Xenharmonic Wiki

Interval size measure: Difference between revisions

← Older edit
VisualWikitext
→Logarithmic: rework backslash notation: set the logic straight by reversing the direction of causation
Update
 
(21 intermediate revisions by 8 users not shown)
Line 1: Line 1:
'''Interval size measure''' or '''interval size unit''' means the ''distance'' between pitches. Intervals can be measured [[#logarithmic|logarithmic]] or by frequency [[#ratio|ratios]].
'''Interval size measure''' or '''interval size unit''' means the ''distance'' between pitches. Intervals can be measured [[#logarithmic|logarithmic]] or by frequency [[#ratio|ratios]].


== Logarithmic ==
== Logarithmic ==
Line 8: Line 7:
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 &minus; 6\19 {{=}} 5\19}} hold. In general, we have
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''}}


Line 15: Line 14:
Or equivalently, for subtraction/division:
Or equivalently, for subtraction/division:


: {{nowrap|''a''\''N'' &minus; ''b''\''N'' {{=}} (''a'' &minus; ''b'')\''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>.}}
Line 26: Line 25:
Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.
Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.


In serial music, all intervals were measured by the number of 12edo [[semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using [[#Backslash notation|backslash notation]] if the degree itself isn't sufficiently clear in context.
In serial music, all intervals were measured by the number of 12edo [[semitone (interval size measure)|semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using [[#Backslash notation|backslash notation]] if the degree itself isn't sufficiently clear in context.


=== Fine ===
=== Fine ===
Line 32: Line 31:


==== Octave-based fine measures ====
==== Octave-based fine measures ====
The following table demonstrates a list of measures derived from the logarithmic division of the octave:
The following table demonstrates a list of measures derived from the logarithmic division of the octave: {{todo|complete table|research|comment=Add all missing citations.}}


{| class="wikitable sortable"
{| class="wikitable sortable"
|+ style="font-size: 105%;" | List of Octave-Based Fine Measures (Logarithmic)
|+ style="font-size: 105%;" | List of octave-based fine measures (logarithmic)
|-
|-
! Unit Name (Symbol):
! Unit name (symbol):
! Divisions of Octave
! Divisions of octave
! Prime Factors
! Prime factors
! Origin / Significance
! Origin/significance
|-
|-
| [[Eka]]
| [[Eka]]
| [[16edo|16]]
| [[16edo|16]]
| 2<sup>4</sup>
| 2<sup>4</sup>
| From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16edo Theory{{Citation needed}}.  
| From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16edo theory<ref>[http://www.armodue.com/risorse.htm Armodue: le risorse di un nuovo sistema musicale]</ref>.
|-
| [[Normal shruti]]
| [[22edo|22]]
| 2 × 11
| Proposed by [[User:Tristanbay|Tristan Bay]] (2025) in reference to the Indian tradition of dividing the octave into 22 unequal parts.
|-
|-
| [[Normal diesis]]
| [[Normal diesis]]
Line 51: Line 55:
| 31 (prime)
| 31 (prime)
| See the dedicated page.  
| See the dedicated page.  
|-
| [[Dea]]
| [[41edo|41]]
| 41 (prime)
| Proposed by [[User:Tristanbay|Tristan Bay]] (2025) to reflect that a mina is a "minute" (1/60 the width) of a 1\41 "degree".
|-
|-
| [[Méride]]
| [[Méride]]
| [[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 Stichting 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>.  
| Proposed by [[Joseph Sauveur]], as 7 heptaméride units<ref name="measure">[http://www.huygens-fokker.org/docs/measures.html Stichting 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]]
| [[53edo|53]]
| [[53edo|53]]
| 53 (prime)
| 53 (prime)
| <ref name="measure"/>
| See the dedicated page.
|-
| [[Holdrian comma|Mercator's old comma]]
| [[55edo|55]]
| 5 × 11
| Not to be confused with [[Mercator's comma]].
|-
|-
| [[Decitone]]
| [[Decitone]]
| [[60edo|60]]
| [[60edo|60]]
| 2<sup>2</sup> × 3 × 5
| 2<sup>2</sup> × 3 × 5
|
| Standard SI prefix + 12edo tone
|-
|-
| [[Morion]]
| [[Morion]]
| [[72edo|72]]
| [[72edo|72]]
| 2<sup>3</sup> × 3<sup>2</sup>
| 2<sup>3</sup> × 3<sup>2</sup>
| See dedicated page.  
| See the dedicated page.  
|-
|-
| [[Farab]]
| [[Farab]]
| [[144edo|144]]
| [[144edo|144]]
| 2<sup>4</sup> × 3<sup>2</sup>
| 2<sup>4</sup> × 3<sup>2</sup>
| 1/12 of [[12edo]] semitone; Proposed by [[al-Farabi]] in 10th century<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/farab.aspx Tonalsoft | ''Farab''].</ref>.  
| 1/12 of [[12edo]] semitone; proposed by [[al-Farabi]] in 10th century<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/farab.aspx Tonalsoft | ''Farab''].</ref>.  
|-
|-
| [[Mem]]
| [[Mem]]
Line 110: Line 124:
| [[494edo|494]]
| [[494edo|494]]
| 2 × 13 × 19
| 2 × 13 × 19
| {{Citation needed}}
| Named after [[729/728]], the squbema, due to its similar size.
|-
|-
| Great [[iring]] / [[centitone]]
| [[Great iring]] / [[great centitone|centitone]]
| [[500edo|500]]
| [[500edo|500]]
| 2<sup>2</sup> × 5<sup>3</sup>
| 2<sup>2</sup> × 5<sup>3</sup>
Line 125: Line 139:
| [[600edo|600]]
| [[600edo|600]]
| 2<sup>3</sup> × 3 × 5<sup>2</sup>
| 2<sup>3</sup> × 3 × 5<sup>2</sup>
| [[Relative cent]] of [[6edo]] ([[12edo]] tone); Proposed by [[Widogast Iring]] (1898), later by [[Joseph Yasser]] as a "centitone" (1932)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/c/centitone.aspx Tonalsoft | ''Centitone, iring'']</ref>.
| [[Relative cent]] of [[6edo]]; proposed by [[Widogast Iring]] (1898), later by [[Joseph Yasser]] as a "centitone", a standard SI prefix + 12edo tone (1932)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/c/centitone.aspx Tonalsoft | ''Centitone, iring'']</ref>.  
|-
|-
| [[Skisma]] (Sk)
| [[Nil]] / [[skisma]] (Sk)
| [[612edo|612]]
| [[612edo|612]]
| 2<sup>2</sup> × 3<sup>2</sup> × 17
| 2<sup>2</sup> × 3<sup>2</sup> × 17
| Edo representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"<ref name="measure"/><ref>[http://tonalsoft.com/enc/s/sk.aspx Tonalsoft | ''Sk, 612-edo'']</ref>.  
| Proposed by [[James Paul White]] (1894) as ''nil'', and by Gene Ward Smith (2007) as ''skisma''<ref name="measure"/><ref>[http://tonalsoft.com/enc/s/sk.aspx Tonalsoft | ''Sk, 612-edo'']</ref>. Edo representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina".  
|-
|-
| [[Delfi]]
| [[Delfi]]
Line 137: Line 151:
| <ref name="measure"/>
| <ref name="measure"/>
|-
|-
| Small [[iring]] / [[centitone]]
| [[Small iring]] / [[small centitone|centitone]]
| [[700edo|700]]
| [[700edo|700]]
| 2<sup>2</sup> × 5<sup>2</sup> x 7
| 2<sup>2</sup> × 5<sup>2</sup> x 7
| {{Citation needed}}
| {{Citation needed}}
|-
|-
| [[Woolhouse]]
| [[Woolhouse unit]]
| [[730edo|730]]
| [[730edo|730]]
| 2 × 5 × 73
| 2 × 5 × 73
Line 157: Line 171:
| See the dedicated page.  
| See the dedicated page.  
|-
|-
| Greater muon
| Dingle
| [[1224edo|1224]]
| [[1395edo|1395]]
| 2<sup>3</sup> × 3<sup>2</sup> × 17
| 3<sup>2</sup> × 5 × 31
| {{Citation needed}}
| Proposed by [[User:Tristanbay|Tristan Bay]] (2026) as a 31edo-friendly fine-grain measure, shortened from "'''di'''esis a'''ngle'''".
|-
| Triangular cent
| [[1260edo|1260]]
| 2<sup>2</sup> × 3<sup>2</sup> × 5 × 7
| {{Citation needed}}
|-
| Pion
| [[1272edo|1272]]
| 2<sup>3</sup> × 3 × 53
| {{Citation needed}}
|-
| Pound
| [[1344edo|1344]]
| 2<sup>6</sup> × 3 × 7
| {{Citation needed}}
|-
| Neutron
| [[1392edo|1392]]
| 2<sup>4</sup> × 3 × 29
| {{Citation needed}}
|-
| Lesser muon
| [[1428edo|1428]]
| 2<sup>2</sup> × 3 × 7 × 17
| {{Citation needed}}
|-
|-
| Decifarab
| Decifarab
| [[1440edo|1440]]
| [[1440edo|1440]]
| 2<sup>5</sup> × 3<sup>2</sup> × 5
| 2<sup>5</sup> × 3<sup>2</sup> × 5
| 1/10 of [[Farab]] unit<ref name="measure"/>.  
| Standard SI prefix + [[farab]]<ref name="measure"/>.  
|-
| Quadratic cent
| [[1452edo|1452]]
| 2<sup>2</sup> × 3 × 11<sup>2</sup>
| {{Citation needed}}
|-
| Ksion
| [[1476edo|1476]]
| 2<sup>2</sup> × 3<sup>2</sup> × 41
| {{Citation needed}}
|-
| Cubic cent
| [[1500edo|1500]]
| 2<sup>2</sup> × 3 × 5<sup>3</sup>
| {{Citation needed}}
|-
|-
| Heptamu (7mu)
| Heptamu (7mu)
Line 211: Line 185:
| 2<sup>9</sup> × 3
| 2<sup>9</sup> × 3
| Seventh MIDI-resolution unit, 1/128 (1/(2<sup>7</sup>)) of [[12edo]] semitone<ref>[http://tonalsoft.com/enc/number/7mu.aspx Tonalsoft | ''7mu / heptamu'']</ref>
| Seventh MIDI-resolution unit, 1/128 (1/(2<sup>7</sup>)) of [[12edo]] semitone<ref>[http://tonalsoft.com/enc/number/7mu.aspx Tonalsoft | ''7mu / heptamu'']</ref>
|-
| Rhoon
| [[1560edo|1560]]
| 2<sup>3</sup> × 3 × 5 × 13
| {{Citation needed}}
|-
|-
| śata
| śata
| [[1600edo|1600]]
| [[1600edo|1600]]
| 2<sup>6</sup> × 5<sup>2</sup>  
| 2<sup>6</sup> × 5<sup>2</sup>  
| From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue 16edo Theory{{Citation needed}}
| From Sanskrit ''śatam'': hundred; [[relative cent]] of Armodue 16edo theory{{Citation needed}}
|-
| Tile
| [[1632edo|1632]]
| 2<sup>5</sup> × 3 × 17
| {{Citation needed}}
|-
|-
| [[Iota]]
| [[Iota]]
Line 256: Line 220:
| 2<sup>2</sup> × 5<sup>2</sup> x 43
| 2<sup>2</sup> × 5<sup>2</sup> x 43
| {{Citation needed}}
| {{Citation needed}}
|-
| [[4320edo|Click]]
| [[4320edo|4320]]
| 2<sup>5</sup> × 3<sup>3</sup> × 5
| Proposed by [[User:Eliora|Eliora]]. See the dedicated page.
|-
|-
| [[Major tina]]
| [[Major tina]]
Line 270: Line 239:
| [[9900edo|9900]]
| [[9900edo|9900]]
| 2<sup>2</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 11
| 2<sup>2</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 11
| [[Relative cent]] of [[99edo]]; Suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.  
| [[Relative cent]] of [[99edo]]; suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.  
|-
|-
| [[Türk sent]] / [[Turkish cent]]
| [[Türk sent]] / [[Turkish cent]]
Line 290: Line 259:
| [[30103edo|30103]]
| [[30103edo|30103]]
| 30103 (prime)
| 30103 (prime)
| 30103 ≃ 100,000 × log<sub>10</sub>2; Proposed by [[Augustus de Morgan]] (1864)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/j/jot.aspx Tonalsoft | ''Jot'']</ref><ref name="equal"/>.  
| 30103 ≃ 100,000 × log<sub>10</sub>2; proposed by [[Augustus de Morgan]] (1864)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/j/jot.aspx Tonalsoft | ''Jot'']</ref><ref name="equal"/>.  
|-
|-
| [[Imp]]
| [[Imp]]
Line 305: Line 274:
| [[78005edo|78005]]
| [[78005edo|78005]]
| 5 × 15601
| 5 × 15601
| Name proposed by Tristan Bay in 2023; 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system. {{Citation needed}}
| Proposed by Tristan Bay (2023); 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system.
|-
|-
| [[MIDI Tuning Standard unit]] (14mu)
| [[MIDI Tuning Standard unit]] (14mu)
Line 312: Line 281:
| 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>*</nowiki> More to be added regarding the Heptaméride/Savart units


==== 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)
|-
|-
|+List of Non-Octave Fine Measures (Logarithmic)
! Unit name (symbol)
|-
! Base interval
! Unit Name (Symbol):
! Divisions of base interval
! Base Interval:
! Origin/significance
! Parts of Base Interval:
! Origin/Significance
|-
|-
| Hekt
| Hekt
| 3/1 (twelfth)
| 3/1 (twelfth)
| 1300
| 1300
| 1/100 of 13edt (Bohlen-Pierce) scale step
| 1/100 of 13edt (Bohlen–Pierce) scale step
|-
|-
| Euhekt
| Euhekt
| 3/1 (twelfth)
| 3/1 (twelfth)
| 1900
| 3900
| 1/100 of 19edt (OnlyPure) scale step
| 1/100 of 39edt (Triboh) scale step
|-
|-
| Grad
| Grad
Line 352: Line 318:


=== Relative measures ===
=== Relative measures ===
Within a given [[Equal-step tuning|equal-stepped]] tonal system, the [[relative cent]] (rct, r¢) can be used to describe properties of pitches (for instance the approximation of [[Just intonation|JI]] intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.
Within a given [[equal-step tuning|equal-stepped tuning system]], the [[relative cent]] (rct, r¢) can be used to describe properties of pitches (for instance the approximation of [[Just intonation|JI]] intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.


== Ratio ==
== Ratio ==
Line 361: Line 327:
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>&minus;4</sup> × 3<sup>4</sup> × 5<sup>&minus;1</sup>}}), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.
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 ==
Line 369: Line 335:
* [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]]


== Notes ==
== References ==
<references />
<references />


[[Category:Interval]]
[[Category:Interval]]
Retrieved from "https://en.xen.wiki/w/Interval_size_measure"