Height: Difference between revisions
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The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s. | The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s. | ||
== Definition == | == Definition == | ||
A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement (see [[Wikipedia: Height function]]). For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes. | A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement (see [[Wikipedia: Height function]]). For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes. | ||
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== Examples of height functions == | == Examples of height functions == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Name | ! Name | ||
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| Height | | Height | ||
| <math>\text{sopfr}(n d)</math> | | <math>\text{sopfr}(n d)</math> | ||
| <math>2^{\large{\text{sopfr}( | | <math>2^{\large{\text{sopfr}(q)}}</math> | ||
| <math>\text{sopfr}(q)</math> | | <math>\text{sopfr}(q)</math> | ||
|- | |- | ||
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== History == | == History == | ||
The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488 Yahoo! Tuning Group | ''Super Particular Stepsize'']</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia: Height function]]). It is not to be confused with the musical notion of [[Wikipedia: Pitch (music) #Theories of pitch perception|''pitch height'' (as opposed to ''pitch chroma'')]]<ref>Though it has also been used to refer to the size of an interval in cents. On page 23 of [https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theor of Harmony''], Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it ''as opposed to'' his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".</ref>. | The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488 Yahoo! Tuning Group | ''Super Particular Stepsize'']</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia: Height function]]). It is not to be confused with the musical notion of [[Wikipedia: Pitch (music) #Theories of pitch perception|''pitch height'' (as opposed to ''pitch chroma'')]]<ref>Though it has also been used to refer to the size of an interval in cents. On page 23 of [https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theor of Harmony''], Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it ''as opposed to'' his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".</ref>. | ||
== See also == | == See also == | ||
* [[Commas by taxicab distance]] | * [[Commas by taxicab distance]] | ||
* [[Harmonic entropy]] | |||
== References == | == References == | ||