Interval size measure: Difference between revisions
backslash notation, strictly speaking, is neither logarithmic nor linear (as it can be both depending on context), and the amount of confusing and irrelevant info to the meaning of backslash notation included in "Gross" (which is also unnecessarily technical) is ridiculous so that it deserves a separate section linked for clarity; prior it did not even define k\N precisely, just refers back to the ambiguous term "steps" which causes confusion to a beginner (recalling from my own experience) |
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A common shorthand in use in the microtonal community is ''k''\''N'', written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2<sup>''k''/''N''</sup>. ''k''\''N'' is pronounced "''k'' steps of ''N'' [[edo]]", and suffices to define the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo). | A common shorthand in use in the microtonal community is ''k''\''N'', written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2<sup>''k''/''N''</sup>. ''k''\''N'' is pronounced "''k'' steps of ''N'' [[edo]]", and suffices to define the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo). | ||
It is logarithmic because it's common to see expressions like 11\19 | It is logarithmic because it's common to see expressions like {{nowrap|11\19 − 6\19 {{=}} 5\19}}, which is an additive notation, though the frequency ratios referred to in the definition are linear, which is to say: | ||
: ''a''\''N'' + ''b''\''N'' = (''a'' + ''b'')\''N'' expresses the same thing as 2<sup>''a''/''N''</sup> × 2<sup>''b''/''N''</sup> = 2<sup>(''a'' + ''b'')/''N''</sup> | : {{nowrap|''a''\''N'' + ''b''\''N'' {{=}} (''a'' + ''b'')\''N''}} expresses the same thing as {{nowrap|2<sup>''a''/''N''</sup> × 2<sup>''b''/''N''</sup> {{=}} 2<sup>(''a'' + ''b'')/''N''</sup>.}} | ||
Or equivalently, for subtraction/division: | Or equivalently, for subtraction/division: | ||
: ''a''\''N'' | : {{nowrap|''a''\''N'' − ''b''\''N'' = (''a'' − ''b'')\''N''}} expresses the same thing as {{nowrap|2<sup>''a''/''N''</sup> / 2<sup>''b''/''N''</sup> {{=}} 2<sup>(''a'' - ''b'')/''N''</sup>.}} | ||
=== Gross === | === Gross === | ||
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{| class="wikitable sortable" | {| class="wikitable sortable" | ||
|+ List of Octave-Based Fine Measures (Logarithmic) | |+ style="font-size: 105%;" | List of Octave-Based Fine Measures (Logarithmic) | ||
|- | |- | ||
! Unit Name (Symbol): | ! Unit Name (Symbol): | ||
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Intervals can be measured also giving their [[ratio]]. For instance the major third as [[5/4]] or the pure fifth [[3/2]]. When combining sizes given in ratios, you have to multiply or divide: | Intervals can be measured also giving their [[ratio]]. For instance the major third as [[5/4]] or the pure fifth [[3/2]]. When combining sizes given in ratios, you have to multiply or divide: | ||
a pure fifth increased by a major third gives the major seventh 3/2 × 5/4 = [[15/8]], | a pure fifth increased by a major third gives the major seventh {{nowrap|3/2 × 5/4 {{=}} [[15/8]]}}, | ||
which is a diatonic semitone below an octave ([[2/1]]) / (15/8) = 2/1 × 8/15 = [[16/15]]. | which is a diatonic semitone below an octave {{nowrap|([[2/1]]) / (15/8) {{=}} 2/1 × 8/15 {{=}} [[16/15]]}}. | ||
Another notation for ratios is a vector of prime factor exponents, often called a [[monzo]], such as {{monzo| -4 4 -1 }} (for the syntonic comma, 81/80 = 2<sup> | Another notation for ratios is a vector of prime factor exponents, often called a [[monzo]], such as {{monzo| -4 4 -1 }} (for the syntonic comma, {{nowrap|81/80 = 2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors. | ||
== See also == | == See also == | ||
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== Notes == | == Notes == | ||
<references/> | <references /> | ||
[[Category:Interval]] | [[Category:Interval]] | ||