In music theory

Music theory

Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

 and tuning

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

, the kleisma, or semicomma majeur, is a minute and barely perceptible comma

Comma (music)

In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...

 type interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 important to musical temperaments

Musical temperament

In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

. It is the difference between six justly

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

 tuned minor third

Minor third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

s (each with a frequency ratio of 6/5) and one justly tuned tritave or perfect twelfth (with a frequency ratio of 3/1, formed by a 2/1 octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 plus a 3/2 perfect fifth

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

). It is equal to a frequency ratio of 15625/15552 = 2^-6 3^-5 5⁶, or approximately 8.1 cent

Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

s . It can be also defined as the difference between five justly tuned minor third

Minor third

In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

s and one justly tuned major tenth (of size 5/2, formed by a 2/1 octave plus a 5/4 major third).

The interval was named by Shohé Tanaka

Shohé Tanaka

was a Japanese physicist, music theorist, and inventor. He graduated from Tokyo University in 1882 as a science student. On an imperial scholarship, he was sent to Germany for doctoral studies in 1884, together with Mori Ōgai...

, who noted that it was tempered to a unison

Unison

In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

 by 53 equal temperament

53 equal temperament

In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...

. Interestingly, it is also tempered out by 19 equal temperament

19 equal temperament

In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

 and 72 equal temperament

72 equal temperament

In music, 72 equal temperament, called twelfth-tone, 72-tet, 72-edo, or 72-et, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps...

, but it is not tempered out in 12 equal temperament. Namely, in 12 equal temperament the difference between six minor thirds (18 semitones) and one perfect twelfth (19 semitones) is not a comma, but a semitone (100 cents). The same is true for the difference between five minor thirds (15 semitones) and one major tenth (16 semitones).

Larry Hanson independently discovered this interval which also manifested in a unique mapping using a generalized keyboard capable of accommodating all the above temperaments as well as Just intonation constant structures (periodicity blocks) with these numbers of scale degrees

The kleisma is also an interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 important to the Bohlen–Pierce scale.