Bear with me, this will take a bit of doing…
The size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (unison), 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Intervals with small-integer ratios are often called just intervals, or pure intervals.
Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called 12-tone equal temperament, in which the main intervals are typically perceived as consonant, but none is justly tuned and as consonant as a just interval, except for the unison (1:1) and octave (2:1). As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, although it is very close to the size of the corresponding just intervals. For instance, an equal-tempered fifth has a frequency ratio of 27/12:1, approximately equal to 1.498:1, or 2.997:2 (very close to 3:2).
There are some instruments that can be played in a just intonated fashion (voice, violin) and many that cannot easily do so (piano, guitar.) The tradeoff in using an equal temperament system is that you can easily change keys which enabled all sorts of wondrous music. However the intervals aren’t pure, and before you dismiss the value of that… part of the excitement and wonder of a choir, string quartet, etc. is that they can change keys and yet sing pure intervals (or not) as they wish. Remarkable flexibility.
The wonderful little volume “Ratio” by Michael Ruhlman displays the ratios behind cooking. In the book one can learn that pasta dough is 3:2 ratio of flour and eggs. No wonder we all love pasta—it’s a perfect fifth of a food! Cookies are 3:2:1 (flour, fat, sugar). There’s little surprise that ratios are there to be found once you start digging into ratios and they’re place in the cosmos. And more importantly, they’re far more useful.
If you have grandma’s pasta recipe (I looked up one by Mario Batalli) you get something like this: 3 1/2 cups unbleached all-purpose flour, 4 extra-large eggs Now we can argue our way through whether this is represented by the ratio above, and how much it deviates, but my point here is if you memorize the ingredients and all the “use half the flour unless it’s not dough or too sticky stuff” you still will only be able to make that one recipe. But if you understand that the ratio of 3:2 makes a pasta dough then you have different starting point. You have *information* that you can use to create other variations… or explore the boundary between pasta and cookies etc. Same is true of music.
If you know how to hack your way through a song on a guitar and sing along that’s cool. But if you understand the intervals, the chord progressions and the meter, you have the tools that will allow you play 1000s of songs or make up your own.
One more example before I’m done torturing all this to a fare the well.
The lovely volume “By Hand & Eye” discusses ratio, although in this instance as it applies to furniture and proportion. As it says on the site…
…George R. Walker and Jim Tolpin show how much of the world is governed by simple proportions, noting how ratios such as 1:2; 3:5 and 4:5 were ubiquitous in the designs of pre-industrial artisans. And the tool that helps us explore this world, then as now, are dividers.
Something like a step stool is one handspan high by two handspans wide… or the same ratio as an octave, which at this point should be no surprise. And when you begin to pin the ratios together you can find them in the subdivisions of our hands and bodies, in the spiral of a nautilus shell, and in the fractal nature of so many things, where the thing up close repeats the pattern of something of greater distance.
And all of this leads us to the Fibonacci series, which is building block that we seek at eh foundation of so many ratio related conversations (which at least for me is a good enough source from which to crib.)
A piano keyboard makes this somewhat clear…
…scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2. While some might “note” that there are only 12 “notes” in the scale, if you don’t have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes. The word “octave” comes from the Latin word for 8, referring to the eight tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C.
So look… I’m not telling you that your whole life should be constructed around Fibonacci, The Golden Ratio, etc. (although many things already are…). Or that the art of music, cooking, design, and creating in general might be in how and when you break or bend that cosmic sense of proportion. But it might well be the case.
The real point of all this is that recipes are for students. They are a constraint that you can embrace in order to begin producing results. Follow these drawings and you’ll create a reasonable staircase. This plan and you’ll produce reasonable tasting food. That sheet of music and maybe something Bach like will be heard, or maybe some ‘Stones or Coltrane.
But if you embrace the knowledge behind how how the universe orders itself engrained in all we use to create, you need only apply is a little bit of inspiration about where to bend the lines.