Tanto
“So much”.
What it means
Tanto means “so much, so very” and is encountered in notated text above all in the negated turn of phrase non tanto, “not so much”. Allegro non tanto accordingly calls for an Allegro that should not be taken too fast; Adagio ma non tanto a slow tempo that is not stretched too far.
In this function, non tanto is nearly synonymous with non troppo, “not too much”. Both restrict the preceding word and warn against exaggeration. Where a distinction is made at all, non tanto counts as the somewhat milder restriction — meaning “not quite so”, while non troppo expressly sets a limit.
Without negation, tanto appears rarely; it then intensifies similarly to molto.
In the notation
The word stands in italics together with the base word, mostly in the turn of phrase ma non tanto after a tempo word, above the first staff of a movement or section. It becomes part of the movement's name.
Musical use
Restricting additions have belonged to the stock of tempo markings since the eighteenth century. They allow fine gradations and at the same time show which exaggeration the composer feared. An Allegro ma non tanto often stands over movements whose notated text tempts toward a fast tempo but whose character demands weight.
Tanto intensifies or limits a marking, for instance non tanto for not so much or allegro non tanto for not too fast. It mostly stands in negated form. Its meaning resembles troppo, which is likewise often used negated. The instruction warns against moderation being lost and helps avoid exaggerated tempos. The turn of phrase allegro ma non tanto is found, for instance, in Beethoven.
Performance practice
Read the restriction as a hint of a danger: ask what could make the movement too fast or too slow, and set the tempo accordingly. Check the tempo at the densest passage. And maintain the moderation through the whole movement — the warning does not apply only to the beginning.