Vals and tuning space: Difference between revisions
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{{Expert|Val}} | {{Expert|Val}} | ||
A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest. | A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest. | ||
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<math>\displaystyle | <math>\displaystyle | ||
\left<31 \; \frac{49}{\log_2(3)} \; \frac{72}{\log_2(5)} \; \frac{87}{\log_2(7)}\right| | \left< 31 \; \frac{49}{\log_2(3)} \; \frac{72}{\log_2(5)} \; \frac{87}{\log_2(7)}\right| | ||
</math> | |||
which is approximately {{val| 31.000 30.916 31.009 30.990 }}. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt (3838.694), or 61.957. To use the RMS we divide that by sqrt (4) = 2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31; any val closely approximating JI is expected to have the TE norm close to its division of the octave. | which is approximately {{val| 31.000 30.916 31.009 30.990 }}. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt (3838.694), or 61.957. To use the RMS we divide that by sqrt (4) = 2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31; any val closely approximating JI is expected to have the TE norm close to its division of the octave. | ||