The scale of harmonics is a

musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...

based on the noded positions of the natural

harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...

s existing on a string. This musical scale is present on the

guqin The ''guqin'' (; ) is a plucked seven-string Chinese musical instrument. It has been played since ancient times, and has traditionally been favoured by scholars and literati as an instrument of great subtlety and refinement, as highlighted b ...

, regarded as one of the first

string instrument String instruments, stringed instruments, or chordophones are musical instruments that produce sound from vibrating strings when a performer plays or sounds the strings in some manner. Musicians play some string instruments by plucking the ...

s with a musical scale.Yin, Wei. ''Zhongguo Qinshi Yanyi'' 【中国琴史演义】 (Chinese). Pages 1-10. Most

fret A fret is any of the thin strips of material, usually metal wire, inserted laterally at specific positions along the neck or fretboard of a stringed instrument. Frets usually extend across the full width of the neck. On some historical instrume ...

positions appearing on Non-Western string instruments (

lute A lute ( or ) is any plucked string instrument with a neck and a deep round back enclosing a hollow cavity, usually with a sound hole or opening in the body. It may be either fretted or unfretted. More specifically, the term "lute" can ref ...

s) are equal to positions of this scale. Unexpectedly, these fret positions are actually the corresponding undertones of the

overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...

s from the harmonic series. The distance from the

nut Nut often refers to: * Nut (fruit), fruit composed of a hard shell and a seed, or a collective noun for dry and edible fruits or seeds * Nut (hardware), fastener used with a bolt Nut or Nuts may also refer to: Arts, entertainment, and media Co ...

to the fret is an integer number lower than the distance from the fret to the

bridge A bridge is a structure built to span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, which is usually somethi ...

(see:

superparticular number In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers. More particularly, the ratio takes the form: :\frac = 1 + \frac where is a positive integer. Thu ...

Origin

On the guqin, the left end of the dotted scale is a mirror image of the right end. The instrument is played with flageolet tones (harmonics) as well as pressing the strings on the wood. The flageolets appear on the harmonic positions of the

overtone series A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''. Pitched musical instruments are often based on an acoustic resonator su ...

, therefore these positions are marked as the musical scale of this instrument. The flageolet positions also represent the harmonic

consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...

relation of the pressed string part with the open string, similar to the calculations

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...

did on his

monochord A monochord, also known as sonometer (see below), is an ancient musical and scientific laboratory instrument, involving one (mono-) string ( chord). The term ''monochord'' is sometimes used as the class-name for any musical stringed instrument h ...

. The guqin has one anomaly in its scale. The guqin scale represents the first six harmonics and the eighth harmonic. The

seventh harmonic The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinar ...

is left out. However this tone is still consonant related to the open string (otherwise it would not be a harmonic) and has a lesser consonant relation to all other harmonic positions. This is the main reason all the ratios of the sevenths family (7:1, 7:2, 7:3, 7:4, 7:5 and 7:6) also often are not present in other musical scales like for instance the just intoned

major Major (commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicators ...

and

minor scale In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which also ...

or the major scale in the

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...

A Vietnamese monochord, called the

đàn bầu The đàn bầu (; "gourd zither"; chữ Nôm: ), also called độc huyền cầm (獨絃琴, "one-string zither") is a Vietnamese stringed instrument, in the form of a monochord (one-string) zither. History While the earliest written records o ...

, also functions with the scale of harmonics. On this instrument only the right half (from the view of the musician) of the scale is present up to the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

of the first seven

s (see

7-limit 7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14 ...

). The dots are on the string lengths , , , , , of the whole string length. The reason for this half scale is because the left half creates the same tones as the right half when played as a flageolet tone and therefore the extra dots on the left half are useless for how this instrument is played. The scale of harmonics was, together with the book of

Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...

an inspiration for

Harry Partch Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...

to switch to

just intonation In music, just intonation or pure intonation is the tuning of musical intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to ...

and alternate tuning systems to create more consonant music than possible with the equal temperament. Partch's tone selection

otonality ''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone (identity), respectively. For example: , , ,... or , , ,.... Definition ...

from his utonality and otonality concept are the complement pitches of the overtones. For instance: the frequency ratio 5:4 is equal to of the string length and is the complement of , the position of the fifth harmonic (and the fourth overtone). The

Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe * Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including ...

composer

Eivind Groven Eivind Groven (8 October 1901 – 8 February 1977) was a Norwegian composer and music-theorist. He was from traditional region of Vest-Telemark and had a background in the folk music of the area. Biography Groven was born in the village of L� ...

also wrote a thesis on the scale of harmonics, claiming this to be the oldest usable scale, frequent in

Norwegian folk music Much has been learned about early music in Norway from physical artifacts found during archaeological digs. These include instruments such as the lur. Viking and medieval sagas also describe musical activity, as do the accounts of priests and pil ...

, and seemingly in other folk musical traditions as well. Groven used the '' seljefløyte'' as basis for his research. The flute uses only the upper harmonic scale. The scale is also present on the

Moodswinger The Moodswinger is a twelve-string electric zither with an additional third bridge designed by Yuri Landman. The rod which functions as the third bridge divides the strings into two sections to cause an overtone multiphonic sound. One of the copi ...

. Although this functions quite differently than a Guqin, oddly enough the scale occurs on this instrument while it is not played in a

tuning but a regular

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

References

External links

"3rd Bridge Helix"
''PerfectSoundForever''. Article about the overtoning positions and their relation to musical scales. {{Scales Hexatonic scales Just tuning and intervals Hemitonic scales Acoustics Sound

Origin

Related

See also

References

Further reading

External links