Harmonotonic tuning: Difference between revisions

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

== 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.

== ~~Credits~~ ==

== Notes on monotonicity ==

~~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].

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

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

In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same if the values are increasing or decreasing. In this sense, all monotonic tunings classified here are strictly monotonic.

Sometimes also absolutely monotonic sequences are distinguished, whose derivatives are all also monotonic. This is true for all tunings classified here. So all tunings here are absolutely and strictly monotonic by step size.

== History ==

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

@@ Line 388: / Line 388: @@
 [[File:Length.svg|700px|link=https://en.xen.wiki/images/d/dc/Length.svg]]
-== A note on etymology ==
+== 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.
-== Credits ==
+== Notes on monotonicity ==
-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].
+Pitches of any tuning could be sorted in order to be monotonic, so for monotonicity to be meaningful for tunings, it must be applied to their steps after sorting their pitches.
+Monotonicity must also be restricted to within the interval of repetition to have much value. Otherwise, only scales with equal step sizes would qualify as monotonic, because any increase or decrease in step size would be countered with an opposite change once the scale repeated.
+In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same if the values are increasing or decreasing. In this sense, all monotonic tunings classified here are strictly monotonic.
+Sometimes also absolutely monotonic sequences are distinguished, whose derivatives are all also monotonic. This is true for all tunings classified here. So all tunings here are absolutely and strictly monotonic by step size.
+== History ==
+Monotonic tunings are not at all new concepts. However, the classifications and manners of specification for monotonic and arithmetic tunings described here were developed by Douglas Blumeyer in March 2021, incorporating existing ideas and further significant input and guidance from Billy Stiltner, [[Shaahin Mohajeri]], [[Paul Erlich]], and [[Dave Keenan]]. Discussion occurred on Facebook on [https://www.facebook.com/groups/497105067092502/permalink/1980938532042474 this comment thread].