Interval size measure: Difference between revisions
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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 can be derived from 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 can be derived from the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo). | ||
Steps are linear in the log-frequency domain, so expressions like {{nowrap|11\19 | Steps are linear in the log-frequency domain, so expressions like {{nowrap|11\19 − 6\19 {{=}} 5\19}} hold. In general, we have | ||
: {{nowrap|''a''\''N'' + ''b''\''N'' {{=}} (''a'' + ''b'')\''N''}} | : {{nowrap|''a''\''N'' + ''b''\''N'' {{=}} (''a'' + ''b'')\''N''}} | ||
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Or equivalently, for subtraction/division: | Or equivalently, for subtraction/division: | ||
: {{nowrap|''a''\''N'' | : {{nowrap|''a''\''N'' − ''b''\''N'' {{=}} (''a'' − ''b'')\''N''}} | ||
which expresses the same thing as {{nowrap|2<sup>''a''/''N''</sup> / 2<sup>''b''/''N''</sup> {{=}} 2<sup>(''a'' - ''b'')/''N''</sup>.}} | which expresses the same thing as {{nowrap|2<sup>''a''/''N''</sup> / 2<sup>''b''/''N''</sup> {{=}} 2<sup>(''a'' - ''b'')/''N''</sup>.}} | ||
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| [[43edo|43]] | | [[43edo|43]] | ||
| 43 (prime) | | 43 (prime) | ||
| Proposed by [[Joseph Sauveur]], as 7 heptaméride units<ref name="measure">[http://www.huygens-fokker.org/docs/measures.html | | Proposed by [[Joseph Sauveur]], as 7 heptaméride units<ref name="measure">[http://www.huygens-fokker.org/docs/measures.html Stitching Huygens–Fokker: Logarithmic Interval Measures]</ref><ref>[http://tonalsoft.com/enc/m/meride.aspx Tonalsoft | ''Méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament'']</ref>. | ||
|- | |- | ||
| [[Holdrian comma]] | | [[Holdrian comma]] | ||
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| Fourteenth MIDI-resolution unit, 1/16384 (1/(2<sup>14</sup>)) of [[12edo]] semitone<ref name="measure"/>. | | Fourteenth MIDI-resolution unit, 1/16384 (1/(2<sup>14</sup>)) of [[12edo]] semitone<ref name="measure"/>. | ||
|} | |} | ||
<nowiki />* More to be added regarding the Heptaméride/Savart units | |||
<nowiki>* | |||
==== Non-octave fine measures ==== | ==== Non-octave fine measures ==== | ||
There are other fine measurements based upon the logarithmic division of other intervals (e.g. 3/1 (twelfth)), a few of which are listed below: | There are other fine measurements based upon the logarithmic division of other intervals (e.g. 3/1 (twelfth)), a few of which are listed below: | ||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
|- | |+ style="font-size: 105%;" List of non-octave fine measures (logarithmic) | ||
|- | |- | ||
! Unit name (symbol) | ! Unit name (symbol) | ||
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| 3/1 (twelfth) | | 3/1 (twelfth) | ||
| 1300 | | 1300 | ||
| 1/100 of 13edt ( | | 1/100 of 13edt (Bohlen–Pierce) scale step | ||
|- | |- | ||
| Euhekt | | Euhekt | ||
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which is a diatonic semitone below an octave {{nowrap|([[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, {{nowrap|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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* [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small''] by [[Don N. Page]] | * [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small''] by [[Don N. Page]] | ||
== | == References == | ||
<references /> | <references /> | ||
[[Category:Interval]] | [[Category:Interval]] | ||