Audio Frequency Chart Memorization with 99.2% Accuracy

That's neat and handy method. Very cool! Not to hijack your thread, and you obviously have a solid understanding, but I'll add my viewpoint of cents VS Freq (long winded as it may be:) as I've seen this confused before.

Yes, absolute frequency (in Hz) and "cents" can be confused, but with a fundamental understanding, the difference is quite clear. One observation is that our musical note intervals increase exponentially. A mathematically easy example is that octaves double. Approximately speaking, G1 = 50Hz, G2 = 100 Hz, G3 = 200Hz, G4 = 400 etc. Semitones are a little more "mathy" expressed as the "1/12 root of 2". So in any given octave, take for example the G2 - G3 octave between 100 and 200 Hz we can count 100 each whole number frequencies (you know of course that in reality there are and infinite number of frequencies...but let's not go there right now). So when we put our 12 semitones in any octave, their absolute frequencies and interval widths will be progressively increasing.

Cents are simply a further division of semitones, expressed as the "1/1200 root of 2". There are 100 cents between every semitone, no matter the number of whole number frequencies between the semitone in question and it's neighbors. Actually, as this turns out, the lower pitch neighbor will have fewer frequencies in it's interval than the higher neighbor's interval. The cents however, will always be 100. But, just like the semitone's varying number of absolute whole number frequencies increase with increasingly higher pitch, so do all the individual cents. The upside and value of using cents over absolute frequency, especially when discussing musical intervals and tuning is that a given number of cents offset (take 5 cents as in Hexspa's example) will always be proportional with respect to any other note across the musical scale.

Yes, it is wrong. It is close for adjacent notes but it deviates as they get farther apart so no, it's not the same. Cent intervals are precisely expressed as the 1/1200 root of 2; just as equally tempered semitone intervals are expressed as the 1/12 root of 2. Cents (and semitone and octave interval...and all other musical intervals ) are non-linear intervals.

Please note that I refer to cents as intervals. They, cents, can also be sub-divided into smaller intervals for finer accuracy. 1/10 cent is not an unusual division and is required for fine piano tuning accuracy etc.

I'm no math wiz either but I use this stuff every day (it's built into my tools so I don't actually have to run the calculations). Actually, my tuning app can display resolution down to .01 cent (a hundredth of a cent) but for practical every day use, resolution down to a tenth of a cent (0.1 cent) is as much as I need. But if I only had resolution to 1 cent, that would not be enough.

BTW to calculate to .01 cent or .1 cent, just add zeros! Examples: 1/120,000 root of 2 for .01 cent or 1/12,000 root of 2 for .1 cent! That's it.

Oh, and about the statement that 5 cents is the least noticeable difference is not true when you have a reference tone you are matching pitch to. When you tune two (or three) piano strings to the same note (a unison), you CAN hear beats down to a tenth of a cent (difference) but two tenths is pretty good and often acceptable. On the best pianos you can nail it to ZERO! (on a resolution of 1/10 cent) but that's really hard to do... except for the "Big Boys". I'm sure the same is true with guitars, violins etc. but I never measured them that finely.

But as far as when two or more instruments playing together, a 5 cent difference between instruments may be unnoticeable. I've done subjective listening experiments and I have found that it can be acceptable all the way up to a 20 cent difference! But at that point it just sounds really bad... almost all of a sudden!? You can try this for yourself with a friend. See what your tolerance for "out of tuneness" is.

Here are some sites I found:

hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html

members.efn.org/~qehn/global/building/cents.htm

Cheers,
Rock

You're kidding right? I just googled and found those links, that's what's incredible Ha Ha.

You're right about ET, it is out of tune with just intonation but the big problem with just intonation is that the pure consonance of perfect ratio intervals in in the base key (which is usually C) preclude other keys from having truly perfect intervals to the extent that some intervals are unplayable (wolf tones). ET solves this problem by distributing the error evenly across all intervals in all keys.

True dat. But I've heard some pretty sweet guitars too...but never in my hands

From the wiki article: "While intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords."[8] The above corresponds to what I meant with "Oh, and about the statement that 5 cents is the least noticeable difference is not true when you have a reference tone you are matching pitch to." I have to admit when I started professional piano tuning about 20 years ago (after amateur tuning for about 18 years) , I met another tuner who demonstrated his ability to hear 1 cent difference between two strings played separately. At the time, didn't believe it was possible but he did it! These days even I can do too! (but I don't rely on it, I use electronics). But I do agree that music in context is much harder or impossible to hear such minutia. The "How to play with Intervals" guy IS right. He just calls semitones "half-steps" same thing.

Yeah, for EQ ing, I would imagine if you could get within a few notes you would be doing great but if you can get closer, so much the better.

Yeah, the wave shape/harmonic content makes a difference. What you should be listening for is the beating or "tremolo like" sound. At A4 (440Hz), if you compare to a note at 441Hz, the difference is 1Hz so you should hear the beat at 1 beat per second. BTW, that difference is about 4 cents. As you bring the two pitches closer together, the beat frequency gets slower until when they are exactly the same freq. the beating stops exactly! But the thing is, how slow can you start to hear the beat? When we're piano tuning, that's what we're listening for when tuning unisons. You do the same thing when tuning any other instrument to another instrument or tone source (like tuning fork or another string).

Do you mean when I said "But if I only had resolution to 1 cent, that would not be enough."? What I meant there is that at 440Hz a 1 cent difference would be about 440.25Hz. That would be a beat rate of about .25 beats per sec. You can definitely hear that! So you have to get closer. So if we get down to 1 tenth of a cent, at 440 the difference is 440.025Hz. That's .025Hz so that's a "tremolo" that is really slow and you can't really hear it "move" so you're pretty much as good as you're gonna get.

You asked: "Does that mean, say, the interval between cents 1 and 2 is smaller than the interval between cents 55 and 56?" Yes, but only in relation to the number of absolute frequencies that the intervals span. The fact that cents are musically proportional means that even though there are more frequencies "in" the higher one, they will sound musically the same. (if you could hear it).

I hope this all makes sense.

Cheers,
Rock

You're welcome Hex,

You wrote: "I understand about ET being limited to one key. Jacob Collier talks about this a lot." I think you might have that flipped around. ET, because the error is distributed equally among all note intervals, treats all keys equally whereas Just Intonation is the one that has limitations. I'll mention here that dozens of different temperaments have been devised through the past centuries because of the inflexibility of fixed note instruments. The study of temperaments is an entire subject in itself. If you have a link of Jacob Collier, please shoot it out, I'd like to read.

Thanks for the kudos but as tuners go, I'm kind of in the middle of the pack; I know some guys & gals who have been tuning for over 40 years! One of my colleagues once said that you need to do this for 10 years to really be good at it; I think he was right! Art, science, craft? Some say that craft is the combination of art and science so that best describes it. For tuning, there are "rules" and specifications for a properly tuned piano, but because of variables and limits of measurements (aurally or electronically) there is what I will call "artistic license". So you can tune the same piano using the same temperament rules but with different minute variations to achieve different, but still "correct" and pleasing results.

I can visualize a graphic that will help explain the math of the entire musical note system. I'll see if I can find one. But in the meantime, it wound look like an exponential curve showing octaves, semitones and cents on the X axis and frequency on the Y axis. You would see that every division going up has more frequencies than the one before.

Cheers,
Rock