Scale of harmonics

The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale. Most fret positions appearing on Non-Western string instruments (lutes) are equal to positions of this scale. Unexpectedly, these fret positions are actually the corresponding undertones of the overtones from the harmonic series. The distance from the nut to the fret is an integer number lower than the distance from the fret to the bridge (see: superparticular number). 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, therefore these positions are marked as the musical scale of this instrument. The flageolet positions also represent the harmonic consonant relation of the pressed string part with the open string, similar to the calculations Pythagoras did on his monochord. The guqin has one anomaly in its scale. The guqin scale represents the first six harmonics and the eighth harmonic. The seventh harmonic 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 and minor scale or the major scale in the Pythagorean tuning. A Vietnamese monochord, called the đàn bầu, 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 of the first seven overtones (see 7-limit). The dots are on the string lengths , , , , , of the whole string length.