Piano Tuning
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Piano Tuning
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Re: Piano Tuning
#2yeah, great, but what about adjusting for inharmonicity in the overtones? (partially /s)
a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
Re: Piano Tuning
#3yeah, great, but what about adjusting for inharmonicity in the overtones? (partially /s) a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
I've been down that rabbit hole. Math and music aren't really in such harmony as advertised. Yeah, equal tuning gives your instrument ability to play in any key, but it will always be slightly off.
For more enjoyable tuning, your frequency ratios should actually be fractions of small integers. For example, note E to note C ratio should be 5/4. This is called "just intonation", you can hear some examples on youtube when compared to equal temperament described in article. It sounds much better to trained ear, but doesn't work for changing keys.
It would be nice for your digital instrument to be aware of key you are in (much harder than it sounds) and to re-tune all notes into just intonation. This would give you best of both tunings.
Re: Piano Tuning
#4Anyone who enjoyed this article might also like “Music for Geeks and Nerds”[1]. It’s a free eBook describing musical structure using Python code.
Re: Piano Tuning
#5yeah, great, but what about adjusting for inharmonicity in the overtones? (partially /s) a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
I've been down that rabbit hole. Math and music aren't really in such harmony as advertised. Yeah, equal tuning gives your instrument ability to play in any key, but it will always be slightly off. For more enjoyable tuning, your frequency ratios should actually be fractions of small integers. For example, note E to note C ratio should be 5/4. This is called "just intonation", you can hear some examples on youtube wh…
Temperament is unrelated/orthogonal to inharmonicity. Temperament deals with picking frequencies for the notes that make up the scale within an octave.
Inharmonicity affects how you tune the octaves themselves. The reason a simple doubling rule doesn't work is a result of the properties of real physical strings which are not perfectly elastic causing the harmonics of a single string vibrating to be slightly sharp. To avoid "beating", the ratio of frequencies between two piano notes an octave apart needs to be slightly greater than 2.
This article is half of what a "what every musician should know about piano tuning" article should contain - it's the "set-up" part where you derive a nice simple rule that is widely known. The second part would then deal with temperament (first) and then inharmonicity.
Re: Piano Tuning
#6yeah, great, but what about adjusting for inharmonicity in the overtones? (partially /s) a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
I've been down that rabbit hole. Math and music aren't really in such harmony as advertised. Yeah, equal tuning gives your instrument ability to play in any key, but it will always be slightly off. For more enjoyable tuning, your frequency ratios should actually be fractions of small integers. For example, note E to note C ratio should be 5/4. This is called "just intonation", you can hear some examples on youtube wh…
It's easy to set up just intonation digitally.
The problem is that most classical music modulates to other keys. So why not just set up some switches or programmed changes?
Because as you modulate there's a grey area in which you're not fully in one key or the other. If you interpolate the intervals as you go through this area, it sounds wrong. If you switch to a new tuning when you land in the new key, that sounds wrong too.
Equal temperament solves the problem by being a good-enough compromise. All the intervals are slightly off, but they're off by a consistent amount, so - paradoxically - key changes become smoother.
Re: Piano Tuning
#7This is certainly a fun way to introduce twelve-tone equal temperament to someone who knows enough math to understand logarithms but is new to music theory. It works well, because the only elementary music theory claims that we need to accept without explanation are that there are twelve different note names that repeat in the same order and that each note of a given name is twice the frequency of the previous note of the same name.
Okay, not quite. You actually also need to know that the frequency ratios of all pairs of keys n keys apart should be the same, for all n. This article kind of sneaks that one in silently. But once you’ve accepted that, it leaves you with only one option: tuning the keys to frequencies spaced evenly along the logarithmic axis.
Of course, there are so many more details on every side of this topic. Why 12 divisions per octave (this certainly isn’t the case for all musical traditions)? Why do we want all pairs of notes n steps apart to have the same ratio (this certainly isn’t the case for some instruments)?
And, most importantly to me, since it’s the question I can find the least solid information on: why do we take for granted that one note with twice the frequency of another note sounds so similar that we call both notes by the same name? I have a pretty good understanding of other details of music theory, but I’ve never gotten a straight answer on why (and indeed to what extent) octave equivalence exists.
I’ve always speculated that it has something to do with our auditory systems evolving to interpret sounds from roughly simple harmonic oscillators, because those exist in important roles in nature (like the vibrating chords and air columns of human and animal vocalization).
But that’s still not a great explanation to me. Yeah, there’s some important noise-makers in nature that have roughly the overtone series. But why would it be important to hear the first overtone as equivalent (in some strong but not absolute sense) to the fundamental? Could it have something to with hearing vocalizations from a distance such that the fundamental would be more attenuated than overtones? That’s really grasping at straws.
Re: Piano Tuning
#8yeah, great, but what about adjusting for inharmonicity in the overtones? (partially /s) a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
When I saw the article title that's what I expected it to be about.
Then I saw your comment and thought "oh, it's going to be about the subtleties of different temperaments and how you get to choose which intervals are how far off from sounding right".
Turns out it's actually telling you that there are 12 semitones per octave, all representing the same ratio of frequencies. Ah well.
Re: Piano Tuning
#9Since the frequency doubles every 12 half steps, that means that the frequency of any note is the twelfth root of two times the frequency of the one before it. That's really all you need to know.
(of course this doesn't account for octave stretching that is typically done on acoustic pianos)
Re: Piano Tuning
#10This is certainly a fun way to introduce twelve-tone equal temperament to someone who knows enough math to understand logarithms but is new to music theory. It works well, because the only elementary music theory claims that we need to accept without explanation are that there are twelve different note names that repeat in the same order and that each note of a given name is twice the frequency of the previous note o…
> And, most importantly to me, since it’s the question I can find the least solid information on: why do we take for granted that one note with twice the frequency of another note sounds so similar that we call both notes by the same name? I have a pretty good understanding of other details of music theory, but I’ve never gotten a straight answer on why (and indeed to what extent) octave equivalence exists.
I don't know what the true answer is, but I'd consider the overtones. Vibrating strings produce overtones whose frequencies are multiples of the fundamental. So the first overtone is twice the frequency; an octave higher. When you play that note an octave higher, you're again producing overtones that perfectly line up with the overtones that were produced by the lower octave (if we just ignore the effect of string stiffness). So it's like the same note, only missing the fundamental frequency and odd multiples of it.
This is also why the feedback from a heavily overdriven guitar amp (which emphasizes overtones) can morph the sound to the same note at a higher octave. It's simply a matter of letting the fundamental frequency grow quiet in comparison to the powerful overtones which are further amplified by feedback. I don't know if that can ever produce a different note; I don't think it could. That would require actual change in frequencies.
Different notes, by comparison, produce overtones that only sometimes (or never) line up with the overtones of another note. I'm not sure but I think that also explains why some notes together sound dissonant while other combinations make good chords. If you're trying to transcribe a piece of music by analyzing its spectrogram (which I do because my ear isn't very good), you'll find that the notes that are usually hard to tell apart are the ones that harmonize well and have overtones that line up.