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A piccolo bass refers to either a bass instrument in the guitar family or a bass instrument in the violin family. Both instruments are most common to the jazz genre. They were developed to get instruments of a lower range with increased facility and a different tonal color. Very broadly, piccolo bass can mean any instrument that falls between the bass and tenor members of an instrument family in size or tuning.
When referencing the member of the guitar family, piccolo bass means a bass guitar that is tuned one octave higher than usual. Four strings is probably most common, but it is not unusual to see models with up to eight. Standard tuning for a regular four-string bass is E1, A1, D2 and G2. Piccolo basses with four strings thus are tuned E2, A2, D3, and G3.
The uptuning on a bass guitar is possible by shortening the length of the neck of the guitar. A musician also can convert his regular bass to a piccolo bass simply by putting on thinner strings, which is probably more common. This works because thicker strings vibrate at a slower rate, producing lower pitches, whereas a thin string can vibrate faster and produce a higher pitch. Thus, in theory, any standard bass also can be a piccolo bass with the proper string set.
Even though a piccolo bass guitar does not sound very bass-like, musicians like this type of bass because it has a different tonal color. The instrument is mellow but does not have the muddiness of a regular bass. The spacing of the strings on a bass allows guitarists to perform techniques such as slapping that are not possible on regular guitars. They can play the bass as a primary, virtuosic lead instrument as a result rather than simply supporting the bass line of a work.
The term "piccolo bass" less commonly refers to an instrument similar to but slightly larger than a cello. The range of the instrument is one octave above that of the standard double bass. Ron Carter, known for his work as a jazz cellist, normally is credited for the development of the instrument. He tuned his version A1, D2, G2 and C3, or a perfect fourth above the standard bass. These instruments have a mellower, richer sound compared to the cello but are not as dark as the double bass.
Even though people attribute Ron Carter for developing the piccolo bass, in reality, similar instruments were developed during the development of the violin family in the 16th century. These instruments were built as experiments in sizing and tone and were not standardized. They were used as classical instruments, however, not for jazz, which developed in the 20th century.
It is very common to see people extend the range of standard basses to incorporate some of the sound made by piccolo bases. Les Claypool of Primus, for example, generally totes a custom bass with an additional eight frets added to reach those higher tones generally associated with a piccolo bass (listen to "Jerry Was a Race Car Driver," for example, to see how he effectively uses those higher notes).