User:TNTErick/和聲限度

第前16個泛音的頻率及其對數(非完全照比例)。

和聲限度，或簡稱限度( 英語：Harmonic Limit)，是音樂理論中用於描述一個有理數音程、和聲或音階的複襍度的度量。該詞彙最早是由美國樂理學家哈里帕奇 (Harry Partch)提出，用於描述和聲的複襍度上界。

泛音列與音階的演進

上泛音列的前五個。範例中基音是C2 (約65.41Hz)。 Play^ⓘ.

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

Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music. In conventional music theory pedagogy, these seventh chords 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 12-ET approximates 4:5:6:7, while the major seventh chord approximates 8:10:12:15.

和聲限度的量度

在纯律之中，一個音程是以兩個音的有理数比例來表示。和聲限度的度量方式有兩種：奇數限度和質數限度。兩種

Odd limit

一個音程或合聲具有奇數限度n，若且惟若分子分母皆不大於正奇數n。

In Genesis of a Music, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves. 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 海姆荷茲, William Sethares, and Paul Erlich.

見下方#範例。

Identity

An identity is each of the odd numbers 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 and may thus be considered an identity:

C  C  G  C  E  G  B  C  D  E  F  G  ...
1  2  3  4  5  6  7  8  9  10 11 12 ...

According to Partch: "The number 9, though not a prime, is nevertheless an identity in music, simply because it is an odd number." Partch defines "identity" as "one of the correlatives, 'major' or 'minor', in a tonality; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".

Odentity and udentity are short for over-identity and under-identity, respectively. According to music software producer Tonalsoft: "An udentity is an identity of an utonality".

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.

p-Limit Tuning. Given a prime number p, the subset of $\mathbb {Q} ^{+}$ consisting of those rational numbers x whose prime factorization has the form $x=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}...p_{r}^{\alpha _{r}}$ with $p_{1},...,p_{r}\leq p$ forms a subgroup of ($\mathbb {Q} ^{+},\cdot$). ... We say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup.

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 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]. 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

ratio	interval	odd-limit	prime-limit	audio
3/2	perfect fifth	3	3	Playⓘ
4/3	perfect fourth	3	3	Playⓘ
5/4	major third	5	5	Playⓘ
5/2	major tenth	5	5	Playⓘ
5/3	major sixth	5	5	Playⓘ
7/5	lesser septimal tritone	7	7	Playⓘ
10/7	greater septimal tritone	7	7	Playⓘ
9/8	major second	9	3	Playⓘ
27/16	Pythagorean major sixth	27	3	Playⓘ
81/64	ditone	81	3	Playⓘ
243/128	Pythagorean major seventh	243	3	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-ET) have several valid tunings in just intonation, their harmonic limit may be ambiguous.

See also

• 3-limit (五度相生律)
• Five-limit tuning
• 7-limit tuning
• Numerary nexus
• Otonality and Utonality
• Tonality diamond
• Tonality flux

External links

• "Limits: Consonance Theory Explained", Glen Peterson's Musical Instruments and Tuning Systems.
• "Harmonic Limit", Xenharmonic.

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