5-limit: Difference between revisions

The '''5-limit''' consists of all [[just intonation]] intervals whose [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5; these are sometimes called {{w|regular number}}s. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]].  

Recently, composers [[Catherine Lamb]] and [[Marc Sabat]] have adopted ''quintal'' for the [[harmonic class]] 5 since the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit.<ref>See e.g. liner notes for Lambs' ''Curva Triangulus'', "Curva Triangulus is an ever-fluctuating modal space - shifting between Tertial (3 prime qualitative), Quintal (5 prime qualitative), and Septimal (7 prime qualitative)."</ref> ''Pental'' is less consistent due to its Greek origins. However, that creates a conflict of usage as ''quintal'' has been the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.) [[User:Lériendil|Lériendil]] suggests the term ''quinary'' as opposed to ''quintal'' (seeing as the pent- root is still overloaded with various terms referring to fifths and pentatonic scales), though there is a minor conflict in naming with [[Arity|quinary]] scales.

A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://masa.plainsound.org/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.  

The term '''ptolemaic''' could also refer to the 5-limit<ref>[https://masa.plainsound.org/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.

Another list of edos that do relatively well in approximating harmonics 3 and 5: {{EDOs| 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, … }} ({{OEIS|A060525}}<ref>The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5. It is yet to be found out what criterion is used. </ref>)

Another list of edos that do relatively well in approximating [[5-odd-limit]] intervals: {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … }} ({{OEIS|A054540}}<ref>The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3. It is yet to be found out what criterion is used. </ref>)