Hey, everyone. I've been working on this for the past few days. Actually, it's been years but it all kind of hit me a few days ago. I came up with a way to find every note frequency value from C0 to B8 - and beyond - with at least a 99.20% accuracy. If you're half-way decent at math then it's better than 99.72% accurate. The best part about it is that you just need to remember 15 numbers and be able to add, subtract and multiply & divide by 2.
Here's the chart:
It's based on whole numbers and simple offsets. Everyone knows that A4=440Hz so finding all the values for A is already easy. However, there are other whole numbers in the chart as well as some that are extremely close. By rounding F4 up by 0.77Hz to 350Hz, which is incidentally 99.78% accurate, I found a memorable base from which I could build an entire system which I call the 'F350 Method'. I tried to make it 'Easy AF' - AF meaning 'and fun' in this case - and I think it is.
By using this, anyone can find at least seven octaves with the stated accuracy and learn it within a week if not a day. You can read more about the details, including the actual differences and accuracy percentages for each value in this blog post. Please feel free to tear this apart and/or share it with whoever and wherever you think it might be helpful.
Quick update: I realize I made some mistakes in the video. Specifically, I use the terms 'Hertz' and 'cents' interchangeably although they're different. Also, the correct value, within the confines of this system, for F1 is 43.75Hz and not 'fourty-three and a quarter'. I'll be making updates to this concept as time goes on.
To clarify the difference between Hz and cents, let's look at the difference in the first and seventh octaves. While the frequency difference between G1 and G#1 is 1.91Hz, each cent is equal to 0.091Hz - a very small absolute amount. However, should you transpose this difference to the seventh octave where the G7 and G#7 are 372.95Hz apart then the single-cent value is 3.726Hz.
In the video I say that 'wikipedia says we can only hear to about a 5 cent difference' while showing the Hz difference chart. Again, I'll be making updates. For now, keep these things in mind should this F350 Method be of any use to you.
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.
Thanks for the reply. I was going to make a video on this but I'll have to defer further questions to you. For now, a cent is the 1/1200 root of 2 and I'll just have to take that at face value since it exceeds my mathematical expertise.
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.
Thanks for the resources. I didn't realize you were a developer - that's incredible.
I guess where I get curious is about the equal temperament system. It itself is out of tune by up to 17.49 cents from just intonation - which I understand to be closest to the harmonic series. That's not even mentioning something like the guitar which is rarely, if ever, perfectly in tune with that. I haven't tested my pitch perception accuracy but I think I should.
Since we're talking about guitar, in my experience, it's one of the least in-tune instruments of all time - excluding my own voice, of course.
Here's the wikipedia article which talks about the threshold of human pitch perception and how even music students are only about 12 cents accurate. It says that timbre, among other factors, play into one's ability to distinguish pitch. Here's another, under 'Comparison with Just Intonation', which displays how far equal temperament is from that system. This article, under 2. How to Play with Intervals, must be wrong since he defines a cent as 1/100th of a half-step.
The main reason I wanted to learn audio frequency note values was to use EQ more confidently. Being only 10 cents out of tune is sufficient for me, in that regard - though I hope to improve it. This whole notion of intonation is just bizarre and almost a dark art.
I should mention that the first article you linked to talks about the Just Noticeable Difference, JND, and they consider it to be 5 cents. The last thing I'll say right now is that I'm familiar with detuning synth oscillators by single cent intervals but I can't remember at what point it sounds out of tune. Maybe it depends on the interval, waveform, octave and whatever else.
Sorry, one more thing. I'm rereading your response about 1/100th of a semitone not being accurate since cents are non-linear. Does that mean, say, the interval between cents 1 and 2 is smaller than the interval between cents 55 and 56?
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." 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 understand about ET being limited to one key. Jacob Collier talks about this a lot.
Good point about guitars. Since they tend to be somewhat individual in their intonation - due to setup, string gauge, construction, player, etc - it makes sense that some would sound better than others, all else being equal. It reminds me that I've heard 'piano tuning is as much an art as a science'. What truth is there to this? I can think of no better person to ask than a pro with over two decades of experience. What's the art? Is it similar to how a vocalist prefers a certain intonation - in cents - than another?
Regarding 0.10 cent accuracy, you must be right. With synths, even a 1 cent detune creates a 'swirling' sound which is a very slow 'beat'. Surely this is an undesirable quality of a freshly-tuned piano! You make a good point about it being perhaps less audible within the context of music, though. It's kind of like dither in Ethan's arguments about the top 25dB or so being the most important parts of a signal.
Ok, I think I understand this math-cent relationship with regard to absolute frequencies. This is why you're saying 'the 1/1200th root of 2' whereas 1/100th of a semitone is musically equally valid. I get that this is the way to arrive at the proper frequency in Hertz but, probably due to my limited math, I don't quite get why dividing the difference between two semitone frequencies by 100 wouldn't give you the exact same result.
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.
Yep, you're right. Not sure what I was thinking. Here's the Collier video at the point where he mentions his use of just tuning. I think I can visualize that graph since I understand that, despite there being an infinite number of frequencies, higher up the notes are further apart in Hertz. I think the main thing is that the 12th root of 2 is the exact value and it's an irrational number whereas dividing by 100 is, by definition, attempting to make it into one. Sonically, they are equivalent but only if you don't write it down. I mean, it's still sonically equivalent even if you write it down but only if you don't try to play that result and it's less than the JND.
Thanks for the links. Yeah, Collier is talking about way more than piano tuning and temperaments. I'll admit there's a lot more to tuning, harmony and theory than piano tuning, it's just that I'm usually thinking and dealing more with piano stuff.
That Horowitz thread is pretty advanced especially since he had his piano's touch set up so extremely light. I had the pleasure of meeting Franz Mohr at a PTG meeting during his "book tour" some years ago. He told some interesting stories about the artists and their pianos.
Piano action "regulation" (the name we use for set-up and adjustment) is a subject completely independent of tuning but is something one needs to learn to be a complete piano technician. All pianos will tend to have a unique touch or feel. With a typical action regulation, one can bring a piano back to factory specs and/or possibly make subtle changes or improvements to the touch. One can also however tear the whole thing apart and change key weights, hammer weights, lever fulcrum positions etc. This kind of work can be very invasive and can cost thousands. It's a specialized kind of work usually not done by your typical tech (like me), although it is quite interesting and I do enjoy attending classes on the subject at conventions when they are offered.
When selecting a piano for yourself, I think it's best to find a piano that suits your preference rather than buy something that's not quite right and then modify it to your specs later. If you do find a piano you think needs tweaking, I suggest asking the dealer to set it up before you buy it so you can check it out and make sure it's what you like. The seller may tell you, "Oh, don't worry, you can easily have your tuner fix or adjust (X, Y, or Z)" If it's that easy, the dealer should have HIS tech take care of whatever.
BTW, if you don't already have a piano, a weighted action digital is a good place to start. If you recall the thread about piano sampling editing, I suggested modeling SW. I'm not much into editing samples but the Modart Pianoteq sounds pretty good to me. The most advanced version allows you to tune the piano and adjust about every parameter you can think of...and many that you can't think of or even knew existed The basic (Stage) version is not nearly as adjustable but after running the FREE demo, I decided that it had enough control for fun playing. (Disclaimer: I did not buy it yet but it's only about $150 or so I'll get around to it one of these days.)
Right on. Great advice about buying a piano. Actually, I'm just using a synth-action controller - not looking to become a legit pianist though a fully-weighted option wouldn't hurt. I guess by 'learning piano' I meant using actual piano samples to play the one Bach piece I know lol.
Funny you should mention Pianoteq - I just listened to a bunch of demos and, at least in those examples, I really preferred the Arturia V and the Waves Grand. There's no doubt that modelling offers more flexibility, though.
I'll just have to live with my Komplete libraries for now.