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I looked up molto allegro and allegro di molto and they both mean on the fast side of allegro.
What is the difference then? Is it like "more allegro" vs "Even more allegro" where "more allegro" is molto allegro and "Even more allegro" is allegro di molto?
Example of where molto allegro is used:
Symphony no. 40 by Mozart
Example of where allegro di molto is used:
Pathetique Sonata by Beethoven
Tempo indications are more suggestions than hard-and-fast rules. This is even true of metronome markings. There's room for interpretation, especially with slower tempos.
To use the Pathetique example, listen to Rubinstein play it. Then listen to Richter. You'll find that Richter's tempos are noticeably faster. However, both performances work musically.
So, these two indications are pretty much the same, and in practice, you'll find as much variation in different performances of the same piece as you will with different pieces with these two different tempo markings. For a great example of this, compare Geiseking's and Rubinstein's performances of the first movement of the Pathetique Sonata. Gieseking plays it in 7:44 to Rubinstein's 8:45, all the more striking because Geiseking actually plays the slow parts slower than Rubinstein does! (I measured the opening section at 1:36 for Gieseking and 1:28 for Rubinstein.) So, tempo is a guideline. You'll find all sorts of arguments about tempo choices of performers. Gieseking's sounds too fast to me, but then, I grew up with the Rubinstein recording and it's still my favorite.
Both of the phrases translate "very lively" in English. Imagine that they are both marked "very lively" and understand it that way, and you'll be thinking about tempo in the way the composer intended.
Interestingly, the word allegro in Italian translates to words such as cheerful, lively, brisk, upbeat, and even "tipsy". It comes from the same Latin root (alacer) from which we get our word "alacrity". While there are plenty of pieces marked allegro which can hardly be characterized as cheerful (the Pathetique being an example), it's probably a good idea to keep in mind (whether molto or not) that sense of energy that one has when taking a short walk on a cold day in a light jacket. :)
This article is a good overview of tempo markings.
One of my metronomes has allegro as 120-168 bpm. Another, 150-180 bpm.The former seems to be the generally accepted.This alone shows that tempo markings are vague, at least. Taking something like a symphony, the interpretation can make a difference of up to 50% extra on the length of a movement, comparing one conductor to another (Toscanini, 1953 - Eroica). So I think composers, then at least, kept it vague on purpose, to allow artistic licence for future performances.
So with this sort of discrepancy, the quick side of allegro can vary. However, allegro di molto is quite quick, while molto allegro is pretty quick.(The clue is in the wording). There is no specific difference, particularly given the vagaries of tempo signs anyway.
Unless a composer states actual bpm.,there is no absolute. In fact, even then, listening (and counting) recordings of the writer playing his own pieces, there is often no strict adherence to a stated tempo.Any number of other factors can and will make that change, so keeping it (a little) vague can be seen as a good thing.
As I have stated before on this site, terms like "molto allegro" were never intended to convey an exact measurement of beats per minute.
Verbal descriptions of tempo, like "allegro", "adagio" and "andante", were used by composers in previous times before the invention of accurate machines that could measure beats per minute. These verbal descriptions are nothing more than approximations. Mozart composed his music before the invention of the metronome.
The metronome, which can be set to beat out a specific number of beats per minute with accuracy, was patented in 1815 and quickly became popular. After that time, composers began to label sections of their music with a precise measurement of beats per minute if they wanted to, and they began to rely less on verbal descriptions like "allegro". If a composer after around 1820 continues to use words like "molto allegro" then that means that the composer wants you, the performer, to make your own decision about how fast to play it.
In classical music, the notes on the page are there for you to play precisely, but the tempo and the phrasing, and the speeding up and the slowing down when you see markings like "accelerando", "ritardando" and "rallentando", are there for you to interpret as you see fit. This is what makes classical music interesting -- no two performers (if they are musically adept) will play it exactly the same way every time.
Footnote: My favorite tempo marking in classical music is Beethoven's Mass in C, Opus 86 (1807), first movement, where Beethoven's tempo indication is "Andante con moto assai vivace quasi Allegretto ma non troppo."
That translates to "Andante with motion, almost like a very lively Allegretto, but not too much."
Not long after Beethoven published this piece, the metronome was invented and became popular. My suspicion is that everybody decided that it would be really useful to have a means of measuring beats per minute accurately if composers were going to go on like that!
