Why is it that there are two ways of writing the octave lower treble clef?
Is there some sort of history behind this or are they used in different circumstances or something?
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As reported by Borland in The Journal of the Society of Arts Vol. 53, No. 2727 (FEBRUARY 24, 1905), pp. 349-374, the double treble clef was conceived by Otto Goldschmidt and published in the “Bach Choir Magazine”, a publication of the Bach Choir music society. The double treble clef was used for the tenor voices, but the symbol did not gain a very wide following. Borland does not report any usage of qualifying arabic numerals as part of his assessment, so that convention likely developed later on. Before the 21st century it was common for the tenors to read an unmodified treble clef, singing an octave lower.
It is also interesting to note that a C clef placed on the third space was also used to allow tenor voices to read as if in a transposed treble staff.
The little 8 above or below the treble clef indicates that the melody is written transposed by an octave, above or below.
For example, guitar music on the treble staff is written one octave higher than it sounds. (and a little 8 above the treble staff is added, or implied)
Why? Because if the range of the instrument is mostly above or below the staff, you would end up writing most notes on ledger lines (additional horizontal lines above or below the main lines of the staff). To make it easier, i.e. to make most notes fall on the main staff lines, the music is written transposed by an octave. Instruments with a low range will be written an octave higher, and instruments with high pitched notes will be written an octave lower.
In case of the guitar, for example, most notes would end up below the treble staff, but by writing them one octave higher they end up on the staff and are much easier to read and play.
The double treble staff, on the other hand, is a bit of an oddity as far as I can tell.