Chord complexity: Difference between revisions

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= ''N''-adic simple complexity/otonalness =

Suppose that {{nowrap|''x''1 : ''x''2 : ''x''3 : ... : ''x''''N''}} is some ''N''-note JI chord. Then some function {{nowrap|''f''(''x''1, ''x''2, ..., ''x''''N''}} can be said to measure the '''simple complexity''' or '''otonalness''' of our N-note chord if it is monotonically increasing in each variable separately. We will sometimes use the notation {{nowrap|'''X''' {{=}} ''x''1 : ''x''2 : ''x''3 : ... : ''x''''N''}}, so that our simple complexity function can be written as ''f''('''X''').

Suppose that {{nowrap|''x''1 : ''x''2 : ''x''3 : ... : ''x''''N''}} is some ''N''-note JI chord. Then some function {{nowrap|''f''(''x''1, ''x''2, ..., ''x''''N'')}} can be said to measure the '''simple complexity''' or '''otonalness''' of our N-note chord if it is monotonically increasing in each variable separately. We will sometimes use the notation {{nowrap|'''X''' {{=}} ''x''1 : ''x''2 : ''x''3 : ... : ''x''''N''}}, so that our simple complexity function can be written as ''f''('''X''').

We may also look at functions ''f'' which are defined for chords with a variable number of notes. This can also be thought of as a family of functions {{nowrap|''f''1, ''f''2, ''f''3, ...}}, each giving the simple complexity for chords of size 1, 2, 3, etc.

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 = ''N''-adic simple complexity/otonalness =
-Suppose that {{nowrap|''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}} is some ''N''-note JI chord. Then some function {{nowrap|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub>}} can be said to measure the '''simple complexity''' or '''otonalness''' of our N-note chord if it is monotonically increasing in each variable separately. We will sometimes use the notation {{nowrap|'''X''' {{=}} ''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}}, so that our simple complexity function can be written as ''f''('''X''').
+Suppose that {{nowrap|''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}} is some ''N''-note JI chord. Then some function {{nowrap|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub>)}} can be said to measure the '''simple complexity''' or '''otonalness''' of our N-note chord if it is monotonically increasing in each variable separately. We will sometimes use the notation {{nowrap|'''X''' {{=}} ''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}}, so that our simple complexity function can be written as ''f''('''X''').
 We may also look at functions ''f'' which are defined for chords with a variable number of notes. This can also be thought of as a family of functions {{nowrap|''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>, ...}}, each giving the simple complexity for chords of size 1, 2, 3, etc.