Consistency: Difference between revisions
m →Mathematical definition: use proof template |
|||
| Line 23: | Line 23: | ||
If for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', and if V (''r'') is ''r'' mapped by a val V, then ''N'' is consistent with respect to a set of intervals S if there exists a val V such that ''N''-edo (''r'') = V (''r'') for any ''r'' in S. | If for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', and if V (''r'') is ''r'' mapped by a val V, then ''N'' is consistent with respect to a set of intervals S if there exists a val V such that ''N''-edo (''r'') = V (''r'') for any ''r'' in S. | ||
{{ | {{Template:Proof|title=Proof for equivalence| | ||
Let us denote the monzo of any ratio ''r'' by m. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in S, their monzos are m<sub>''i''</sub>, m<sub>''j''</sub>, and m<sub>''i''</sub> + m<sub>''j''</sub>, respectively. | contents=Let us denote the monzo of any ratio ''r'' by m. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in S, their monzos are m<sub>''i''</sub>, m<sub>''j''</sub>, and m<sub>''i''</sub> + m<sub>''j''</sub>, respectively. | ||
The ratio ''r'' mapped by the val V is the tempered step number V (''r'') = V·m. There is the following identity: | The ratio ''r'' mapped by the val V is the tempered step number V (''r'') = V·m. There is the following identity: | ||