My mom felt it important to prepare us for the two day process to be without our piano and to tread lightly as he worked through his vast toolset of tuning forks on each key with his most amazing one, his ear.
To watch him act as one with our piano, positioning his head to absorb its vibrations and set it right for us, made our most prized possession seem like part of an intricate world of students connected and tuned to the sound motions of the Universe.
As far as I undesrtand it has to do with the thickness (i.e. the real world physicallity) of the lower octave strings.
The strings are not trully one-dimensional. this makes their timbre deviate from the ideal harmonic series. This is specially true for the thicker strings of the lower notes.
If we could make a piano using very thin strings we could get away with a less stretched tuning.
As an aside I wonder if you could make a digital piano sound better than a real one by correcting for that? And also perhaps auto tuning to the key the music is in, in the manner that a singer will do automatically. So far the digital pianos I've heard don't sound as good as a decent concert grand.
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.
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.
It would be an interesting (and perhaps cruel) experiment to expose babies to synthesized sounds that have a harmonic structure with different mathematics. For example, sounds with lots of sqrt(2) frequency ratios.
In jazz it's often left out to open up chords more (i.e. not muddy the frequency distribution with too many adjacent notes unless you explicitly want the powerful sound). In pop the nondescript nature is used to create a strong sense of grounding in the scale; the root note of the scale and its fifth fit over every possible diatonic root note (play c-g in your right hand with root notes c,f,a,g in the left one after the other, it will sound familiar). In rock they literally call the perfect fifth a 'power chord' and they play melodies with it. It fattens up the melody note without adding any harmonic identity that would create tension with the rest of the harmonic content of the song.
Come to think of it, a lot of western harmony is based on a two-octave "gamut". Like for instance alterations to chords such as dominants take place close to this "3f octave" (dodecade?) 12th interval, like the like 11th, 13th.
The usual explanation is that if these alterations are in the higher octave, it prevents certain dissonances. But from the "3f octave" view, we can just regard them as different notes; that the 13th is not simply the 6th, only one octave higher, but rather an "augmented dodecave" interval.
Also overtones don't have to be even multiples of the fundamental. Many organs have stops that are not multiples of the fundamental. 2-2⁄3′, 1-3⁄5' are common flute stops.
Sure we can. Does the explanation given above require that we shouldn't? Of course, laying sines an octave apart is a good first step to building a simple additive synth that doesn't sound as dull as a pure sine. In that case, are you adding a note or an overtone? If anything, to me this only makes the given explanation more credible. The relationship between common naturally occurring harmonic overtones and the octave.
> Also overtones don't have to be even multiples of the fundamental. Many organs have stops that are not multiples of the fundamental. 2-2⁄3′, 1-3⁄5' are common flute stops.
Sure, there are dissonant sounding instruments as well as instruments that can sound more like a chord when a single note is played. Of course, we can synthesize anything. On the other hand, many instruments are prone to producing overtones that increasingly deviate from the harmonic series. And many tricks are employed to try and reduce that effect, because it doesn't sound good.
We could have made a different second assumption instead, and remembered that not only is the note an octave higher twice the frequency, but also, the note a perfect fifth higher is one-and-a-half times the frequency. Propagating these axioms to tune all the notes would lead to Pythagorean tuning, which has some nicer-sounding intervals but makes playing in certain keys sound bad, since you can't make all n-key intervals exactly the same size.
Earlier systems had you tune instruments so that notes within an octave would be tuned to a few different simple harmonic relationships to a root key. This has the downside that playing a tune in a different key yields entirely different frequency relationships to the root. Twelve tone equal temperament is a sort of compromise in that it produces pitches that only roughly correspond to these simple relationships, which it trades for consistent relationships in any key. A few of these simple relationships are common across multiple musical traditions. In the Arab tone system there are musically significant intervals that don't have any nears in 12-TET. At some point it was modernized to an equally tempered scale, but with 24 tones per octave instead of 12.
If you listen to a string ensemble or a choir, you can however often hear that they tend towards the simple harmonic relationships rather than their corresponding equal-tempered frequencies.
> 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.
Maybe thinking of sounds in terms of sets of overtones is useful. If we define sound similarity (of largely harmonic and overtone-rich sounds) as the size of the intersection of their sets of harmonic overtones, the octave is the most similar you'll get to the root because half of the harmonic overtones of the root also appear in the octave. The next best interval in these terms is the perfect 12th, which has a third of all the overtones in the octave.
Of course this is oversimplifying and the overtones should probably be weighed according to timbre and the limits of hearing, but I think it gets the general idea across.
These integer harmonic intervals (octave and perfect 12th) are also unique in that played together with the root, they don't produce any undertones.
The relationships within the octave are more complex and don't have integer harmonic relationships to the root. They occur early in the harmonic series, but the root does not coincide with the base frequency of the series, so they produce undertones. For example, a perfect fifth (3:2) played together with the root will produce an undertone an octave below the root. If you sum two sine waves with this frequency relationship together, you'll see that the resulting waveform cycles at half the frequency of the root. For very small ratios like that of the minor second (16:15 in some tuning systems) the overtone series in which both appear starts at a 1/15 of the frequency of the root note, so the length of this cycle is much longer, thus appearing dissonant. If the interval is really small, though, or if you play a very low root note, this undertone ends up at an inaudible frequency and will be perceived rather as a slow timbral modulation than a dissonance.
Of course, the perfect fifth and the perfect 12th don't appear at all in 12-TET.
I bought the piano and bit by bit collected all the other pieces from wherever they had ended up. What really didn't help is that they had ripped out all of the strings except for the basses, and in the process had scratched the soundboard quite badly.
Over the course of a year I rebuilt it bit by bit with a lot of knowledge gleaned from a local piano tuner, Marek Koczy (he died some years ago). Finally at the end of all that I had a really nice piano, except for one little detail: it had to be tuned up from scratch.
Tuning up a piano from zero is a lot harder than it seems. As the tension on the cast iron frame increases it deforms a bit, enough to de-tune everything you've done up to that point. So the only way to get this done is to tune the whole thing up gradually and to pace yourself so you don't end up overstretching part of the frame or end up in a never ending cycle of de-tunings.
It took me a month, I could probably do it much faster a second time. Never realized that your ears could be tired either, after a couple of hours of tuning I was unable to hear the subtle beats that tell you that you are getting close, very low frequency and soft you need to really pay attention.
Keeping the rest of the piano quiet (especially when it is still a mess) with sympathetic resonances each of which will have their own harmonics is also quite a trick, in the end I used strips of felt woven through the strings that were not 'in scope'.
All in all a fantastic experience and I would be happy to just work on restoring musical instruments. The piano when it was done got donated to the Conservatory of Poznan where it still is in use today, with my lack of skills in play I did not feel that keeping it was right, an instrument like that should be used as much as possible.
edit: HAH! I am so happy. So, the message that Marek had died reached me in a pretty roundabout way through friends from long ago, I did not verify in any way that it was true. So, after writing this little bit above I decided to type his name into google to see if there was an obituary, but instead I found he's alive and well!! https://spsf.pl/pl/marek-koczy/
I will definitely send him a message and maybe go visit, he's one of the nicest people I ever met and had endless patience teaching me all the tricks of the trade.
Like tensioning a bicycle wheel! Easy and rewarding process, highly recommended.
It's a fun job and this too will take you some time the first time around.
We hit one key. "That's D!", he said.
We hit another. "That's G!"
With a smirk, my friend pressed three keys at once.
"Uh, Uh, B, D and F#!"
We were amazed.
I'm really happy I picked computers over music because I think that it was more in line with my skills and it is super hard to make a living in music if you are not absolutely top grade, and even then it is hard. But I've always enjoyed music very much.
Super impressive what your blind friend did there, especially to be able to name them like that, I could pick out the chord after a number of tries but to be able to decompose a chord like that without being able to try whether you have it right or not is quite a skill. I'd love to be able to do that but I am not sure how you would train that particular ability.
The number of blind piano tuners is probably much larger than what you'd expect from statistics, I know of two myself, one in Toronto at Lowrey's and another in Amsterdam who used to work for Cristofiori. That's one profession where lack of sight is not an obstacle, and where a better developed sense of hearing is an asset.
I sing with someone that has perfect pitch (not just good relative pitch like I have). He can immediately identify a note at random, but has a very hard time when we want to transpose a piece—he's never really had to learn how to figure out intervals.
Minute physics does a great video on it:
It really does mean equal geometric steps between successive tones.
Each key is equally "out of tune" under equal temperament; e.g. C# minor and D minor are basically the same, modulo pitch.
a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
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.
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.
I've also never heard about a systematic strategy string players use to a) analyze a tonal piece for pivot points in a modulation, b) temper all the notes going forward from that pivot to fit whatever tuning system they are using.
I've seen various approaches for microtones and idiosyncratic tunings for single pieces of modern music, but those are fairly static things that the players practice and perform. It's not a system they apply dynamically to the standard rep.
Also keep in mind that string players have an added constraint that the keyboard does not-- if you give a cellist a sudden, large leap it can be difficult for them to even find the note at all.
A professional cellist? No. They'll find the note. And that goes for all pro (or just good) string players, fretless instrument or fretted instrument.
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.
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.
A wonkier, somewhat more no-nonsense treatment is J. Murray Barbour's _Tuning and Temperament_ (Dover 2004).
(of course this doesn't account for octave stretching that is typically done on acoustic pianos)
Question for piano tuners: Does tuning the piano require multiple iterations? When re-tuning my 12-string to or from an open (where some or all of the strings are tuned to a specific chord) I sometimes have to go back and tweak each string a couple times. I think this is due to the change in stress on the top and neck of the guitar.
The G# minor fugue in Book II of Bach's WTC has a rather long sequence around the circle fifths. It starts on E# minor.
Did the well-tempered tuning system open up the possibility for Bach to start writing longer chromatic sequences like that? Would that sequence have sounded out of tune in the meantone tuning system?
And really, it's 96 because there is both a prelude and a fugue for each key.
And if you think about it, it's 48 + (48 * x) where x is the average number of voices in the fugue.
And then at least a few of those are double and triple fugues, so I guess a forEach statement in there to multiple by 2 or 3 for those cases.
So if you were an organist who was opposed to this tuning method, you'd have a pretty difficult time making a persuasive counterargument against all that.
"Why It's Impossible to Tune a Piano"