I'm with you, I don't see how a particular note frequency being an integer (let alone one that's 2^n ) has to do with anything. No need to say more as you have already said it.
If there is someting "special" Grover doesn't cast any light on it.
and I don't see
https://wikipedia.org/wiki/Scientific_pitch
as being relevant to bafflement.
Hey folks. Just some clarification - what I'm specifically getting at here is that Frequency is logarithmic while chromatic divisions of the octave are linear. The systems do not inherently line up. Add on to this the problem of tuning, which can either refer to frequency or tuning within a relative temperament system and it gets rather confusing. I hope this will clear up some of the confusion:
1) While seconds may be a subjective measure, Hertz are not. As a unit of measure it is specifically designed to give us numbers that we can work with that represent a natural phenomena. We often pick even numbers for their simplicity. With two tones such as 440hz and 880hz there should be no inharmonicity - they should be completely just/in-tune from a frequency perspective. Same with 379.126hz and 758.252hz and so on. At first glance, this shouldn't make any difference in how we tune. Just pick a pitch and tune.
However, because (primarily western) music works on an equal-tempered chromatic scale, changing our tuning frequency shifts or linear system in relation to our logarithmic system. This leads us to some complicated interactions which we don't currently have a good unit to describe - frequency obviously is logarithmic, and cents, the smallest unit of tuning in western systems, is just a subdivision of our linear, Equal-temperament system. I'll make a note here as well that while it doesn't matter what notes the octaves are tuned at for the sake of the octaves, it does have an effect on how dissonance is spread between those different octaves. Another reason we divide notes in 'perfect' 'major' 'minor' and other designations.
2) Because ET is spread across roughly the range of a piano for any given western ensemble, changing our tuning frequency effects primarily two things: the tuning of the keys, scales and intervals we are most commonly playing in, and the spread of dissonance across the octaves we are using. What I mean by the latter is that ET is an attempt to 'manage' dissonance if you will. It spreads the difference between the logarithmic and linear systems across the entirety of our used octaves. The result is that every note is a little out of tune, instead of having several notes perfectly in tune with others wildly out of tune. (wolf tones) The point is, that by choosing our tuning frequency carefully, different keys, scales and chords will have either more similar or more different characters (a la baroque tunings such as well temperament, in which Db is rarely used) It also means that specific frequencies are less likely to aggravate our problem of managing dissonance. With A4 = 440hz as our tuning frequency, we assume a certain amount of dissonance we'll call X. Whenever we shift our tuning frequency, because the difference between logarithmic and linear systems only needs to reconcile inside a certain range (in our case from about 60hz-20khz + for most western orchestral instruments) The practical value of X can actually be increased or decreased slightly because we are changing our operating parameters. Because C4 is a lower frequency than A4, we've moved 'down' the logarithmic scale, the frequency has a closer to linear relationship, or rather, one that breaks into twelve chromatic notes a little more cleanly. Why wouldn't we just tune to the lowest note possible then? Well, if you remember that we're spreading our dissonance from all octaves between all octaves, we'll want to pick a frequency that's close to the middle of whatever instrument we're playing so that we're not accounting for dissonance at far higher or lower frequencies than necessary, which would cause our value of X to be far higher than necessary for whatever our audible musical range needs to be.
3) So why C256? As I mentioned above, its close to the center, is lower than A4 so helps shift our parameters a bit, and this is the big one which I agonized over whether to include in the paper or not: because we play in C diatonic more often. This is of course a huge value judgement, and keen readers will be asking if the Keys of B for example still display dissonance (as does the key of Ab in A440) and the answer is yes. The 7th and tritone are the most dissonant notes in the 12-note chromatic scale. C256 does not change this, but rather puts it into starker contrast because the key of C and other diatonic major keys are significantly more in tune. The result is that each key or scale more individual 'character' or tuning qualities in relation to each other (somewhat similar though to a lesser degree than changing temperament systems) I struggled with whether or not to put this in the paper (which I consider a work in progress) because it represents a very sticky musicological discussion to say the least. My subjective impressions are that this changes my chordal and textural perception of the music. While I can pinpoint the exact reasons why, if you want my full thoughts on the matter send me a PM.
4) Inharmonicity in this case is the word I'm using to describe frequency-based dissonance. When two frequencies are not perfect intervals (doubled, halved, etc.) they create a 'beating' against each other, which is the result of conflicting periods. As I note in the paper, managing this beating or inharmonicity is the goal of our current tuning and temperament systems, but actually reconciling the two remains an unsolved problem. You could do it by only dividing an octave into perfect intervals, although you'd have limited notes available to you at audible frequencies. I have an idea for how one might approximate this in a twelve-tone system, but it's incredibly tedious and likely irrelevant to the general audiophile population.
5) I'm using iZotope's mastering spectrogram plug-in. For synthesis I'm using Absynth by Native Instruments, pure sine tones. The genesis of this paper was my observation many years ago that frequency beating can be viewed as spectral information. The beating effect is quite audible, and has been a part of binaural and monaural beats psychoacoustic phenomena for a while.
I hope this makes more sense? This is a rather complex topic that we don't have great tools to describe because it lies at the crossroads of a value-laden linear system (western classical music) and the logarithmic system of physical acoustics. I am not making any claims as to whether C=256hz are superior, inferior, etc. This is my understanding of the issue. I provide a few subjective impressions only because I realize this is in the context of a proposed audio device for purchase, and wish to provide some nonbiased sonic notes from my own recreation of said device.
Happy listening!
