In a string instrument, the scale length (often simply called the "scale") is the sounding length of the strings. On instruments with strings which are not stopped (harp, piano) and on most fretless instruments it is the length of string between the nut and the bridge. On most modern fretted instruments the actual string length is a bit longer than the scale length, to provide some compensation for the sharping effect caused by the string being stretched slightly as it is fretted. On such instruments it is not possible to determine the scale length by simply measuring the speaking length of the string.
In many but not all instruments, all the strings are roughly the same length, so the scale can be expressed as a single length measurement, as for example in the case of the violin or guitar. In others, the strings are of different lengths, as for example in the case of the harp or piano.
All other things being equal, increasing the scale length of an instrument requires an increase in string tension for a given pitch.
A musical string may be divided by 17.816, and the quotient taken as the location of the next semitone pitch from the nut of the instrument. The remainder is again divided by 17.816 to locate the next semitone pitch higher, and so on. This technique is used when fretting instruments and is known as "The Eighteen Rule".
In many instruments, for example the violin, the scale of a full-sized instrument is very strictly standardised. Smaller scale instruments are still often used:
Larger scale instruments are rare, but may be used by experimental and avant-garde players, or specially made for soloists with particularly extended reach.
Smaller scale instruments are used extensively to teach younger players. The size of these is described by a "conventional" fraction that has no mathematical significance. For example, a 7/8 violin has a scale of about 317 mm, a 3/4-size instrument a scale of 307 mm, a half-size one 287 mm, and a quarter-size one 267 mm. 1/8, 1/10, 1/16 and 1/32 and even 1/64 violins also exist, becoming progressively smaller, but again in no proportional relationship. (A full-size instrument is described as 4/4.)
Cellos exist in a smaller range of sizes than violins, with 4/4, 3/4, 1/2, 1/4, 1/8, and 1/10 being reasonably common. As with the violin, the Stradivarius scale is regarded as standard for orchestral work; This is about 27.4 in (695 mm).
Violas are commonly described in terms of their body length rather than by a conventional fraction. There are two reasons for this. Firstly, unlike that of the violin and the cello, the viola scale length has not standardised, but rather an advanced player will use whatever scale length best suits them, see viola. Secondly, student sizes are not as often required, as most viola players who start learning at a young age would start on the violin. Common sizes include 17 in (43 cm), 16.5 in (42 cm), 16 in (41 cm), 15.5 in (39 cm), 15 in (38 cm), 14 in (36 cm), and less commonly 12 in (30 cm), smaller than a standard violin; These measurements are nominal and approximate. At least one of the surviving Stradivarius violas has a scale length of 14.25 in (362 mm).
The system of conventional fractions is taken to its logical conclusion with string bass sizes, in that a full size (4/4) bass is uncommon. Most basses are 3/4 or 7/8, and younger players can use 1/2 or even 1/4 size instruments.
Unlike Stradivarius, Torres had no strong tradition on which to build regarding scale length, so the 25.6 in figure can be attributed to him with confidence.
Gibson uses a scale length of 24.75 in (63 cm) on many of its electric guitars, including the Les Paul, Flying V, Explorer, SG, and ES-335. Gibson has used other scale lengths on various models through the years.
On a modern bass guitar, 30" (76 cm) or less is considered short scale, standard (also called long) scale is 34" (86 cm) for a 4-string and 35" (89 cm) for a B-E-A-D-G 5-string, and extra-long scale basses of 36" (91 cm) also exist.
Smaller grand pianos vary in naming. The larger models, about 6 ft (183 cm) or more in scale length, may have the full grand piano action, and are used in smaller concert spaces. Others are intended for larger homes, and may have a simplified action lacking the repeat lever that is only useful for advanced players.
Baby grand pianos are the smallest, intended for homes, restaurants and similar applications where the grand style of piano is desired even at the expense of the longer scale and better sound that an upright format would permit in the available space.
Pythagorean scale refers to the relative lengths of strings within an instrument. This kind of scaling dictates that the ratio of string lengths should be equal to the harmonic ratio of their pitches. It is a logarithmic scale which doubles at each octave. This type of scaling was offered under the assumption that by keeping all other factors consistent (esp. string thickness and tension) and changing only length, the sound of the instrument would be homogeneous across its full range.
Because the length of strings increases so quickly in the bass register, it often would produce an instrument of such length as to be impractical. If pythagorean scaling were applied to the stringed instruments, the double bass would be extremely cumbersome. In pianos and harpsichords, generally less tension or thicker strings are used in the lower register to avoid the need for such large dimensions as demanded by a Pythagorean scale.