harmonics

On the traditional scale, middle C = 256hz. On today's "scientific" scale, the A on the treble clef = 440hz. That should be all you need to do the math. You must know the ratios of the notes on both the harmonic and the chromatic scales to be this far into the course.

 

Always use your basic Pythagorean scale as a benchmark:

G0=24hz
A0=27
B0=30
C1=32
D1=36
E1=40
F#1=45
G1=48

Obviously your instructor is not using my intonation because his D1 works out to 36.75hz instead of 36. No big deal, there was no "standard" tuning for centuries, we just use G0=24 for demonstration purposes so everything comes out in integers and makes the arithmetic easy. You create your scale by multiplying every pitch in the demonstration scale by 36.75/36 and write it down like I did for easy reference.

The way you name any note is to keep dividing its pitch by 2--which lowers it by one octave--until you get it down into the demonstration octave of G0-G1. The number of divisions equals the number of octaves.

For example (using Pythagorean pitch and a different note so I'm not doing your homework for you), if the note you're trying to identify is 6,912hz:

Divide by 2 once = 3,456
Divide by 2 twice = 1,728
Divide by 2 three times = 864
Divide by 2 four times = 432
Divide by 2 five times = 216
Divide by 2 six times = 108
Divide by 2 seven times = 54
Divide by 2 eight times = 27

27hz = A0. It's eight octaves down from the note I started with because I divided by two eight times. Therefore my original note was eight octaves above A0, which is A8.

Don't forget that the demonstration scale switches octaves in the middle from B0 to C1, because it's a G scale instead of a C scale. (That's the awkwardness we accept in the model in return for the convenience of working with all integers. You can't build a C scale in which the notes are all integers because F-natural can never be an integer in any octave.) So watch yourself. If I had ended up on E1 instead of A0, then going up eight octaves would have taken me to E9 instead of A8. Make sure you notice which octave you bottom-out in so you get the right octave in your answer.

BTW, I don't know what tools you're using to do the arithmetic. A problem like this is perfect for Excel. You can lay out your scale as a column, and put the associated frequencies in the next column to the right. Build a formula of multiplying the frequency by two in the top cell of the next column to the right. Then copy down into all eight cells in the new column. Then copy the formulas in that column into the next eight or ten columns. You will have the frequencies for every note in the G-major scale in every octave from zero to way off the upper end of the piano keyboard near the threshold of human hearing. Then you can just solve your problem by inspection.

I'm not telling you how to use Excel to quickly convert the Pythagorean scale to the scale your teacher is using. I'm leaving that for you to figure out. It's easy and shouldn't take you more than a couple of minutes--depending on what grade you're in, sorry I have no idea how old you are and how much of this stuff you have already learned.

Whatever level you're at in school, or even if you're doing a lot of arithmetic in a job, I recommend that everyone learn to use Excel this way. It makes math quick and easy, leaves you with all of your work not only visible but completely organized so you can find a mistake in two seconds, and it minimizes the number of times you have to type in any number or any formula to greatly reduce the opportunity for error. It even makes sense out of all of it by arranging it in a logical, understandable table structure with headings you can choose for yourself. I personally think Excel is one of the most useful software programs ever invented for use in daily life and I'm sorry it's not more widely recommended for stuff like this.

 

The problem with the Pythagorean scale is that there are no notes in between. There is only the major scale with no accidentals.

You picked a bad example because 54hz is not an in-between frequency. It's 2*27 and is therefore just A1. But let's assume you picked 57hz.

In general, the only way to create in-between frequencies is to work your way around the circle of keys. Let's go up by fourths starting with C1, since going down by fourths toward D0 is going to give us very small numbers with very long decimals.

Use the same ratios.

C1=32hz, we already have that. But we do NOT assume that all the others except F, which we don't have, are the same as they were in the G scale. We calculate them by multiplying the frequency of our root tone by the ratios of each of the notes in the G0 scale.

D1=32*(27/24)=36hz. Same as in the original scale. So far so good.
E1=32*(30/24)=40hz. Still no problem.
F1=32*(32/24)=41 1/3 hz. We expected a problem here since there's no F natural in the G scale.
G1=32*(36/24)=48hz. That's 2*24, exactly one octave above the G0 we started with, so we're cool here.
A1=32*(40/24)=53 1/3 hz. Oops. This is a little lower than one octave above 27hz. It turns out that A in a C scale is not quite the same note as A in a G scale. I pointed out in the first post that in the Pythagorean scale, every whole tone is not quite the same ratio, and here we are presented with the fallout from that. This is starting to get messy.
B1=32*(45/24)=60hz. Well we're okay here. This is one octave above 30hz, the B0 in the G scale.
And of course C2=32*(48/24)=64hz. That's a no-brainer, a perfect octave about our root tone, but it's nice to do the math and make sure there's no more craziness.

Every time you take one step around the circle of keys, you introduce one new note because obviously you're flattening (in my example) or sharpening (if you go the other direction) one note. But you get two non-integer frequencies. We expecte the F since that note didn't exist in the original key, but the new frequency for A is quite a surprise.

It's obvious that by the time we go three more steps around the circle through F and Bb into Eb, we will have introduced three more flattened notes, but six more non-integer frequencies. That means that every note in the Eb scale will be a non-integer. And of course it will keep getting worse. We'll never get back to integers because we're dividing by three every time.

We could try circling the other way and go from G to D to A to E etc., but I'll leave that as an exercise for you. You'll discover that fractions are inevitable there too.

So if you're looking for an integer frequency like 57hz that wasn't on the original scale, you will never find it! Not even its octaves!

I'll also leave it to you as an exercise to go around the circle completely, through thirteen transitions. That should get you back to the key of G, shouldn't it? You will be interested to see what frequency your G8 has! It will not be eight exact octaves above G0! Not only will it be an annoying decimal, it may be closer to a Gb than a G!

So you have to fall back on the chromatic scale to find an accidental note. Define a half tone as 2**(1/12) and a whole tone as 2**(1/6) and calculate new frequencies for all the notes including the sharps/flats. Then see by inspection which one is closest to your 57hz. It will be Bb1. 57hz is either a slightly sharp Bb1 or a slightly flat Bb1, I'll let you figure that out.

Also take into account that for the past century or so, "scientific" tuning has been in standard use, based on A4=440hz instead the old Pythagorean tuning of C4=256hz. Any professor who is introducing you to the modern chromatic scale is probably not using obsolete Pythagorean tuning. This will change your frequencies by a little bit.

If your teacher asks you to figure out something like this without first explaining the difference between the Pythagorean scale and the chromatic scale, then either he's leaving out an important step and confusing a lot of people, or else I'm not understanding your assignment.