

! rowspan="2" |division

|[[~~ELD~~|'''n-~~ELDp~~:''' n ~~equal <~~u>l~~ength~~ divisions of interval p]]

|[[Overtone scale#Next steps|'''subharmonic mode, or under-n scale''' ''(equivalent to n-UDO)'']]

|

! rowspan="2" |

| rowspan="2" |[[ELD|'''n-ELDp:''' n equal length divisions of irrational interval p]]

! rowspan="2" |

| rowspan="2" |

|-

|[[UD|'''n-UDp:''' n utonal divisions of rational interval p]]

|-

|[[US|'''(n-)USp:''' (n pitches of a) utonal sequence adding by p]]

!sequence

|[[ALS|'''(n-)ALSp:''' (n pitches of an) arithmetic length sequence adding by p]]

|[[US|'''(n-)USp:''' (n pitches of a) utonal sequence adding by rational interval p]]

!

|[[ALS|'''(n-)ALSp:''' (n pitches of an) arithmetic length sequence adding by irrational interval p]]

!

|

|-

|

! rowspan="2" |other open

|

| rowspan="2" |

! rowspan="2" |

| rowspan="2" |

! rowspan="2" |

|'''[[Powharmonic series|c-subpowharmonic series]]''' exponent c

|-

|

|'''[[Logharmonic series|b-sublogharmonic series]]''' base b

|}

[[Shaahin Mohajeri]] has previously developed some tunings which qualify as ~~monotonic~~. His [[ADO|n-ADO]] is equivalent to n-ODO, and his [[EDL|n-EDL]] is equivalent to n~~-UDn~~.

[[Shaahin Mohajeri]] has previously developed some tunings which qualify as harmonotonic. His [[ADO|n-ADO]] is equivalent to n-ODO, and his [[EDL|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 ==

~~== Example monotonic tuning charts~~ and ~~graphs for comparison ==~~

Typically, the subharmonic 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.

~~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 harmonotonic tunings which have equal steps in the given quantity kind.

{| class="wikitable"

|+comparison of example ~~monotonic~~ tunings

|+comparison of example harmonotonic tunings

! colspan="2" rowspan="2" |~~monotonic~~ tuning

! colspan="2" rowspan="2" |harmonotonic tuning

! colspan="9" |frequency

! colspan="9" |frequency (f)

! colspan="9" |pitch

!

! colspan="9" |length

! colspan="9" |pitch (log₂f)

!

! colspan="9" |length (1/f)

|-

! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 ~~!! 9~~

! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8

! !! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8

|-

! style="background-color: #00DB00;" |

! 4-ODO = 4th ~~overtone~~ mode = 4-ADO

! 4-ODO = 4th harmonic mode = 4-ADO

| 1/1 || 5/4 || 3/2 || 7/4 || 2/1 || 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

| (4/4) || 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''

|-

! style="background-color: #34A853;" |

! 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189

| ~~1.00~~ || ~~1.19~~ || ~~1.41~~ || ~~1.68~~ || ~~2.00~~ || 2.38 || 2.83 || 3.36 || 4~~.00 |~~| 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

| (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''

|-

! style="background-color: #45818E;" |

! 4-UDO = 4th ~~undertone~~ mode

! 4-UDO = 4th subharmonic mode

| 1/1 || 8/7 || 4/3 || 8/5 || 2/1 || 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

| (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''

|-

! style="background-color: #B4A7D6;" |

! 4-EFDφ

| 1~~.00~~ || 1~~.15~~ || 1~~.31~~ || 1~~.46~~ || 1~~.62~~ || 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

| (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''

|-

! style="background-color: #674EA7;" |

! 4-ELDφ

| 1~~.00~~ || 1.11 || 1.24 || 1.40 || ~~1.62~~ || 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

| (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''

|-

! style="background-color: #D2EB00;" |

! ~~overtone~~ series segment = 9-OS = 9-OD9

! harmonic 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

| (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

|-

! style="background-color: #A1A200;" |

! ~~undertone~~ series segment = 9-US = 9-UD9 ~~= 9-EDL~~

! subharmonic series segment = 8-US = 8-UD9

| 1/1 || 9/8 || 9/7 || 3/2 || 9/5 || 9/4 || 3/1 || 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

| (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

|-

! style="background-color: #FF7700;" |

! 9-OS(3/4)

! 8-OS(3/4)

| 1/1 || 7/4 || 5/2 || 13/4 || 4/1 || 19/4 || 11/2 || 25/4 || 7/~~1 |~~| 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

| (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

|-

! style="background-color: #C95E00;" |

! 9-US(3/4)

! 8-US(3/4)

| 1/1 || 28/25 || 14/11 || 28/19 || 7/4 || 28/13 || 14/5 || 4/1 || 7/~~1 |~~| 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

| (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

|-

! style="background-color: #FFC000;" |

! (1/⁴√2)-shifted ~~overtone~~ series segment = 9-AFS(1/⁴√2)

! (1/⁴√2)-shifted harmonic series segment = 8-AFS(1/⁴√2)

| 1~~.00~~ || 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) || 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

|-

! style="background-color: #C59500;" |

! (1/⁴√2)-shifted ~~undertone~~ series segment = 9-ALS(1/⁴√2)

! (1/⁴√2)-shifted subharmonic series segment = 8-ALS(1/⁴√2)

| 1~~.00~~ || 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

| (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

|-

! style="background-color: #00CFA9;" |

! AS5/4 = 1D JI lattice of 5/4 = 5/4 chain

| 1/1 || 5/4 || 25/16 || ~~125~~/64 || ~~625~~/~~256~~ || ~~3125~~/~~1024~~ || ~~15625~~/~~4096~~ || ~~78125~~/~~16384~~ || ~~390625~~/~~65536 |~~| 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

| (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

|-

! style="background-color: #F4CCCC;" |

! 9 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2

! 8 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2

| ~~1.00~~ || ~~1.41~~ || ~~1.73~~ || ~~2.00~~ || ~~2.24~~ || ~~2.45 |~~| ~~2.65~~ || ~~2.83~~ || ~~3.00~~ || 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

| (√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

|-

! style="background-color: #E06666;" |

! 9 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2

! 8 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2

| 1~~.00~~ || 1.06 || 1.13 || 1.22 || 1.34 || 1.50 || 1.73 || 2.12 || 3~~.00 |~~| 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

| (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

|-

! style="background-color: #00B6FF;" |

! 9 pitches of 2-logharmonic series

! 8 pitches of 2-logharmonic series

| 1~~.00~~ || 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

| (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

|-

! style="background-color: #006D99;" |

! 9 pitches of 2-sublogharmonic series

! 8 pitches of 2-sublogharmonic series

| 1~~.00~~ || 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

| (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

|}

[[File:Frequency.svg|~~700px~~|~~link=https://en.xen.wiki/images/7/71/Frequency.svg~~]]

[[File:Frequency (1).svg|alt=|700x700px]]

[[File:Pitch.svg|~~700px~~|~~link=https://en.xen.wiki/images/0/09/Pitch.svg~~]]

[[File:Pitch (1).svg|alt=|700x700px]]

[[File:Length.svg|~~700px~~|~~link=https://en.xen.wiki/images/d/dc/Length.svg~~]]

[[File:Length (1).svg|alt=|700x700px]]

~~== Derivation of OS ==~~

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

~~Begin with the overtone series:~~

~~<math>1, 2, 3, 4...</math>~~

~~Shift it by <math>\frac 13</math>:~~

~~<math>~~

~~1\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\~~

~~</math>~~

~~Convert to improper fractions by first expanding the whole number:~~

~~<math>~~

~~\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\~~

~~</math>~~

~~...then consolidating numerators:~~

== Notes on etymology ==

~~<math>~~

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.

~~\frac 43~~, ~~\frac 73~~, ~~\frac{10}{3}~~, ~~\frac{13}{3}..~~.

~~</math>~~

~~Resize to start at <math>\frac 11</math> by multiplying every term by~~ the ~~reciprocal of~~ the ~~first term, <math>\frac 43</math>, which~~ is ~~<math>\frac 34</math>:~~

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.

~~<math>~~

== Notes on monotonicity ==

~~\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:~~

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.

~~<math>~~

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

~~\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:~~

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.

~~<math>~~

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.

~~\frac 44~~, ~~\frac 74~~, ~~\frac{10}{4}~~, ~~\frac{13}{4}~~...

~~</math>~~

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

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.

== ~~A note on etymology~~ ==

== History ==

~~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~~.

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 [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread]. Other discussion occurred over email and private Facebook messages.

~~== Credits ==~~

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).

~~The classifications~~ and ~~manners of specification for monotonic and arithmetic tunings here were developed by Douglas Blumeyer in March 2021, with significant input and guidance from Billy Stiltner,~~ [[~~Shaahin Mohajeri~~]], [[~~Paul Erlich~~]]~~, and~~ [[~~Dave Keenan~~]]~~. Discussion occurred on Facebook on~~ [~~https~~:~~//www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread~~].

[[Category:Otonality and utonality]]

[[Category:Otonality]]

[[Category:Harmonic]]

[[Category:Harmonic series‏‎]]

[[Category:Utonality]]

[[Category:Subharmonic]]

[[Category:Subharmonic series‏‎]]

[[Category:Equal-step tuning‏‎]]

[[Category:Equal divisions of the octave‏‎]]

@@ Line 1: / Line 1: @@
-A '''monotonic tuning''' is one whose ''step sizes'' are [https://en.wikipedia.org/wiki/Monotonic_function monotonic]: they do not both increase and decrease.
+A '''harmonotonic tuning''', or '''step-monotonic tuning''', is one whose ''step sizes'' are [https://en.wikipedia.org/wiki/Monotonic_function 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 diatonic tuning is ''not'' harmonotonic 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).
+* A segment of the harmonic series ''is'' harmonotonic 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.
+* An EDO tuning ''is'' harmonotonic because the steps are all the same size.
-{|
+<span>[[File:Diatonic scale not monotonic.svg|400px]]</span><span>[[File:Overtone series segment monotonic.svg|400px]]</span><span>[[File:EDO monotonic.svg|400px]]</span>
-|-
-| [[File:Diatonic scale not monotonic.svg|thumb]] || [[File:Overtone series segment monotonic.svg|thumb]] || [[File:EDO monotonic.svg|thumb]]
+Essentially, "harmonotonic" references the mathematical concept of monotonicity to form an umbrella term for tunings which are closely related to the harmonic series.
-|}
 == Categorization ==
-Monotonic tunings can be categorized in two different ways: by '''shape''', and by '''type'''.
+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:
-# '''decreasing''' step size (e.g. overtone series)
+# '''decreasing''' step size (e.g. harmonic series)
 # '''equal''' step size (e.g. EDO)
-# '''increasing''' step size (e.g. undertone series)
+# '''increasing''' step size (e.g. subharmonic series)
 And here are the three different '''types''':
-# '''[[Monotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. overtone or undertone series)
+# '''[[Harmonotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. harmonic or subharmonic series)
 # '''arithmetic & irrational''' (e.g. EDO)
 # '''non-arithmetic & irrational'''
+Because different acronyms are used to distinguish rational (JI) tunings from general tunings which include irrational (non-JI) tunings, while the acronyms used for general tunings technically include the JI tunings, these general acronyms are more useful when reserved for non-JI tunings, and this is what is typically done. So when "irrational" is used on this page, it more accurately means "probably 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).
+See [[arithmetic tunings]].
-All arithmetic tunings are monotonic tunings.
-Basic examples of arithmetic tunings:
-# the '''overtone''' series has equal steps of '''frequency''' (1, 2, 3, 4, etc.; adding 1 each step)
-# any '''EDO''' has equal steps of '''pitch''' (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step)
-# the '''undertone''' series has equal steps of '''length''' (to play the first four steps of the undertone 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)
-=== Sequences ===
-Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <span><math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math></span>, or in other words, a class iii [[isoharmonic_chords|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: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> etc. we could have the AFSφ.
-OS and AFS are equivalent to taking an overtone 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 the later section on the [[Monotonic tunings#Derivation of OS|derivation of OS]].
-The same principles that were just described for frequency are also possible for length: by varying the undertone 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 arithmetic length sequence (ALS). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.
+== Non-arithmetic harmonotonic tunings ==
-=== Divisions ===
+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 harmonotonic harmonic series.
-So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
-The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
-But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.
-=== Comparing Arithmetic Tunings ===
-We can state a few helpful analogies:
-'''(n-)AQSp : n-EQDp'''
-An arithmetic sequence of some kind of quantity Q is analogous to an equal division. Both require an interval '''p''' to be specified. The key difference is that a sequence is potentially open-ended — proceeding forever without repeating (such as the overtone series) — so its parameter '''n''', for the total number of pitches, is optional.
-'''OS : AFS :: OD : EFD'''
-'''AS : APS :: __ : EPD'''
-'''US : ALS :: UD : ELD'''
-Each of these rows has the form rational sequence : irrational division :: rational division : irrational division. The first row is for frequency, the second for pitch, and the third for length.
-We haven't looked in detail at the middle row, for pitch. EPD, again, is long for simply ED. AS stands for ambitonal sequence; these are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. There is one blank space in the system of analogies for rational divisions of pitch; these are theoretically impossible.
-Every rational arithmetic tuning is a subtype of its corresponding irrational arithmetic tuning:
-* An OS is a specific (rational) type of AFS.
-* An OD is a specific (rational) type of EFD.
-* An AS is a specific (rational) type of APS.
-* A US is a specific (rational) type of ALS.
-* A UD is a specific (rational) type of ELD.
-== Non-arithmetic monotonic tunings ==
-We've shown that 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.
@@ Line 87: / Line 39: @@
 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]].
+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]].
 The next operation above exponentiation is tetration. Tetrating frequency is equivalent to exponentiating pitch. This operation and beyond will not be explored here.
@@ Line 102: / Line 54: @@
 !shifting
 |addition
-|[[wikipedia:Gaussian_logarithm|Gaussian logarithm]]
+|[[wikipedia:Gaussian_logarithm|Gaussian logarithmic addition]]
 |-
 !transposition
@@ Line 116: / Line 68: @@
 |exponentiation
 |}
-All powharmonic tunings are monotonic, but non-arithmetic and ir-rational.
+All powharmonic tunings are harmonotonic, but non-arithmetic and ir-rational.
-== Table of monotonic tunings ==
+== Table of harmonotonic tunings ==
 {| class="wikitable"
-|+Table of monotonic tunings
+|+Table of harmonotonic tunings
-! colspan="2" rowspan="2" style="background-color: white; border-left: 1px solid white; border-top: 1px solid white;" |
+! colspan="3" rowspan="2" style="background-color: white; border-left: 1px solid white; border-top: 1px solid white;" |
-! colspan="3" |tuning type
+! colspan="5" |tuning type
 |-
 ! arithmetic
 rational
+!
 ! arithmetic
 irrational
+!
 ! non-arithmetic
 irrational
 |-
-! rowspan="15" |'''tuning'''
+! rowspan="19" |'''tuning'''
 '''shape'''
-! rowspan="6" |'''decreasing'''
+! rowspan="7" |'''decreasing'''
 '''step size'''
-|'''[[Overtone series|overtone series, or harmonic series]]'''||[[AFS|'''shifted overtone series''' (± frequency) ''(equivalent to AFS)'']]
+! rowspan="2" |basic
-| [[Powharmonic series|'''stretched/compressed overtone series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'']]
+|'''[[harmonic series]]'''
+! rowspan="2" |
+| rowspan="2" |[[AFS|'''irrationally shifted harmonic series''' (± frequency) ''(equivalent to AFS)'']]
+! rowspan="2" |
+| rowspan="2" | [[Powharmonic series|'''stretched/compressed harmonic series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'']]
+|-
+|[[OS|'''rationally shifted harmonic series''' (± frequency) ''(equivalent to OS)'']]
 |-
-|[[Overtone scale#Over-n Scales|'''overtone mode, or over-n scale''' ''(equivalent to n-ODO)'']]
+! rowspan="2" |division
-|
+|[[Overtone scale#Over-n scales|'''harmonic mode, or over-n scale''' ''(equivalent to n-ODO)'']]
-|
+! rowspan="2" |
+| rowspan="2" |[[EFD|'''n-EFDp:''' <u>n</u> <u>e</u>qual <u>f</u>requency <u>d</u>ivisions of irrational interval <u>p</u>]]
+! rowspan="2" |
+| rowspan="2" |
 |-
-|[[OD|'''n-ODp:''' <u>n</u> <u>o</u>tonal <u>d</u>ivisions of interval <u>p</u>]]
+|[[OD|'''n-ODp:''' <u>n</u> <u>o</u>tonal <u>d</u>ivisions of rational interval <u>p</u>]]
-|[[EFD|'''n-EFDp:''' <u>n</u> <u>e</u>qual <u>f</u>requency <u>d</u>ivisions of interval <u>p</u>]]
-|
 |-
-|[[OS|'''(n-)OSp:''' (<u>n</u> pitches of an) <u>o</u>tonal <u>s</u>equence adding by <u>p</u>]]
+!sequence
-|[[AFS|'''(n-)AFSp:''' (<u>n</u> pitches of an) <u>a</u>rithmetic <u>f</u>requency <u>s</u>equence adding by <u>p</u>]]
+|[[OS|'''(n-)OSp:''' (<u>n</u> pitches of an) <u>o</u>tonal <u>s</u>equence adding by rational interval <u>p</u>]]
+!
+|[[AFS|'''(n-)AFSp:''' (<u>n</u> pitches of an) <u>a</u>rithmetic <u>f</u>requency <u>s</u>equence adding by irrational interval <u>p</u>]]
+!
 |
 |-
-|
+! rowspan="2" |other open
-|
+| rowspan="2" |
+! rowspan="2" |
+| rowspan="2" |
+! rowspan="2" |
 |'''[[Powharmonic series|c-powharmonic series]]''' exponent c
 |-
-|
-|
 |'''[[Logharmonic series|b-logharmonic series]]''' base b
 |-
-! rowspan="3" |'''equal'''
+! colspan="7" |
-'''step size'''
+|-
+! rowspan="3" |'''[[Equal-step tuning|equal]]'''
+'''[[Equal-step tuning|step size]]'''
+!basic
 |'''1D [[Harmonic Lattice Diagram|JI lattice]]'''
-| [[Tour of Regular Temperaments#Equal temperaments .28Rank-1 temperaments.29|'''rank-1 temperament''']] || rowspan="3" style="background-color: white; border-right: 1px solid white;" |
+!
+|[[Tour of Regular Temperaments#Equal temperaments .28Rank-1 temperaments.29|'''rank-1 temperament''']] [[Equal-step tuning#Equal multiplications|(equivalent to equal multiplications)]]
+!
+| rowspan="3" style="background-color: white; border-right: 1px solid white;" |
 |-
+!division
 |
+!
 |[[EPD|'''n-EDp:''' <u>n</u> <u>e</u>qual (pitch) <u>d</u>ivisions of interval <u>p</u> (e.g. 12-EDO) ''(equivalent to rank-1 temperament of p/n)'']]
+!
 |-
-|[[AS|'''(n-)ASp:''' (<u>n</u> pitches of an) <u>a</u>mbitonal <u>s</u>equence adding by <u>p</u> ''(equivalent to 1D JI lattice of p)'']]
+!sequence
-|[[APS|'''(n-)APSp:''' (<u>n</u> pitches of an) <u>a</u>rithmetic <u>p</u>itch <u>s</u>equence adding by <u>p</u> ''(equivalent to rank-1 temperament with generator p)'']]
+|[[AS|'''(n-)ASp:''' (<u>n</u> pitches of an) <u>a</u>mbitonal <u>s</u>equence adding by rational interval <u>p</u> ''(equivalent to 1D JI lattice of p)'']]
+!
+|[[APS|'''(n-)APSp:''' (<u>n</u> pitches of an) <u>a</u>rithmetic <u>p</u>itch <u>s</u>equence adding by irrational interval <u>p</u> ''(equivalent to rank-1 temperament with generator p)'']]
+!
 |-
-! rowspan="6" |'''increasing'''
+! colspan="7" |
+|-
+! rowspan="7" |'''increasing'''
 '''step size'''
-| '''[[wikipedia:Undertone_series|undertone series, or subharmonic series]]''' || [[ALS|'''shifted undertone series''' (± frequency) ''(equivalent to ALS)'']] || [[Powharmonic series|'''stretched/compressed undertone series''' (exponentiated frequency, multiplied pitch)  ''(equivalent to subpowharmonic series)'']]
+! rowspan="2" |basic
+|'''[[subharmonic series]]'''
+! rowspan="2" |
+| rowspan="2" |[[ALS|'''irrationally shifted subharmonic series''' (± frequency) ''(equivalent to ALS)'']]
+! rowspan="2" |
+|[[Powharmonic series|'''stretched/compressed subharmonic series''' (exponentiated frequency, multiplied pitch)  ''(equivalent to subpowharmonic series)'']]
 |-
-|[[Overtone scale#Next Steps|'''undertone mode, or under-n scale''' ''(equivalent to n-UDO)'']]
+|[[US|'''rationally shifted subharmonic series''' (± frequency) ''(equivalent to US)'']]
-|
 |
 |-
-|[[UD|'''n-UDp:''' <u>n</u> <u>u</u>tonal <u>d</u>ivisions of interval <u>p</u>]]
+! rowspan="2" |division
-|[[ELD|'''n-ELDp:''' <u>n</u> <u>e</u>qual <u>l</u>ength <u>d</u>ivisions of interval <u>p</u>]]
+|[[Overtone scale#Next steps|'''subharmonic mode, or under-n scale''' ''(equivalent to n-UDO)'']]
-|
+! rowspan="2" |
+| rowspan="2" |[[ELD|'''n-ELDp:''' <u>n</u> <u>e</u>qual <u>l</u>ength <u>d</u>ivisions of irrational interval <u>p</u>]]
+! rowspan="2" |
+| rowspan="2" |
+|-
+|[[UD|'''n-UDp:''' <u>n</u> <u>u</u>tonal <u>d</u>ivisions of rational interval <u>p</u>]]
 |-
-|[[US|'''(n-)USp:''' (<u>n</u> pitches of a) <u>u</u>tonal <u>s</u>equence adding by <u>p</u>]]
+!sequence
-|[[ALS|'''(n-)ALSp:''' (<u>n</u> pitches of an) <u>a</u>rithmetic <u>l</u>ength <u>s</u>equence adding by <u>p</u>]]
+|[[US|'''(n-)USp:''' (<u>n</u> pitches of a) <u>u</u>tonal <u>s</u>equence adding by rational interval <u>p</u>]]
+!
+|[[ALS|'''(n-)ALSp:''' (<u>n</u> pitches of an) <u>a</u>rithmetic <u>l</u>ength <u>s</u>equence adding by irrational interval <u>p</u>]]
+!
 |
 |-
-|
+! rowspan="2" |other open
-|
+| rowspan="2" |
+! rowspan="2" |
+| rowspan="2" |
+! rowspan="2" |
 |'''[[Powharmonic series|c-subpowharmonic series]]''' exponent c
 |-
-|
-|
 |'''[[Logharmonic series|b-sublogharmonic series]]''' base b
 |}
-[[Shaahin Mohajeri]] has previously developed some tunings which qualify as monotonic. His [[ADO|n-ADO]] is equivalent to n-ODO, and his [[EDL|n-EDL]] is equivalent to n-UDn.
+[[Shaahin Mohajeri]] has previously developed some tunings which qualify as harmonotonic. His [[ADO|n-ADO]] is equivalent to n-ODO, and his [[EDL|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 ==
-== Example monotonic tuning charts and graphs for comparison ==
+Typically, the subharmonic 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.
-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 harmonotonic tunings which have equal steps in the given quantity kind.
 {| class="wikitable"
-|+comparison of example monotonic tunings
+|+comparison of example harmonotonic tunings
-! colspan="2" rowspan="2" |monotonic tuning
+! colspan="2" rowspan="2" |harmonotonic tuning
-! colspan="9" |frequency
+! colspan="9" |frequency (f)
-! colspan="9" |pitch
+!
-! colspan="9" |length
+! colspan="9" |pitch (log₂f)
+!
+! colspan="9" |length (1/f)
 |-
-! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
+! (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
 |-
 ! style="background-color: #00DB00;" |
-! 4-ODO = 4th overtone mode = 4-ADO
+! 4-ODO = 4th harmonic mode = 4-ADO
-| 1/1 || 5/4 || 3/2 || 7/4 || 2/1 || 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
+| (4/4) || 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''
 |-
 ! style="background-color: #34A853;" |
 ! 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189
-| 1.00 || 1.19 || 1.41 || 1.68 || 2.00 || 2.38 || 2.83 || 3.36 || 4.00 || 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
+| (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''
 |-
 ! style="background-color: #45818E;" |
-! 4-UDO = 4th undertone mode
+! 4-UDO = 4th subharmonic mode
-| 1/1 || 8/7 || 4/3 || 8/5 || 2/1 || 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
+| (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''
 |-
 ! style="background-color: #B4A7D6;" |
 ! 4-EFDφ
-| 1.00 || 1.15 || 1.31 || 1.46 || 1.62 || 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
+| (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''
 |-
 ! style="background-color: #674EA7;" |
 ! 4-ELDφ
-| 1.00 || 1.11 || 1.24 || 1.40 || 1.62 || 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
+| (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''
 |-
 ! style="background-color: #D2EB00;" |
-! overtone series segment = 9-OS = 9-OD9
+! harmonic 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
+| (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
 |-
 ! style="background-color: #A1A200;" |
-! undertone series segment = 9-US = 9-UD9 = 9-EDL
+! subharmonic series segment = 8-US = 8-UD9
-| 1/1 || 9/8 || 9/7 || 3/2 || 9/5 || 9/4 || 3/1 || 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
+| (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
 |-
 ! style="background-color: #FF7700;" |
-! 9-OS(3/4)
+! 8-OS(3/4)
-| 1/1 || 7/4 || 5/2 || 13/4 || 4/1 || 19/4 || 11/2 || 25/4 || 7/1 || 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
+| (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
 |-
 ! style="background-color: #C95E00;" |
-! 9-US(3/4)
+! 8-US(3/4)
-| 1/1 || 28/25 || 14/11 || 28/19 || 7/4 || 28/13 || 14/5 || 4/1 || 7/1 || 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
+| (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
 |-
 ! style="background-color: #FFC000;" |
-! (1/⁴√2)-shifted overtone series segment = 9-AFS(1/⁴√2)
+! (1/⁴√2)-shifted harmonic series segment = 8-AFS(1/⁴√2)
-| 1.00 || 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) || 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
 |-
 ! style="background-color: #C59500;" |
-! (1/⁴√2)-shifted undertone series segment = 9-ALS(1/⁴√2)
+! (1/⁴√2)-shifted subharmonic series segment = 8-ALS(1/⁴√2)
-| 1.00 || 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
+| (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
 |-
 ! style="background-color: #00CFA9;" |
 ! AS5/4 = 1D JI lattice of 5/4 = 5/4 chain
-| 1/1 || 5/4 || 25/16 || 125/64 || 625/256 || 3125/1024 || 15625/4096 || 78125/16384 || 390625/65536 || 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
+| (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
 |-
 ! style="background-color: #F4CCCC;" |
-! 9 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2
+! 8 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2
-| 1.00 || 1.41 || 1.73 || 2.00 || 2.24 || 2.45 || 2.65 || 2.83 || 3.00 || 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
+| (√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
 |-
 ! style="background-color: #E06666;" |
-! 9 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2
+! 8 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2
-| 1.00 || 1.06 || 1.13 || 1.22 || 1.34 || 1.50 || 1.73 || 2.12 || 3.00 || 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
+| (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
 |-
 ! style="background-color: #00B6FF;" |
-! 9 pitches of 2-logharmonic series
+! 8 pitches of 2-logharmonic series
-| 1.00 || 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
+| (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
 |-
 ! style="background-color: #006D99;" |
-! 9 pitches of 2-sublogharmonic series
+! 8 pitches of 2-sublogharmonic series
-| 1.00 || 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
+| (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
 |}
-[[File:Frequency.svg|700px|link=https://en.xen.wiki/images/7/71/Frequency.svg]]
+[[File:Frequency (1).svg|alt=|700x700px]]
-[[File:Pitch.svg|700px|link=https://en.xen.wiki/images/0/09/Pitch.svg]]
+[[File:Pitch (1).svg|alt=|700x700px]]
-[[File:Length.svg|700px|link=https://en.xen.wiki/images/d/dc/Length.svg]]
+[[File:Length (1).svg|alt=|700x700px]]
-== Derivation of OS ==
-The tuning OS3/4 is the sequence <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math></span> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <span><math>\frac 13</math></span>. Let's show how.
-Begin with the overtone series:
-<math>1, 2, 3, 4...</math>
-Shift it by <span><math>\frac 13</math></span>:
-<math>
-\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\
-</math>
-Convert to improper fractions by first expanding the whole number:
-<math>
-\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\
-</math>
-...then consolidating numerators:
+== Notes on etymology ==
-<math>
+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.
-\frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}...
-</math>
-Resize to start at <span><math>\frac 11</math></span> by multiplying every term by the reciprocal of the first term, <span><math>\frac 43</math><span>, which is <span><math>\frac 34</math><span>:
+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.
-<math>
+== Notes on monotonicity ==
-\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:
+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.
-<math>
+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
-\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:
+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.
-<math>
+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.
-\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...
-</math>
-So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it.
+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.
-== A note on etymology ==
+== History ==
-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.
+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 [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread]. Other discussion occurred over email and private Facebook messages.
-== Credits ==
+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).
-The classifications and manners of specification for monotonic and arithmetic tunings here were developed by Douglas Blumeyer in March 2021, with significant input and guidance from Billy Stiltner, [[Shaahin Mohajeri]], [[Paul Erlich]], and [[Dave Keenan]]. Discussion occurred on Facebook on [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread].
+[[Category:Otonality and utonality]]
+[[Category:Otonality]]
+[[Category:Harmonic]]
+[[Category:Harmonic series‏‎]]
+[[Category:Utonality]]
+[[Category:Subharmonic]]
+[[Category:Subharmonic series‏‎]]
+[[Category:Equal-step tuning‏‎]]
+[[Category:Equal divisions of the octave‏‎]]