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Interval size measure: Difference between revisions

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== Logarithmic ==
== Logarithmic ==
All logarithmic measures can be combined by adding and subtracting them.
All logarithmic measures can be combined by adding and subtracting them.
=== Backslash notation ===
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 − 6\19 {{=}} 5\19}} hold. In general, we have
: {{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>.}}
Or equivalently, for subtraction/division:
: {{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>.}}
Backslash notation can be extended to support [[nonoctave]] [[equal tuning]]s by writing the tuning in full after the backslash. For example, 11\13edt means 11 steps of [[13edt]], 14\9edf means 14 steps of [[9edf]], and 7\12ed12/5 means 7 steps of [[12ed12/5]].


=== Gross ===
=== Gross ===
The [[octave]] and the [[decade]] are common coarse units for interval sizes. The {{w|decibel}}, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the {{w|neper}} (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in [[logarithmic approximants]].
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 semitones. 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 the '''backslash notation''', which looks like a frequency ratio but using a backslash (instead of a forward slash) to indicate a logarithmic ratio. For example, 11\15 means 11 steps of 15edo, 4\9edf means 4 steps of 9edf, and 16\21ed12/7 means 16 steps of 21ed12/7.
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 ===
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==== 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"
|+ 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]]
| [[31edo|31]]
| [[31edo|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]]
| 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]]
| 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]]
| [[60edo|60]]
| 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]]
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| [[270edo|270]]
| [[270edo|270]]
| 2 × 3<sup>3</sup> × 5
| 2 × 3<sup>3</sup> × 5
| Proposed by [[Joe Monzo]] (2013)<ref>[http://tonalsoft.com/enc/t/tredek.aspx Tonalsoft | ''Tredek, 270-edo'']</ref>.  
| Proposed by [[Joseph Monzo]] (2013)<ref>[http://tonalsoft.com/enc/t/tredek.aspx Tonalsoft | ''Tredek, 270-edo'']</ref>.  
|-
|-
| [[Savart]]*
| [[Savart]]*
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| [[Gene]]
| [[Gene]]
| [[311edo|311]]
| [[311edo|311]]
| PRIME
| 311 (prime)
| Proposed by Joe Monzo (2007)<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft | ''Gene, 311-edo'']</ref>.
| Proposed by Joseph Monzo (2007)<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft | ''Gene, 311-edo'']</ref>.
|-
|-
| [[Dröbisch Angle]]
| [[Dröbisch Angle]]
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| [[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>
| {{Citation needed}}
| {{Citation needed}}
|-
| Dexl
| [[540edo|540]]
| 2<sup>2</sup> × 3<sup>3</sup> × 5
| Proposed by Joseph Monzo (2023)<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft | ''Dexl, 540-edo'']</ref>.
|-
|-
| [[Iring]] / [[centitone]]
| [[Iring]] / [[centitone]]
| [[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]]
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| <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 128: 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
| Heptamu (7mu)
| [[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)
| [[1536edo|1536]]
| [[1536edo|1536]]
| 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]]
| [[1\1700_octave|1700]]
| [[1700edo|1700]]
| 2<sup>2</sup> × 5<sup>2</sup> × 17
| 2<sup>2</sup> × 5<sup>2</sup> × 17
| [[Relative cent]] of [[17edo]]; proposed by [[Margo Schulter]] (2002) and [[George Secor]]<ref name="measure"/>.  
| [[Relative cent]] of [[17edo]]; proposed by [[Margo Schulter]] (2002) and [[George Secor]]<ref name="measure"/>.  
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| 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]]
| [[8269edo|8269]]
| 8269 (prime)
| Proposed by [[Flora Canou]] (2021)<ref>[https://forum.sagittal.org/viewtopic.php?f=4&t=515 The Sagittal Forum | ''Definition of the tina reviewed'']</ref>.
|-
|-
| [[Tina]]
| [[Tina]]
| [[8539edo|8539]]
| [[8539edo|8539]]
| PRIME
| 8539 (prime)
| Provides good approximations for 41-limit primes except 37; named by [[Dave Keenan]] and [[George Secor]]; edo representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809eda)<ref name="measure"/><ref>[http://tonalsoft.com/enc/t/tina.aspx Tonalsoft | ''Tina'']</ref>.  
| Provides good approximations for 41-limit primes except 37; named by [[Dave Keenan]] and [[George Secor]]; edo representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809eda)<ref name="measure"/><ref>[http://tonalsoft.com/enc/t/tina.aspx Tonalsoft | ''Tina'']</ref>.  
|-
|-
Line 236: 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]]
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| [[Jot]]
| [[Jot]]
| [[30103edo|30103]]
| [[30103edo|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]]
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| [[46032edo|46032]]
| [[46032edo|46032]]
| 2<sup>4</sup> × 3 × 7 × 137
| 2<sup>4</sup> × 3 × 7 × 137
| Proposed by Gene Ward Smith (2005)<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/flu.aspx Tonalsoft | ''Flu'']</ref>.  
| Proposed by Gene Ward Smith (2005)<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/flu.aspx Tonalsoft | ''Flu'']</ref>.
|-
| [[Normal atom]]
| [[78005edo|78005]]
| 5 × 15601
| 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)
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| 2<sup>16</sup> × 3
| 2<sup>16</sup> × 3
| 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"/>.  
|-
| [[Mean free path]]
| ~216,608,494
| 2 × 41 × 2641567
|
|}
|}
 
<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 ====
Line 285: Line 287:


{| 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
! Divisions of base interval
! Origin/significance
|-
|-
! Unit Name (Symbol):
| Hekt
! Base Interval:
| 3/1 (twelfth)
! Parts of Base Interval:
| 1300
! Origin/Significance
| 1/100 of 13edt (Bohlen–Pierce) scale step
|-
|-
| '''[[Hekt]]'''
| Euhekt
| 3/1 (twelfth)
| 3/1 (twelfth)
| 1300
| 3900
| 1/100 of 13-ED3 (Bohlen-Pierce) scale step
| 1/100 of 39edt (Triboh) scale step
|-
|-
| '''[[Grad]]'''
| Grad
| [[Pythagorean comma|531441/524288]] (Pythagorean comma)
| [[Pythagorean comma|531441/524288]] (Pythagorean comma)
| 12
| 12
|
| [[12edo]] flattens [[3/2]] by this amount
|-
|-
| '''[[Tuning unit]]'''
| Tuning unit
| [[531441/524288]] (Pythagorean comma)
| [[531441/524288]] (Pythagorean comma)
| 720
| 720
|
|
|-
| '''[[Wikipedia:Neper|Neper]]''' (Np)
| <math>e</math> ≈ 2.71828
| 1
| the natural unit for logarithmic measurement
|-
| '''Dineper''' (dNp)
| <math>e^2</math> ≈ 7.38906
| 1
| used for [[logarithmic approximants]]
|}
|}


Line 322: 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.
 
see also: Kirnberger Atom http://arxiv.org/abs/0907.5249


== Ratio ==
== Ratio ==
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 {{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>−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.


which is a diatonic semitone below an octave ([[2/1]]) / (15/8) = 2/1 × 8/15 = [[16/15]].
== See also ==
* [[Interval span]]


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>-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.
== Articles ==
* [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small''] by [[Don N. Page]]


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


[[Category:Interval size]]
[[Category:Interval]]
[[Category:Interval size measure]]
[[Category:Measure]]
[[Category:Size]]
[[Category:Theory]]
Retrieved from "https://en.xen.wiki/w/Interval_size_measure"