# Why does the exponential curve that shows up in the orchestra not overwhelm anything?

So, I've noticed these ratios of range in the modern orchestra, which is primarily strings:

• Soprano:Alto - Factor of 4 or 5 more soprano than alto
• Tenor:Bass - About twice as much tenor as bass
• Alto:Low range - About the same amount of Alto as Tenor + Bass

This clearly shows an exponential decay curve as the range gets lower, similar to this curve:

Thus the Violin:Viola:Cello:Double Bass ratio in an orchestra could easily be this:

60:15:8:6 (in fact, I often see ratios similar to this in say a recording of Beethoven's symphonies, but over time, every part of this ratio becomes bigger, yet the ratio in simplest form is almost the same as in Beethoven's symphonies)

Now in this question about a possible woodwind predecessor to an orchestra, Ceremonial flute choirs as a predecessor to the orchestra, how plausible is this? I mentioned in the comments and in the main body of the question these same ratios that I listed above and somebody mentioned that this exponential curve is not ideal, that it would lead to too little bass and too much soprano. Which is why I'm asking this question.

Why is the exponential curve a default for the orchestra if in an entirely woodwind ensemble, it would be unbalanced? What is it about the timbre of multiple string instruments that leads to this exponential curve? I know that the difference between 1 and 3 of the same string instruments is staggering(1 gives you a solo timbre that can be more nerve-wracking, almost painful at high pitches(in the case of the violin and viola), soulful(in the case of the cello and to some extent the high range of the double bass), or intensely deep(in the case of the double bass), 3 gives you a more orchestral-type sound)

But in small string ensembles, the ratio is almost 1:1 and is 1:1 in the case of the Trio and Sextet(that is if the Sextet does not include the Double Bass, which some do). Yet as it gets larger and approaches the 60:15:8:6 ratio of Beethoven symphonies, it goes from this 1:1 or almost 1:1 to an exponential curve.

Why is that? Why is it that the orchestra follows an exponential curve with pitch range and small ensembles have an almost 1:1 ratio for all the ranges?

• The orchestra I see most often uses a linear pattern, eg 4 double basses, 6 cellos, 8 violas, 10 2nd violins, 12 1st violins. Jul 19, 2020 at 8:45
• Can you clarify: it appears that your words "range" and "decay" refer solely to the *number of instruments in each group? The final sound product is easily controlled by means of those cute things called "pianissimo" and "fortissimo" Jul 20, 2020 at 13:58
• I am using them that way because 1) it makes sense to classify instruments by pitch range and 2) I’m asking why an exponential curve shows up, thus my mention of exponential decay. Jul 20, 2020 at 16:04

It is mainly because increasing the number of instruments in a section does not actually make the section much louder, mostly it just makes it sound “denser”. Any doubling of the number of unisono instruments effectively increases their volume by 3 dB. Note that dB is a logarithmic scale, corresponding to the ears' response. In other words, to effect a linear increase in perceived loudness, you need to increase the number of instruments exponentially.

Now, why would you want that in the first place? –I don't think it's only about loudness; the sound silkiness itself that you only get with upwards of a dozen violins is just important part of the aesthetic of symphonic music. But inseparable from that is that those violins serve also as the main melodic role in the orchestra, which does require a robust volume and to get that (with only wooden acoustic fiddles) the vast numbers are also needed – even if, again, increasing numbers is a very expensive way of increasing volume.

The volume-need of the violins is not to persist against the low strings, those could easily just play quieter (as indeed they often do in a chamber music setting), but against the other instrument families which are inherently louder, in particular brass and percussion. Concretely, those instruments are capable of actually transforming a sizeable amount of the power a human player can put in, into sound. That's the reason the bass wind instruments, despite their bigger size, aren't really louder than their treble counterparts. In some cases they are actually quieter, since low pitches need more energy to be perceived as same volume.

On the other hand, on string instruments, except perhaps double bass, you first hit the limitations of the instrument itself. Even at fff, you can't put in all the force into a violin that a trained player's muscles could, because that pressure would just squelch the string.

• Also, as an addendum, for approx. each 10 violins added in the orchestra you likely double the dynamic palette of the instrument. This is important to obtain solid nuances from ppp to fff, not only a forte and a piano sound, with some indefinite state in between both. This, also, is an important musical role to most late Romantic and early Modern musics, bringing the common orchestra up to 300+ musicians on stage. Jul 18, 2020 at 23:19
• @Glorfindel thanks for your edit, but would you care to explain why some people consider uncommon spellings such as æsthetic or naïve as worth “correcting”? Those are perfectly cromulent in English writing, they don't make it harder to read, and it should be obvious that they're chosen deliberately and not as a slip of hand. If they mark me out as a pretentious cutup then so be it, that's my personal choice, isn't it? Jul 19, 2020 at 15:43
• Sure, feel free to rollback the parts you don't agree with. It looks like the Stack Exchange search doesn't care how you spell the word, but more primitive ones might care, and people are more likely to search for 'aesthetic'. Jul 19, 2020 at 16:18

If you don't mind some equations, you might look at "Why do choirs work?" for a bit of mathematical analysis of a closely-related question.

The key observation in the question there is that the mean amplitude of N people playing the same note together, with random phase-shifts, is proportional to the square root of N; the perceived loudness is roughly the log of this, so about half of the log of N. That makes the cellos a little louder than the basses --- say louder by some increment Q; the violas are an increment 2Q louder than the basses; the violins are an increment 3Q louder than the basses. Of course, that assumes that a single violin, a single viola, a single cello, and a single bass all have the same basic amplitude, which I doubt is true, but I'm not a string player. But the main point is that you don't end up with the violins being twice as loud as the violas, etc. --- there's a linear growth in loudness due to the exponential growth in numbers [again, assuming the same single-voice loudness].

This also ignores two other potentially important factors --- the differing rates of attenuation of different frequencies in a concert hall, and the differing perception of loudness as a function of frequency. It also assumes (pretty correctly, I think) that the listener is the same distance from all four string sections (i.e., that the differences in distances are small compared to the overall distance). That's probably wrong in the first 10 rows, but right thereafter.