C256, a numerically-beautiful pitch standard

I've heard about how we should use \latex \text{A}432 over \latex \text{A}440 a lot of times. So many times that famous music YouTubers like Adam Neely have made a video about the \latex \text{A}432 tuning, and another one about its application with temperaments. According to him, \latex \text{A}432 is "basically just a big new age circle jerk" for its notoriety among new age influencers, that 432 Hz taps into the "natural frequencies" of the universe. Using \latex \text{A}432 or not doesn't affect the music its played upon, rather it is the stylistic preferences of the composers or the performers.

History of the pitch standard

Let's set things straight first.

Learners of keyboard instruments are first taught the concept of "Middle C". It is named because it sits in the middle of the keyboard. Depending on the size of the keyboard, it might not be dead center, but its octave should be. The octave is the span of keys from one C to another, between the 12 keys. Since we have 12 keys, we can say there are 12 divisions of the octave. Keyboards are tuned with this divisions equal so keyboardists can play in an key without going out of tune. Thus, our system of pitches in an octave is what we call the 12 divisions of the octave, or 12-EDO. Every "C" is situated in the immediate left of the two black keys in each repeating pattern of the keyboard. Middle C is also called such because it sits at the intersections of the voice registers.

Not all musicians are pianists, so the "Middle C" isn't always in the center of whatever instrument they play. If you play solo in the piccolo, it wouldn't make sense to call Middle C that way since the instrument's range begins higher than the pitch of Middle C.

So there have been proposals for naming absolute pitches like the Helmholtz notation where Middle C is written as c’. The more popular system was the scientific pitch notation in which every octave is assigned a number, and the lowest octave is determined in reference to the human's hearing range. Humans are known to hear at frequencies as low as 20 Hz, therefore the reference "octave zero" should be around this frequency. With this system, the octave where Middle C sits is in the fourth octave, and so we can call Middle C as \latex \text{C}_4.

The modern pitch standard revolves around \latex \text{A}_4, or the A above Middle C. Currently the standard is set to \latex \text{A}_4 = 440 Hz, or simply \latex \text{A}440. This means that the \latex \text{A}_4 key on the piano should generate a wave with a fundamental frequency of 440 Hz. Before, \latex \text{A}_4 wasn't always tuned in 440 Hz. A few instances there were tunings higher than \latex \text{A}_4 = 440 Hz, but most of the time \latex \text{A}_4 had a lower pitch. I think this is why we associate lower tunings with the olden days; we could have existed in the 17th century hearing performers play at lower tunings. But people used to have a limited understanding of frequencies so they weren't able to measure the exact frequencies of how their instruments were tuned. Nevertheless, we could safely say that the tunings of the past inflated as we have accounts of performers wanting to sound brighter by raising their pitches, that led to standardization efforts because people were tuning their instruments so high that music became increasingly difficult to play. The standard \latex \text{A}440 was finally set by ISO in 1955 and further reaffirmed in 1975 as ISO 16.

My take on the pitch standard

It's good to have a standard for the safety of the performers. But what does this mean for the composers?

Personally, I think \latex \text{A}440 is too bright for most music. While I do agree we should control this pitch inflation, what stops composers from writing their works in a different, perhaps a lower pitch standard? There are already tons of music delving into a territory outside the classical tastes of Western music, such as xenharmony where the notes could fall outside the 12 keys we know (you might want to check the Xenharmonic Wiki for the technical stuff). If there is a sound argument to using xenharmony, then I believe there is also a sound argument to using a different pitch standard.

To be honest, I don't have hard feelings over people who prefer using \latex \text{A}432, but seeing the controversy and the crazy conspiracy theories about \latex \text{A}432, I might not want to take that route myself.

Plus, my experience as a software developer gave me a fascination about powers of two. Synthesizing my musicianship with software development, I want to use something that feels neat to use. That's why I decided to use \latex \text{C}_4 = 256 Hz.

Enter C256

\latex \text{C}256, commonly known as "scientific pitch" (not to be confused with scientific pitch notation above), was proposed in 1713 by a French scientist named Joseph Sauveur as a response to the pitch inflation they were experiencing. For some time it was also promoted by the composer Giuseppe Verdi but never caught on among other composers. The next higher octave of a pitch can be computed, such that the standard \latex \text{A}_5 is just twice that of \latex \text{A}_4, which is 880 Hz. The opposite can be said of lower octaves--just halve their frequencies. The beauty of \latex \text{C}256 is that every C will be a power of 2, and so we can already know the octave in which the pitches sit by determining which powers of two they are in between.

g(x) := \lfloor \log_2(x) - 4 \rfloor

In order to get the frequencies of a given key in \latex \text{C}256, use the following formula:

f(x) := 256 {\sqrt[12]{2}}^{(x - 60)}

The value of \latex x is the index of the key we want to use. For this instance, our key starts with 0, which is C0. Continuing from this value, we can say the value of Middle C is 60. The key index for \latex \text{A}_4 is 69 (nice), and plugging this value into our function, we now get 430.5389646099019. We know that under \latex \text{C}256 12-EDO, only the C's get an integer value and the rest of the keys have fractional parts. This is also the same with \latex \text{A}440, except it is all the A's that get integer frequencies. Here is a table of the frequencies in the octave of Middle C under \latex \text{C}256:

Remember, get the lower octaves by halving the frequencies, and higher octaves by doubling them.

If \latex \text{C}256 looks OK to you, you might want to use it. I shall update this post with samples of comparing \latex \text{A}440 to \latex \text{C}256.