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{{short description|Way of characterizing the harmony found in a piece or genre of music}}
[[Image:Normalized harmonic identities, names, and frequencies.svg|thumb|400px|The first 16 harmonics, with frequencies and log frequencies (not drawn to scale).]]

In [[music theory]], '''limits''' or '''harmonic limits''' are a way of characterizing the [[harmony]] found in a piece or [[genre (music)|genre]] of music, or the harmonies that can be made using a particular [[scale (music)|scale]]. The term ''limit'' was introduced by [[Harry Partch]],<ref>
{{citation
| title=Alternative Tunings, Alternative Tonalities
| first= Daniel James
| last= Wolf
| journal= Contemporary Music Review
| volume= 22
| number= 1/2
| date= 2003
| publisher= Routledge
| location= Abingdon, UK
| page= 13
| doi= 10.1080/0749446032000134715
| s2cid= 191457676
}}
</ref> who used it to give an [[upper bound]] on the complexity of harmony; hence the name.

==The harmonic series and the evolution of music==
[[Image:Harmonic series klang.png|thumb|Overtone series, partials 1-5 numbered {{audio|Harmonic series klang.mid|Play}}.]]

Harry Partch, [[Ivor Darreg]], and Ralph David Hill are among the many [[microtonal music|microtonalists]] to suggest that music has been slowly evolving to employ higher and higher [[harmonic series (music)|harmonics]] in its constructs (see [[emancipation of the dissonance]]).{{citation needed|date=August 2014}} In [[medieval music]], only chords made of [[octave]]s and [[perfect fifth]]s (involving relationships among the first three [[harmonic]]s) were considered consonant. In the West, triadic harmony arose ([[contenance angloise]]) around the time of the [[Renaissance]], and [[triad (music)|triads]] quickly became the fundamental building blocks of Western music. The [[major third|major]] and [[minor third]]s of these triads invoke relationships among the first five harmonics.

Around the turn of the 20th century, [[tetrad (music)|tetrads]] debuted as fundamental building blocks in [[African-American music]].{{Citation needed|date=December 2023}} In conventional music theory pedagogy, these [[seventh chord]]s are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5. For example, the [[dominant seventh chord]] in [[Equal temperament|12-ET]] approximates 4:5:6:7 (albeit very poorly), while the [[major seventh chord]] approximates 8:10:12:15.

==Odd-limit and prime-limit==
In [[just intonation]], intervals between pitches are drawn from the [[rational numbers]]. Since Partch, two distinct formulations of the limit concept have emerged: '''odd limit''' and '''prime limit'''. Odd limit and prime limit ''n'' do not include the same intervals even when ''n'' is an odd prime.

===Odd limit===
For a positive odd number ''n'', the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than ''n''.

In ''[[Genesis of a Music]]'', Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves.<ref>Harry Partch, ''Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments'', second edition, enlarged (New York: Da Capo Press, 1974), p. 73. {{ISBN|0-306-71597-X}}; {{ISBN|0-306-80106-X}} (pbk reprint, 1979).</ref> Since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partch's theoretical prediction of the sensory dissonance of intervals (his "One-Footed Bride") are very similar to those of theorists including [[Hermann von Helmholtz]], [[William Sethares]], and [[Paul Erlich]].<ref name="Erlich">Paul Erlich, "[http://lumma.org/tuning/erlich/erlich-tFoT.pdf The Forms of Tonality: A Preview]". ''[http://lumma.org/tuning/erlich/ Some Music Theory from Paul Erlich]'' (2001), pp. 1–3 (Accessed 29 May 2010).</ref>

See {{slink||Examples}}, below.

===Identity<!--[[Identity (tuning)]] and [[Identity (limit)]] redirect directly here.-->===
{{other uses|Identity (music)}}
An '''identity''' is each of the [[odd number]]s below and including the (odd) limit in a tuning. For example, the identities included in 5-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the [[harmonic series (music)|harmonic series]] and may thus be considered an identity:
 {{underline|C}}  C  {{underline|G}}  C  {{underline|E}}  G  {{underline|B}}  C  {{underline|D}}  E  {{underline|F}}  G  ...
 {{underline|1}}  2  {{underline|3}}  4  {{underline|5}}  6  {{underline|7}}  8  {{underline|9}}  10 {{underline|11}} 12 ...

According to Partch: "The number 9, though not a [[prime number|prime]], is nevertheless an identity in music, simply because it is an odd number."<ref>Partch, Harry (1979). ''Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments'', p.93. {{ISBN|0-306-80106-X}}.</ref> Partch defines "identity" as "one of the correlatives, '[[Otonality and Utonality|major]]' or '[[major and minor|minor]]', in a [[tonality]]; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".<ref>Partch (1979), p.71.</ref>

'''''Odentity''''' and '''''udentity''''' are short for ''over-identity'' and ''under-identity'', respectively.<ref>Dunn, David, ed. (2000). ''Harry Partch: An Anthology of Critical Perspectives'', p.28. {{ISBN|9789057550652}}.</ref> According to music software producer Tonalsoft: "An udentity is an identity of an [[utonality]]".<ref>{{cite web |url=http://www.tonalsoft.com/enc/u/udentity.aspx |title=Udentity |author=<!--Staff writer(s); no by-line.--> |website=Tonalsoft |access-date=23 October 2013 |url-status=dead |archive-url=https://web.archive.org/web/20131029222531/http://www.tonalsoft.com/enc/u/udentity.aspx |archive-date=29 October 2013 |df=dmy-all }}</ref>

===Prime limit===
[[File:Missing fundamental rectangles.png|thumb|First 32 harmonics, with the harmonics unique to each limit sharing the same color.]]

For a [[prime number]] ''n'', the n-prime-limit contains all rational numbers that can be factored using primes no greater than ''n''. In other words, it is the set of rationals with numerator and denominator both ''n''-[[smooth number|smooth]].

{{quote|'''p-Limit Tuning.''' Given a prime number ''p'', the subset of <math>\mathbb{Q}^+</math> consisting of those rational numbers ''x'' whose prime factorization has the form
<math>x=p_1^{\alpha_1} p_2^{\alpha_2}... p_r^{\alpha_r} </math> with <math>p_1,...,p_r \le p</math> forms a subgroup of (<math>\mathbb{Q}^+,\cdot</math>). ... We say that a scale or system of tuning uses ''p-limit tuning'' if all interval ratios between pitches lie in this subgroup.<ref>David Wright, ''Mathematics and Music''. Mathematical World 28. (Providence, R.I.: American Mathematical Society, 2009), p. 137. {{ISBN|0-8218-4873-9}}.</ref>}}

In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the [[American gamelan|American gamelan school]]. Inspired by Indonesian [[gamelan]], musicians in California and elsewhere began to build their own gamelan instruments, often tuning them in just intonation. The central figure of this movement was the American composer [[Lou Harrison]]{{Citation needed|date=May 2011}}. Unlike Partch, who often took scales directly from the harmonic series, the composers of the American Gamelan movement tended to draw scales from the just intonation lattice, in a manner like that used to construct [[Fokker periodicity blocks]]. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale.

Prime-limit tuning and intervals are often referred to using the term for the [[numeral system]] based on the limit. For example, 7-limit tuning and intervals are called septimal, 11-limit is called undecimal, and so on.

==Examples==
{| class="wikitable sortable"
|-
! ratio || interval || odd-limit || prime-limit || audio
|-
| 3/2 || [[perfect fifth]] || 3 || 3 || {{audio|Just perfect fifth on C.mid|Play}}
|-
| 4/3 || [[perfect fourth]] || 3 || 3 || {{audio|Just perfect fourth on C.mid|Play}}
|-
| 5/4 || [[major third]] || 5 || 5 || {{audio|Just major third on C.mid|Play}}
|-
| 5/2 || [[major tenth]] || 5 || 5 || {{audio|Just major tenth on C.mid|Play}}
|-
| 5/3 || [[major sixth]] || 5 || 5 || {{audio|Just major sixth on C.mid|Play}}
|-
| 7/5 || lesser [[septimal tritone]] || 7 || 7 || {{audio|Lesser septimal tritone on C.mid|Play}}
|-
| 10/7 || greater septimal tritone || 7 || 7 || {{audio|Greater septimal tritone on C.mid|Play}}
|-
| 9/8 || [[major second]] || 9 || 3 || {{audio|Major tone on C.mid|Play}}
|-
| 27/16 || Pythagorean major sixth || 27 || 3 || {{audio|Pythagorean major sixth on C.mid|Play}}
|-
| 81/64 || [[ditone]] || 81 || 3 || {{audio|Pythagorean major third on C.mid|Play}}
|-
| 243/128 || [[Pythagorean interval|Pythagorean major seventh]] || 243 || 3 || {{audio|Pythagorean major seventh on C.mid|Play}}
|}

==Beyond just intonation==
In [[musical temperament]], the simple ratios of just intonation are mapped to nearby irrational approximations. This operation, if successful, does not change the relative harmonic complexity of the different intervals, but it can complicate the use of the harmonic limit concept. Since some chords (such as the [[diminished seventh chord]] in [[12 equal temperament|12-ET]]) have several valid tunings in just intonation, their harmonic limit may be ambiguous.

==See also==
* [[Pythagorean tuning|3-limit (Pythagorean) tuning]]
* [[Five-limit tuning]]
* [[7-limit tuning]]
* [[Numerary nexus]]
* [[Otonality and Utonality]]
* [[Tonality diamond]]
* [[Tonality flux]]

==References==
{{reflist}}

==External links==
* [https://web.archive.org/web/20081220221910/http://organicdesign.org/peterson/tuning/consonance.html#limits "Limits: Consonance Theory Explained"], ''Glen Peterson's Musical Instruments and Tuning Systems''.
* [http://xenharmonic.wikispaces.com/harmonic+limit "Harmonic Limit"], ''Xenharmonic''.
{{Use dmy dates|date=April 2014}}
{{Microtonal music}}
{{Musical tuning}}

[[Category:Just tuning and intervals|*]]
[[Category:Harmony]]
[[Category:Harry Partch]]