Limit (music)
In
The harmonic series and the evolution of music
Harry Partch,
Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music.[citation needed] 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.
Odd-limit and prime-limit
In
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.[2] 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.[3]
See § Examples, below.
Identity
An identity is each of the
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
Odentity and udentity are short for over-identity and under-identity, respectively.
Prime limit
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 consisting of those rational numbers x whose prime factorization has the form with forms a subgroup of (). ... We say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup.[8]
In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the
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 | ⓘ |
4/3 | perfect fourth | 3 | 3 | ⓘ |
5/4 | major third | 5 | 5 | ⓘ |
5/2 | major tenth |
5 | 5 | ⓘ |
5/3 | major sixth | 5 | 5 | ⓘ |
7/5 | lesser septimal tritone | 7 | 7 | ⓘ |
10/7 | greater septimal tritone | 7 | 7 | ⓘ |
9/8 | major second | 9 | 3 | ⓘ |
27/16 | Pythagorean major sixth | 27 | 3 | ⓘ |
81/64 | ditone | 81 | 3 | ⓘ |
243/128 | Pythagorean major seventh | 243 | 3 | ⓘ |
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 (Pythagorean) tuning
- Five-limit tuning
- 7-limit tuning
- Numerary nexus
- Otonality and Utonality
- Tonality diamond
- Tonality flux
References
- ^
Wolf, Daniel James (2003), "Alternative Tunings, Alternative Tonalities", Contemporary Music Review, 22 (1/2), Abingdon, UK: Routledge: 13, S2CID 191457676
- ISBN 0-306-80106-X(pbk reprint, 1979).
- ^ Paul Erlich, "The Forms of Tonality: A Preview". Some Music Theory from Paul Erlich (2001), pp. 1–3 (Accessed 29 May 2010).
- ISBN 0-306-80106-X.
- ^ Partch (1979), p.71.
- ISBN 9789057550652.
- ^ "Udentity". Tonalsoft. Archived from the original on 29 October 2013. Retrieved 23 October 2013.
- ISBN 0-8218-4873-9.
External links
- "Limits: Consonance Theory Explained", Glen Peterson's Musical Instruments and Tuning Systems.
- "Harmonic Limit", Xenharmonic.