Every time we had our piano tuner scheduled to come was a special occasion. When he'd come, he'd have this aura about him which made his intense disposition seem fitting for the work he was doing.
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.
Two days? Our tuner does the job in about an hour every six months on a modern upright. I'll allow differnces between tuners and pianos but two days seems a lot?
Our old piano tuner explained to me how in practice when tuning you must stretch the high notes higher and the low notes lower so as to make the piano sound right. He didn't explain why. It was obvious if you played octaves and listened to the notes, and he was an excellent piano tuner and the piano did sound good.
The first comment in the article describes it nicely and links to the excellent Wikipedia section which everybody here ought to read:
Indeed, if one tuned a piano like the article describes it wouldn't sound good.
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.
There are also guitars which take a tempered approach to tuning, and typically require a multi-scale fitment of bridge and nut. The Buzz Feiten system [1] is the first I can recall, although now many guitar manufacturers are building guitars with each string using a different scale and fanned fretboards [2].
This 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.
> 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.
> 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.
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.
> 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)?
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.
Once, long ago I landed in a house in Poznan, Poland which had the wreck of a Bosendorfer baby grand piano sitting in one of the rooms. I had a lot of time on my hand and tried to figure out the story behind the piano. After tracing the owner it turned out that he was trying to 'restore' the piano but in the process had butchered it and had sent parts all over Poland for refurbishing. For some strange reason the Action was in Gdansk but the hammers were in Warsaw, and the dampers were in yet another location.
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.
> 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.
Like tensioning a bicycle wheel! Easy and rewarding process, highly recommended.
I had a friend, Jim, who was blind. He went to school to become a piano tuner. Once, we were at another friend's house who had a piano. The friend played a short piece on the piano and Jim mentioned that it could use some tuning and went on to claim he had perfect pitch hearing. Well, of course, we had to test him.
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.
great story- I spent some time watching/discussing piano tuning with the guy that does my yamaha C2. You basically bring it up in 3-5 passes depending on how out of tune it starts in. He started with a tuning fork and then continued with an app/microphone to dial it in. Another cool trick to dial in the harmonics is to play a (13th? my music theory has escaped me for a moment) but basically play a C and then an octave higher E and keep that spacing throughout the piano and you should hear the notes beating in a pleasant manner. If they dont, you need a tune.
Also important to keep in mind that tuning a piano early and often is the best thing for it. Having to make drastic changes to bring it up to pitch is very likely to crack the sound board and or bend/break the harp. If you live in an area with high humidity keep the piano at least 3-5 feet(1-2 meters) from the outside wall of your home.
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.
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.
For those of you that want to go deep on this subject (of which this article only superficially covers), I commend Stuart Isacoff's entertaining book _Temperament_ (Vintage 2009) to your attention.
A wonkier, somewhat more no-nonsense treatment is J. Murray Barbour's _Tuning and Temperament_ (Dover 2004).
Since 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)
I think that knowing frequencies and hearing frequencies still leaves one a long way from being able to tweak a string to match that frequency.
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.
It does. When I tune (warning - I am novice) I tune middle octaves, then go up and down the scale with octave intervals. And then check the whole instruments tuning notes that sound off. String itself stretches and loses the pitch a little sometimes, especially if it was out of tune by quarter tone. But more importantly string peg that holds the string is just screwed into the hardwood board. So when you tune it, you still have some room for it to settle. Even slight press with tuning hammer on the peg can result in big difference in tune. Especially on high notes.
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?
It's 48 because Bach did it all again in book II of the Well-Tempered Clavier.
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.
[+] [-] mitchtbaum|7 years ago|reply
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.
[+] [-] js2|7 years ago|reply
[+] [-] richardhod|7 years ago|reply
https://en.m.wikipedia.org/wiki/Piano_acoustics#The_Railsbac...
[+] [-] naringas|7 years ago|reply
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.
[+] [-] tim333|7 years ago|reply
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.
[+] [-] auiya|7 years ago|reply
[1] https://en.m.wikipedia.org/wiki/Buzz_Feiten#Buzz_Feiten_tuni...
[2] https://images.guitarguitar.co.uk/large/130/170316308819006f...
[+] [-] baddox|7 years ago|reply
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.
[+] [-] clarry|7 years ago|reply
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.
[+] [-] ramshorns|7 years ago|reply
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.
[+] [-] boomlinde|7 years ago|reply
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.
[+] [-] slashdotdash|7 years ago|reply
[1] https://pedrokroger.net/mfgan/
[+] [-] jacquesm|7 years ago|reply
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.
[+] [-] xavieralexandre|7 years ago|reply
Like tensioning a bicycle wheel! Easy and rewarding process, highly recommended.
[+] [-] sureaboutthis|7 years ago|reply
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.
[+] [-] S_A_P|7 years ago|reply
[+] [-] unknown|7 years ago|reply
[deleted]
[+] [-] pjc50|7 years ago|reply
[+] [-] exabrial|7 years ago|reply
Minute physics does a great video on it: https://youtu.be/1Hqm0dYKUx4
[+] [-] kazinator|7 years ago|reply
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.
[+] [-] lumens|7 years ago|reply
a good explanation: https://www.youtube.com/watch?v=b_fU6yVxDZs
[+] [-] veli_joza|7 years ago|reply
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.
[+] [-] gjm11|7 years ago|reply
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.
[+] [-] phlakaton|7 years ago|reply
A wonkier, somewhat more no-nonsense treatment is J. Murray Barbour's _Tuning and Temperament_ (Dover 2004).
[+] [-] frankhorrigan|7 years ago|reply
[+] [-] robbrown451|7 years ago|reply
(of course this doesn't account for octave stretching that is typically done on acoustic pianos)
[+] [-] mnemotronic|7 years ago|reply
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.
[+] [-] yarosv|7 years ago|reply
[+] [-] jancsika|7 years ago|reply
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?
[+] [-] stevehiehn|7 years ago|reply
https://en.m.wikipedia.org/wiki/The_Well-Tempered_Clavier
[+] [-] jancsika|7 years ago|reply
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.
[+] [-] ramanan|7 years ago|reply
"Why It's Impossible to Tune a Piano" https://www.youtube.com/watch?v=1Hqm0dYKUx4
[+] [-] kzrdude|7 years ago|reply
[+] [-] GuillaumeBrdet|7 years ago|reply
[+] [-] kazinator|7 years ago|reply
[+] [-] unknown|7 years ago|reply
[deleted]