Glossary
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This page is for brief descriptions of terms essential to understanding contemporary xenharmonic theory.
**Hertz** - counts per second
**pitch** - property of some sounds that allows them to be ordered on a one-dimensional 'pitch axis'. often measured in terms of the frequency (in Hertz) of a sine wave having the same pitch
**cents** - logarithmic units of measure. for a ratio R, 1200 * log2(R)
**interval** - distance between two pitches, expressed as their ratio (of their associated frequencies in Hertz) or in cents
**scale** - an ordered list of intervals. (exactly what can be described by a Scala file (.scl))
**harmonic distance** - property of intervals other than size; a second type of distance, such that pitches separated by 'short' intervals are more likely to be confused (as by novice singers) than those separated by 'long' intervals. e.g. an octave is 'shorter' than a tritone. related to consonance
**interval of equivalence** - an interval having small or zero harmonic distance, such as an octave (2:1 or 1200 cents)
**periodic scale** - a scale that repeats at an interval of equivalence
**just intonation** - a set of intervals considered consonant and expressible as rational numbers
**generators** - a set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the Pythogorean scale
**temperament** - a mapping from a just intonation to a set of intervals with fewer generators. expressible as a mapping matrix whose columns are generators of the just intonation and whose rows are generators of the temperament. traditionally, any tuning not based on just or integer ratios.
**rank** - the number of generators of a set of intervals. e.g. 12-tone equal temperament is rank-1 because it is generated by the semitone; the Pythagorean scale is rank-2 because it is generated by the primes {2, 3}
**equal temperament** - a rank-1 temperament
**MOS** - a rank-2 scale
**tuning** - the generators of a temperament. a temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents will produce a scale similar to one produced by a semitone of 100.0 cents
**superparticular ratio** - a rational number n/d such that n = d+1
**prime limit** - the largest prime number in the generators of a just intonation
**subgroup** (e.g., 2.3.7 subgroup) - the generators of a just intonation that does not contain all primes smaller than the prime limit. a new name for what some of us see as the norm for many tunings over the last 2500 years.
**odd limit** - for the list of intervals (dyads) formed between all pitches in a just intonation chord, the largest number appearing as either a numerator or denominator after all factors of 2 are discarded
**comma** - a rational number that is tempered out (i.e. maps to 1:1) in a given temperament. Some people feel that accurately observing a comma, e.g. having accurate versions of both 16/9 and 7/4 (and so "observing 64:63," their difference), can be just as important as tempering out a comma when this seems convenient.
**Maqam** - The traditional music and modal system of the Near East, and various xenharmonic spinoffs
**Dastgah** - The classic music and modal system of Persia/Iran, as derived around the 16th-19th centuries from maqam, and spinoffs
**Well-temperament** - A temperament (e.g. in 12 or 17 notes per octave) where all impure fifths are tempered in the same direction
**Temperament ordinaire** - A temperament (e.g. in 12 or 17 notes per octave) where some fifths are tempered in different directions
**Circulation** - The property of a tuning that supports desired intervals on every note or step, with the question of how to define "desired intervals" often in the eye of the beholder, or ear of the listener.
**Wolf interval** - An interval that differs from the expected size for that number of tuning or scale steps in a way deemed to impede or prevent its use as a musical equivalent, e.g. the diminished sixth G#-Eb in a meantone tuning near 1/4-comma as a "wolf fifth." A wolf in this sense might be defined as an interval that prevents circulation, if the same tuning system doesn't also provide a more usual size (e.g. both a diminished sixth and a regular fifth from the same note in an extended meantone system). Sometimes one person's wolf can be another person's delightful alternative consonance, as with F-G#, an augmented second at or near 1/4-comma meantone.Original HTML content:
<html><head><title>glossary</title></head><body>This page is for brief descriptions of terms essential to understanding contemporary xenharmonic theory.<br />
<br />
<strong>Hertz</strong> - counts per second<br />
<br />
<strong>pitch</strong> - property of some sounds that allows them to be ordered on a one-dimensional 'pitch axis'. often measured in terms of the frequency (in Hertz) of a sine wave having the same pitch<br />
<br />
<strong>cents</strong> - logarithmic units of measure. for a ratio R, 1200 * log2(R)<br />
<br />
<strong>interval</strong> - distance between two pitches, expressed as their ratio (of their associated frequencies in Hertz) or in cents<br />
<br />
<strong>scale</strong> - an ordered list of intervals. (exactly what can be described by a Scala file (.scl))<br />
<br />
<strong>harmonic distance</strong> - property of intervals other than size; a second type of distance, such that pitches separated by 'short' intervals are more likely to be confused (as by novice singers) than those separated by 'long' intervals. e.g. an octave is 'shorter' than a tritone. related to consonance<br />
<br />
<strong>interval of equivalence</strong> - an interval having small or zero harmonic distance, such as an octave (2:1 or 1200 cents)<br />
<br />
<strong>periodic scale</strong> - a scale that repeats at an interval of equivalence<br />
<br />
<strong>just intonation</strong> - a set of intervals considered consonant and expressible as rational numbers<br />
<br />
<strong>generators</strong> - a set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the Pythogorean scale<br />
<br />
<strong>temperament</strong> - a mapping from a just intonation to a set of intervals with fewer generators. expressible as a mapping matrix whose columns are generators of the just intonation and whose rows are generators of the temperament. traditionally, any tuning not based on just or integer ratios.<br />
<br />
<strong>rank</strong> - the number of generators of a set of intervals. e.g. 12-tone equal temperament is rank-1 because it is generated by the semitone; the Pythagorean scale is rank-2 because it is generated by the primes {2, 3}<br />
<br />
<strong>equal temperament</strong> - a rank-1 temperament<br />
<br />
<strong>MOS</strong> - a rank-2 scale<br />
<br />
<strong>tuning</strong> - the generators of a temperament. a temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents will produce a scale similar to one produced by a semitone of 100.0 cents<br />
<br />
<strong>superparticular ratio</strong> - a rational number n/d such that n = d+1<br />
<br />
<strong>prime limit</strong> - the largest prime number in the generators of a just intonation<br />
<br />
<strong>subgroup</strong> (e.g., 2.3.7 subgroup) - the generators of a just intonation that does not contain all primes smaller than the prime limit. a new name for what some of us see as the norm for many tunings over the last 2500 years.<br />
<br />
<strong>odd limit</strong> - for the list of intervals (dyads) formed between all pitches in a just intonation chord, the largest number appearing as either a numerator or denominator after all factors of 2 are discarded<br />
<br />
<strong>comma</strong> - a rational number that is tempered out (i.e. maps to 1:1) in a given temperament. Some people feel that accurately observing a comma, e.g. having accurate versions of both 16/9 and 7/4 (and so "observing 64:63," their difference), can be just as important as tempering out a comma when this seems convenient.<br />
<br />
<strong>Maqam</strong> - The traditional music and modal system of the Near East, and various xenharmonic spinoffs<br />
<br />
<strong>Dastgah</strong> - The classic music and modal system of Persia/Iran, as derived around the 16th-19th centuries from maqam, and spinoffs<br />
<br />
<strong>Well-temperament</strong> - A temperament (e.g. in 12 or 17 notes per octave) where all impure fifths are tempered in the same direction<br />
<br />
<strong>Temperament ordinaire</strong> - A temperament (e.g. in 12 or 17 notes per octave) where some fifths are tempered in different directions<br />
<br />
<strong>Circulation</strong> - The property of a tuning that supports desired intervals on every note or step, with the question of how to define "desired intervals" often in the eye of the beholder, or ear of the listener.<br />
<br />
<strong>Wolf interval</strong> - An interval that differs from the expected size for that number of tuning or scale steps in a way deemed to impede or prevent its use as a musical equivalent, e.g. the diminished sixth G#-Eb in a meantone tuning near 1/4-comma as a "wolf fifth." A wolf in this sense might be defined as an interval that prevents circulation, if the same tuning system doesn't also provide a more usual size (e.g. both a diminished sixth and a regular fifth from the same note in an extended meantone system). Sometimes one person's wolf can be another person's delightful alternative consonance, as with F-G#, an augmented second at or near 1/4-comma meantone.</body></html>