Benedetti height: Difference between revisions

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The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.

The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm.

The [[logarithm base two]] of the Benedetti height is the Tenney height, or [[Tenney norm]].

== Computation ==

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

~~<math>~~nd~~</math>~~

$$ nd $$

=== Vector form ===

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

~~<math>~~2^{\lVert H \vec m \rVert_1} \\

$$ 2^{\lVert H \vec m \rVert_1} = 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|} $$

= 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|}~~</math>~~

where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},

~~<math>~~H = \operatorname {diag} (\log_2 (Q))~~</math>~~

$$ H = \operatorname {diag} (\log_2 (Q)) $$

== Examples ==

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|-

| [[1/1]]

| {{~~monzo~~| 0 }}

| {{Monzo| 0 }}

| 1

|-

| [[2/1]]

| {{~~monzo~~| 1 }}

| {{Monzo| 1 }}

| 2

|-

| [[3/2]]

| {{~~monzo~~| -1 1 }}

| {{Monzo| -1 1 }}

| 6

|-

| [[6/5]]

| {{~~monzo~~| 1 1 -1 }}

| {{Monzo| 1 1 -1 }}

| 30

|-

| [[9/7]]

| {{~~monzo~~| 0 2 0 -1 }}

| {{Monzo| 0 2 0 -1 }}

| 63

|-

| [[13/11]]

| {{~~monzo~~| 0 0 0 0 -1 1 }}

| {{Monzo| 0 0 0 0 -1 1 }}

| 143

|}

@@ Line 1: / Line 1: @@
 The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.
-The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm.
+The [[logarithm base two]] of the Benedetti height is the Tenney height, or [[Tenney norm]].
 == Computation ==
@@ Line 7: / Line 7: @@
 The Benedetti height of a ratio ''n''/''d'' is given by
-<math>nd</math>
+$$ nd $$
 === Vector form ===
 The Benedetti height of a [[Harmonic limit|''p''-limit]] [[monzo]] '''m''' = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by
-<math>2^{\lVert H \vec m \rVert_1} \\
+$$ 2^{\lVert H \vec m \rVert_1} = 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|} $$
-= 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|}</math>
 where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},
-<math>H = \operatorname {diag} (\log_2 (Q))</math>
+$$ H = \operatorname {diag} (\log_2 (Q)) $$
 == Examples ==
@@ Line 26: / Line 25: @@
 |-
 | [[1/1]]
-| {{monzo| 0 }}
+| {{Monzo| 0 }}
 | 1
 |-
 | [[2/1]]
-| {{monzo| 1 }}
+| {{Monzo| 1 }}
 | 2
 |-
 | [[3/2]]
-| {{monzo| -1 1 }}
+| {{Monzo| -1 1 }}
 | 6
 |-
 | [[6/5]]
-| {{monzo| 1 1 -1 }}
+| {{Monzo| 1 1 -1 }}
 | 30
 |-
 | [[9/7]]
-| {{monzo| 0 2 0 -1 }}
+| {{Monzo| 0 2 0 -1 }}
 | 63
 |-
 | [[13/11]]
-| {{monzo| 0 0 0 0 -1 1 }}
+| {{Monzo| 0 0 0 0 -1 1 }}
 | 143
 |}