no, the existing "classical" notation system doesn't parallel other forms of measurement in that way. there have been some of other notational methods and extreme extensions to western notation, especially during the 20th century, but relating musical notation to other forms of measurement doesn't work. it'd be like trying to translate a written language into metric.

As far as I know, there was no attempt to do such thing.

If metrical systems are so great, why wasn't it used for music, then? I can see two reasons: 1) there was no motivation for such a thing, 2) it would have been a terrible idea.

Let's start with reason one. Decimal systems do have some nice order and practicality in many contexts, but a big motivation for the radical French changes was removing religious and royal influences. They were not doing it just because they loved the number 10, they wanted to get rid of the links to the two powerful institutions of "the past." A calendar that had nothing in common with the one full of religious festivities distanced the church from daily life, and getting rid of the "pied du roi" ("foot of the king," about 32.48 cm, or 1.066 ft if you still use that other system) and similar units was a good thing because fuck the king and fuck the oppressive ways of the past.

There was no such interest for music. Music theory was not an important part of daily life, commerce, or science. Music was starting to be considered as an art, as opposed to a mere craft, but as far as I know they were not worried about the binary and ternary divisions used in music.

The second reason is a little more complicated. The system used to notate rhythm was based on binary and ternary divisions or rather arbitrary units. However, musicians were not fond of the "all croches are equal." Yes, they were notating a bunch of notes of the same duration in sequence, but that doesn't mean all those notes were equal (see notes inégales)... There was no égalité for musical notes in 18th century France.

There was also the thing that notation was a reference to make music, you were not supposed to play everything exactly as it was notated. As a result, there was no point in being particularly pedantic about each tiny detail.

I don't know if you are not into French baroque music (I think the harpsichord has the most gorgeous repertoire from this time and place), just listen to a few modern interpretations. I'd start with this prélude from around 1716. Notating the exact duration of each note for performance was/is difficult (Marie-Dominique-Joseph Engramelle started working in way to do that in "La tonotechnie").

17th century French musicians were into "unmeasured" music. Some form of that lasted until the late 18th century if not a little more, and 18th century French music in general has a lot of nuance in measurement.

What about, say, intervals and tuning? Not really, they weren't after that, either.

Joseph Sauveur (1653 – 1716) was a mathematician and physicist, and he joined forces with Étienne Loulié (musician and theorist) to teach Louis XIV's nephew. Sauveur started using the word acoustique for the study of sound. He worked on intervals and vibrating string, and came up with some units: méride: (1/43 part of an octave), heptaméride (1/301 part of an octave),
demi-heptaméride (1/602 part of an octave).

Félix Savart (1791 – 1841) proposed a unit derived from Sauveur's work for tuning systems. His proposal was of a more decimal nature. Musicians were not into that, at all... 18th century music theory mostly deals with scales using note names and ratios, just it was done in the past. It is in the 19th century that "acoustics" takes over the study of tuning and musicians mostly forget about tuning systems for a while.

Here's some more French music, just because it's lovely (from a little earlier than the Revolution, though):

• Jacques Duphly (1715 - 1789) La Pothouin

• François Couperin (1668 – 1733) La Forqueray

• Jean-Philippe Rameau (1683 – 1764) Suite