Tenney norm: Difference between revisions

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The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney ~~height~~.

The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm.

== Computation ==

=== Ratio form ===

The Tenney ~~height~~ of a ratio ''n''/''d'' is given by

The Tenney norm of a ratio ''n''/''d'' is given by

=== Vector form ===

The Tenney ~~height~~ of a [[~~Harmonic~~ limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''1 ''m''2 … ''m''π (''p'') }}}} (π being the {{w|prime-counting function}}) is given by

The Tenney norm of a [[harmonic limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''1 ''m''2 … ''m''π (''p'') }}}} (π being the {{w|prime-counting function}}) is given by

<math>

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{| class="wikitable center-2"

|-

! Interval ~~Name~~

! Interval name

! Ratio (''n''/''d'')

! Monzo

! Tenney ~~Height~~

! Tenney norm

|-

| Unison

@@ Line 1: / Line 1: @@
 {{Wikipedia| James Tenney }}
-{{texops}}
+{{Texops}}
-The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.
+The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm.
 == Computation ==
 === Ratio form ===
-The Tenney height of a ratio ''n''/''d'' is given by
+The Tenney norm of a ratio ''n''/''d'' is given by
 <math>\log_2 (nd)</math>
 === Vector form ===
-The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w|prime-counting function}}) is given by
+The Tenney norm of a [[harmonic limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w|prime-counting function}}) is given by
 <math>
@@ Line 25: / Line 25: @@
 {| class="wikitable center-2"
 |-
-! Interval Name
+! Interval name
 ! Ratio (''n''/''d'')
 ! Monzo
-! Tenney Height
+! Tenney norm
 |-
 | Unison