Harmonotonic tuning: Difference between revisions

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A '''harmonotonic 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'' 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).

* A segment of the harmonic 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.

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

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

== Categorization ==

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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''':

# '''[[Harmonotonic 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 ==

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== 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.

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.

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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'''step size'''

! rowspan="2" |basic

|'''[[~~Overtone series|overtone series, or~~ harmonic series]]'''

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

! rowspan="2" |

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

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

! rowspan="2" |

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

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

|-

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

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

|-

! rowspan="2" |division

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

|[[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]]

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'''step size'''

! rowspan="2" |basic

|'''[[~~wikipedia:Undertone_series|undertone series, or~~ subharmonic series]]'''

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

! rowspan="2" |

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

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

! rowspan="2" |

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

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

|-

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

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

|

|-

! rowspan="2" |division

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

|[[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]]

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== 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.

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.

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.

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|-

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

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

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

| (4/4) || 5/4 || 6/4 || 7/4 || 8/4 ||''5/2''||''3/1''||''7/2''||''4/1''

!

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|-

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

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

! 4-UDO = 4th subharmonic mode

| (8/8) || 8/7 || 8/6 || 8/5 || 8/4 ||''16/7''||''8/3''||''16/5''||''4/1''

!

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|-

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

! ~~overtone~~ series segment = 8-OS = 8-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

!

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|-

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

! ~~undertone~~ series segment = 8-US = 8-UD9

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

| (9/9) || 9/8 || 9/7 || 9/6 || 9/5 || 9/4 || 9/3 || 9/2 || 9/1

!

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|-

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

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

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

| (1) || 1.84 || 2.68 || 3.52 || 4.36 || 5.20 || 6.05 || 6.89 || 7.73

!

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|-

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

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

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

| (1) || 1.12 || 1.28 || 1.48 || 1.77 || 2.19 || 2.88 || 4.20 || 7.73

!

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

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

| √1 || √2 || √3 || √4 || √5 || √6 || √7 || √8 || √9

| (√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

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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.

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.

~~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~~.

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, ~~and~~ [[Dave Keenan]]. Discussion mostly occurred on Facebook on [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread].

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.

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).

[[Category:Otonality and utonality]]

~~[[Category:Overtone]]~~

~~[[Category:Overtone‏‎ series]]~~

[[Category:Otonality]]

[[Category:Harmonic]]

[[Category:Harmonic series‏‎]]

~~[[Category:Otonality and utonality]]~~

~~[[Category:Undertone]]~~

~~[[Category:Undertone series]]~~

[[Category:Utonality]]

[[Category:Subharmonic]]

[[Category:Subharmonic series‏‎]]

[[Category:Equal-step tuning‏‎]]

[[Category:Equal divisions of the octave‏‎ ]]

[[Category:Equal divisions of the octave‏‎]]

@@ Line 1: / Line 1: @@
-A '''harmonotonic 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'' 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).
+* A segment of the harmonic 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.
 <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>
+Essentially, "harmonotonic" references the mathematical concept of monotonicity to form an umbrella term for tunings which are closely related to the harmonic series.
 == Categorization ==
@@ Line 13: / Line 15: @@
 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''':
-# '''[[Harmonotonic 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 ==
@@ Line 29: / Line 33: @@
 == 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.
+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.
 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 35: / 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 50: / Line 54: @@
 !shifting
 |addition
-|[[wikipedia:Gaussian_logarithm|Gaussian logarithm]]
+|[[wikipedia:Gaussian_logarithm|Gaussian logarithmic addition]]
 |-
 !transposition
@@ Line 87: / Line 91: @@
 '''step size'''
 ! rowspan="2" |basic
-|'''[[Overtone series|overtone series, or harmonic series]]'''
+|'''[[harmonic series]]'''
 ! rowspan="2" |
-| rowspan="2" |[[AFS|'''irrationally shifted overtone series''' (± frequency) ''(equivalent to AFS)'']]
+| rowspan="2" |[[AFS|'''irrationally shifted harmonic series''' (± frequency) ''(equivalent to AFS)'']]
 ! rowspan="2" |
-| rowspan="2" | [[Powharmonic series|'''stretched/compressed overtone series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'']]
+| rowspan="2" | [[Powharmonic series|'''stretched/compressed harmonic series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'']]
 |-
-|[[OS|'''rationally shifted overtone series''' (± frequency) ''(equivalent to OS)'']]
+|[[OS|'''rationally shifted harmonic series''' (± frequency) ''(equivalent to OS)'']]
 |-
 ! rowspan="2" |division
-|[[Overtone scale#Over-n Scales|'''overtone mode, or over-n scale''' ''(equivalent to n-ODO)'']]
+|[[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>]]
@@ Line 148: / Line 152: @@
 '''step size'''
 ! rowspan="2" |basic
-|'''[[wikipedia:Undertone_series|undertone series, or subharmonic series]]'''
+|'''[[subharmonic series]]'''
 ! rowspan="2" |
-| rowspan="2" |[[ALS|'''irrationally shifted undertone series''' (± frequency) ''(equivalent to ALS)'']]
+| rowspan="2" |[[ALS|'''irrationally shifted subharmonic series''' (± frequency) ''(equivalent to ALS)'']]
 ! rowspan="2" |
-|[[Powharmonic series|'''stretched/compressed undertone series''' (exponentiated frequency, multiplied pitch)  ''(equivalent to subpowharmonic series)'']]
+|[[Powharmonic series|'''stretched/compressed subharmonic series''' (exponentiated frequency, multiplied pitch)  ''(equivalent to subpowharmonic series)'']]
 |-
-|[[US|'''rationally shifted undertone series''' (± frequency) ''(equivalent to US)'']]
+|[[US|'''rationally shifted subharmonic series''' (± frequency) ''(equivalent to US)'']]
 |
 |-
 ! rowspan="2" |division
-|[[Overtone scale#Next Steps|'''undertone mode, or under-n scale''' ''(equivalent to n-UDO)'']]
+|[[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>]]
@@ Line 186: / Line 190: @@
 == 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.
+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.
 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.
@@ Line 208: / Line 212: @@
 |-
 ! style="background-color: #00DB00;" |
-! 4-ODO = 4th overtone mode = 4-ADO
+! 4-ODO = 4th harmonic mode = 4-ADO
 | (4/4) || 5/4 || 6/4 || 7/4 || 8/4 ||''5/2''||''3/1''||''7/2''||''4/1''
 !
@@ Line 224: / Line 228: @@
 |-
 ! style="background-color: #45818E;" |
-! 4-UDO = 4th undertone mode
+! 4-UDO = 4th subharmonic mode
 | (8/8) || 8/7 || 8/6 || 8/5 || 8/4 ||''16/7''||''8/3''||''16/5''||''4/1''
 !
@@ Line 248: / Line 252: @@
 |-
 ! style="background-color: #D2EB00;" |
-! overtone series segment = 8-OS = 8-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
 !
@@ Line 256: / Line 260: @@
 |-
 ! style="background-color: #A1A200;" |
-! undertone series segment = 8-US = 8-UD9
+! subharmonic series segment = 8-US = 8-UD9
 | (9/9) || 9/8 || 9/7 || 9/6 || 9/5 || 9/4 || 9/3 || 9/2 || 9/1
 !
@@ Line 280: / Line 284: @@
 |-
 ! style="background-color: #FFC000;" |
-! (1/⁴√2)-shifted overtone series segment = 8-AFS(1/⁴√2)
+! (1/⁴√2)-shifted harmonic series segment = 8-AFS(1/⁴√2)
 | (1) || 1.84 || 2.68 || 3.52 || 4.36 || 5.20 || 6.05 || 6.89 || 7.73
 !
@@ Line 288: / Line 292: @@
 |-
 ! style="background-color: #C59500;" |
-! (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2)
+! (1/⁴√2)-shifted subharmonic series segment = 8-ALS(1/⁴√2)
 | (1) || 1.12 || 1.28 || 1.48 || 1.77 || 2.19 || 2.88 || 4.20 || 7.73
 !
@@ Line 305: / Line 309: @@
 ! style="background-color: #F4CCCC;" |
 ! 8 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2
-| √1 || √2 || √3 || √4 || √5 || √6 || √7 || √8 || √9
+| (√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
@@ Line 354: / Line 358: @@
 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.
+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.
-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.
+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, and [[Dave Keenan]]. Discussion mostly occurred on Facebook on [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread].
+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.
+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).
 [[Category:Otonality and utonality]]
-[[Category:Overtone]]
-[[Category:Overtone‏‎ series]]
 [[Category:Otonality]]
 [[Category:Harmonic]]
 [[Category:Harmonic series‏‎]]
-[[Category:Otonality and utonality]]
-[[Category:Undertone]]
-[[Category:Undertone series]]
 [[Category:Utonality]]
 [[Category:Subharmonic]]
 [[Category:Subharmonic series‏‎]]
 [[Category:Equal-step tuning‏‎]]
-[[Category:Equal divisions of the octave‏‎ ]]
+[[Category:Equal divisions of the octave‏‎]]