What analysis can be used to determine musical key signature?

Isn't this like guessing a persons' nationality from their name? Sometimes it works, sometimes not. And there may be examples of ambiguity where its impossible to tell for sure. Can you imagine some software to do this for you that has a database of names and associated nationalities? Here you could enter the name and the computer software would output the nationality of the person with that name. But there is a question. What would someone use this for? If someone cannot themselves determine the nationality, what would they be doing with the answer it provides?

Perhaps they will use the result to engage in prejudice. Or perhaps they will engage some stereotype that they don't understand, but know is prevalent and with which they desire some compatibility.

In the analogy of determining musical key, there is a chance that someone might use an automated answer to give the appearance of understanding, the appearance of cultural compliance and willing complicity with a stereotype; but without possessing the understanding themselves or even the desire to understand.

And where there is no desire to understand...there is only a desire to experience.

Here is a question: If you can't determine the key of music yourself, why are you making music? Why would you use a tool instead? Or perhaps it provides a convenient answer in a timely fashion to be considered among many choices?

How could knowing the key of music possibly help you? So you can group music that shares the same key? It would seem reasonable to try this, and then when mixing music together it might have a level of organization that indicates some intention as opposed to their combination being an accident toward key.

The sound of intention. Perhaps this is what we strive for. If you hand over music that cannot be discriminated from randomness then it might be considered unacceptable. Or if what you deliver cannot be separated from a body of pre-existing works you may be accused of plagary or the work regarded as unoriginal.

The musicological discussion of key and invariably time signature across cultures and nationalities is fascinating. The world is full of music that indicates some shared inheritance but also a shared desire to create something new. The question of which keys are most popular across cultures historically tells a story that might indicate how music spreads, but also some influence of the instrumentation and the ensembles the music is written for. For example, why need more than 1 key? Why is music in different keys? In fact why does music exhibit harmony at all since everyone singing in unison would be much easier? Is there a difference between the key of F# and the key of C?

Why on earth or mars are modern orchestra and much western music tuned to A440?

*Equation for Western Music*

To make a formula for tuning modern orchestra in the form of

y = (a^x)(b)

y = (((2)^(1/12))^x)(8.17579892)

where

y = frequency
x = note number
a = 2^(1/12)
b = 8.17579892

stated in words this says that

y is equal to the 12th root of 2 raised to x, times 8.17579892

You may use this formula to calculate the frequency of any note in range 0-127 in the modern orchestra using the calibration A4 = 440Hz. Features? It shows a y intercept of nearly 8 hertz as the lowest note. Fast vibrato is around 6 hz so think Edith Piaf or King of the Foreeeeest from The Wizard of Oz.

Besides the influence of history and culture on the Key question and the ambiguity there, what ambiguity might lie in the theory or terms? Also there is the question of audio vs sound.

For audio, asking the key question would require asking the power distribution of frequency spectra and answering with FFT. This domain is pure and culture immune. FFT may be converted to pitch and further key. Within the pitch domain there is culture.

We may ask the Key question initially of 2 choices: Ask key of a polyphonic instant or of a succession of p instances.

Definition, What is Key?

Key = root + scale For example C major

How does polyphony affect key? It increases ambiguity, but how?

Ambiguity of Key= ambiguity of root (times) ambiguity of scale

Definition, what is ambiguity?

ambiguity = 1 - probability

Where ambiguity is uncertainty, probability is a measure of certainty.

Definition, what is probability?

p = 1 / n where n is the size of a group, set, or degree of freedom. In the english alphabet there are 26 characters or degrees of freedom. For each letter in a word there is a probability just based on degrees of freedom that is 1/n or 1/26. Asking the same question inverted what is the ambiguity possible? 25/26.

In western music there are 12 tones. And asking the scale question involves an 8 note scale. So in a manner of speaking asking what Key and scale is answering what 8 selections from a possible set of 12. So what 8 in 12? It turns out there are quite a few to choose from.

The above graph shows a set size (n=12) and a list of scale sizes within 12 notes. Starting with scale size 1, or a 1 note scale, we find 12 possible 1 note scales to choose from. What is the probability of that 1 note scale from a set of 12? This is p and found in the last column. Predictably at the bottom we may ask how many 12 note scales there are in a set of 12 and the answer of course is 1. Probability 100%.

Because of symmetry and isomerism there are as many 8 note scales as 4 note scales. In analysis it is best to consider fewer combinations then mirror invert, using this principle.

Asking again the key signature and what 8 note major or minor scale, we are asking within the 8 note permutations, those of major or minor quality. There are 24 of these so the probability of any 1 major or minor scale is 1/24 or p = 0.04166.

If some audio is used for analysis, then first the tonal center or root will be determined, followed by the quality. For audio analysis algorithms and people, the question of any audio having a single tonal center is much the same. Proceeding from audio depicting noise, there is no recognized center. With audio depicting singing voices in unison, there will be complete agreement. As voices diverge from unison to complete chromatic separation, there are increasing levels of ambiguity.

Besides the classical drama between consonance and dissonance in music as devices for tension and relaxation, there is very much a dimension of ambiguity that is not equivalent with dissonance. Ambiguity of Key Signature in music may come from the separation of many sounds in pitch, or the complete absence of pitched instrumentation to begin with. Further there are many levels of ambiguity in the qualities of scales and chords.