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''' 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 overtone 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]]

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== 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:

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

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

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

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

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

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See [[arithmetic tunings]].

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

== Non-arithmetic harmonotonic tunings ==

New arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the overtone series. But addition is not the only operation we could try applying to the frequencies of a basic ~~monotonic~~ overtone series.

New arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the overtone series. But addition is not the only operation we could try applying to the frequencies of a basic harmonotonic overtone series.

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

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

! colspan="5" |tuning type

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|'''[[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 a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).

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

== 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 ~~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.

Typically, the undertone series is displayed moving from 1/1, descending in pitch. To better illustrate the analogies between these different types of harmonotonic tunings, however, all tunings are shown starting on 1/1 but ascending in pitch. This means that all utonal tunings here were necessary to truncate, so that the lowest pitch would be known, so that the pitches' order could be reversed and the lowest pitch repositioned to 1/1 and all other pitches relative to that instead.

For a variety of perspectives, the frequency table values have been left unreduced, to illustrate the ideas behind the tuning better; the length values have been reduced.

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The first several examples repeat after 4 steps. Their second repetition is italicized in the tables below.

In the charts, dots indicate places where values are rational with respect to the given quantity kind. Thicker lines indicate ~~monotonic~~ tunings which have equal steps in the given quantity kind.

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 (f)

!

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== Notes on etymology ==

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

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.

== Notes on monotonicity ==

Pitches of any tuning could be sorted in order to be monotonic, so for monotonicity to be meaningful for tunings, it must be applied to their steps after sorting their pitches.

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

Monotonicity must also be restricted to within the interval of repetition to have much value. Otherwise, only scales with equal step sizes would qualify as monotonic, because any increase or decrease in step size would be countered with an opposite change once the scale repeated.

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== History ==

~~Monotonic~~ tunings are not at all new concepts. However, the classifications and manners of specification for ~~monotonic~~ and arithmetic tunings described here were developed by Douglas Blumeyer in March 2021, incorporating existing ideas and further significant input and guidance from Billy Stiltner, [[Shaahin Mohajeri]], [[Paul Erlich]], Joakim Bang Larsen, and [[Dave Keenan]]. Discussion occurred on Facebook on [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, and [[Dave Keenan]]. Discussion mostly occurred on Facebook on [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread].

[[Category:Otonality and utonality]]

@@ 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''' 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 overtone 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>
@@ Line 9: / Line 9: @@
 == 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:
@@ Line 19: / Line 19: @@
 And here are the three different '''types''':
-# '''[[Monotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. overtone or undertone series)
+# '''[[Haronotonic tunings#Arithmetic tunings|arithmetic]] & rational''' (e.g. overtone or undertone series)
 # '''arithmetic & irrational''' (e.g. EDO)
 # '''non-arithmetic & irrational'''
@@ Line 27: / Line 27: @@
 See [[arithmetic tunings]].
-== Non-arithmetic monotonic tunings ==
+== Non-arithmetic harmonotonic tunings ==
-New arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the overtone series. But addition is not the only operation we could try applying to the frequencies of a basic monotonic overtone series.
+New arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the overtone series. But addition is not the only operation we could try applying to the frequencies of a basic harmonotonic overtone series.
 The next operation above addition is multiplication. This operation is not very interesting, however, because multiplying frequency is equivalent to adding pitch, which does not meaningfully change a tuning; this merely transposes it. The reason multiplying frequency is equivalent to adding pitch is because pitch is found by taking the logarithm of frequency, and taking the logarithm of something effectively gears it down one operation lower on the hierarchy of operations: addition, multiplication, exponentiation, tetration, etc.
@@ Line 64: / Line 64: @@
 |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="3" rowspan="2" style="background-color: white; border-left: 1px solid white; border-top: 1px solid white;" |
 ! colspan="5" |tuning type
@@ Line 182: / Line 182: @@
 |'''[[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 a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
+[[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 monotonic tuning charts and graphs for comparison ==
+== 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 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.
+Typically, the undertone series is displayed moving from 1/1, descending in pitch. To better illustrate the analogies between these different types of harmonotonic tunings, however, all tunings are shown starting on 1/1 but ascending in pitch. This means that all utonal tunings here were necessary to truncate, so that the lowest pitch would be known, so that the pitches' order could be reversed and the lowest pitch repositioned to 1/1 and all other pitches relative to that instead.
 For a variety of perspectives, the frequency table values have been left unreduced, to illustrate the ideas behind the tuning better; the length values have been reduced.
@@ Line 192: / Line 192: @@
 The first several examples repeat after 4 steps. Their second repetition is italicized in the tables below.
-In the charts, dots indicate places where values are rational with respect to the given quantity kind. Thicker lines indicate monotonic tunings which have equal steps in the given quantity kind.
+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 (f)
 !
@@ Line 342: / Line 342: @@
 == Notes on etymology ==
-The "-tonic" root of "monotonic" does share etymology with the musical terms "tone" and "tonic". They both come from the Greek word "tonikos" which means "a stretching". This is also the explanation for "tonic" water, which supposedly relaxes you by stretching your muscles. So, the term "monotonic tuning" reunites these divergent applications of stretching — function values, and instrument strings — back into one place.
+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.
 == Notes on monotonicity ==
 Pitches of any tuning could be sorted in order to be monotonic, so for monotonicity to be meaningful for tunings, it must be applied to their steps after sorting their pitches.
+Monotonicity is also used in another sense in xenharmonics, however, for scales, which are not order-agnostic. In these cases, the monotonicity does apply to the pitches, not the steps. See: https://en.xen.wiki/index.php?search=monotone&title=Special%3ASearch&go=Go
 Monotonicity must also be restricted to within the interval of repetition to have much value. Otherwise, only scales with equal step sizes would qualify as monotonic, because any increase or decrease in step size would be countered with an opposite change once the scale repeated.
@@ Line 356: / Line 358: @@
 == History ==
-Monotonic tunings are not at all new concepts. However, the classifications and manners of specification for monotonic and arithmetic tunings described here were developed by Douglas Blumeyer in March 2021, incorporating existing ideas and further significant input and guidance from Billy Stiltner, [[Shaahin Mohajeri]], [[Paul Erlich]], Joakim Bang Larsen, and [[Dave Keenan]]. Discussion occurred on Facebook on [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, and [[Dave Keenan]]. Discussion mostly occurred on Facebook on [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread].
 [[Category:Otonality and utonality]]