Vals and tuning space: Difference between revisions

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It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP|JIP]], which in weighted coordinates is {{val|1 1 1 ... 1}}. It has the property that if M is a monzo in weighted coordinates, then ~~{{val~~|~~JIP}}{{monzo|M}},~~ or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = {{val|1 log2(3) ... log2(p)}}, and applied to a monzo this gives the log base two of the corresponding interval.

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP|JIP]], which in weighted coordinates is {{val|1 1 1 ... 1}}. It has the property that if M is a monzo in weighted coordinates, then ⟨JIP|M⟩ or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = {{val|1 log2(3) ... log2(p)}}, and applied to a monzo this gives the log base two of the corresponding interval.

== Example ==

@@ Line 38: / Line 38: @@
 It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.
-It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP|JIP]], which in weighted coordinates is {{val|1 1 1 ... 1}}. It has the property that if M is a monzo in weighted coordinates, then {{val|JIP}}{{monzo|M}}, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = {{val|1 log2(3) ... log2(p)}}, and applied to a monzo this gives the log base two of the corresponding interval.
+It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP|JIP]], which in weighted coordinates is {{val|1 1 1 ... 1}}. It has the property that if M is a monzo in weighted coordinates, then ⟨JIP|M⟩ or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = {{val|1 log2(3) ... log2(p)}}, and applied to a monzo this gives the log base two of the corresponding interval.
 == Example ==