Kees semi-height: Difference between revisions
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Given a [[ratio]] of positive integers ''p''/''q'', the '''Kees semi-height''' is found by first removing factors of two and all common factors from ''p''/''q'', producing a ratio ''a''/''b'' of relatively prime odd positive integers. Then kees(''p''/''q'') = kees(''a''/''b'') = max(''a'', ''b''). The '''Kees expressibility''' is then the [[logarithm base two]] of the Kees semi-height. | Given a [[ratio]] of positive integers ''p''/''q'', the '''Kees semi-height''' is found by first removing factors of two and all common factors from ''p''/''q'', producing a ratio ''a''/''b'' of relatively prime odd positive integers. Then kees(''p''/''q'') = kees(''a''/''b'') = max(''a'', ''b''). The '''Kees expressibility''' is then the [[logarithm base two]] of the Kees semi-height. | ||
Expressibility can be extended to all vectors in [[Monzos and interval space|interval space]], by means of the formula<math> \lVert |m_2 \, m_3 \, m_5 \ldots m_p \rangle \rVert_{K1} = (|m_3 + m_5 + ... + m_p| + |m_3| + |m_5| + ... + |m_p|)/2</math> where "K1" denotes Kees expressibility and < | Expressibility can be extended to all vectors in [[Monzos and interval space|interval space]], by means of the formula | ||
<math> \lVert |m_2 \, m_3 \, m_5 \ldots m_p \rangle \rVert_{K1} = (|m_3 + m_5 + ... + m_p| + |m_3| + |m_5| + ... + |m_p|)/2</math> | |||
where "K1" denotes Kees expressibility and {{monzo| ''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>p</sub> }} is a vector with weighted coordinates in interval space. | |||
The set of JI intervals with Kees semi-height less than or equal to an odd integer q comprises the [[Odd limit|''q''-odd-limit]]. | The set of JI intervals with Kees semi-height less than or equal to an odd integer q comprises the [[Odd limit|''q''-odd-limit]]. | ||