I often huff and puff at articles like TFA, but here I found myself nodding: the method is sound and makes sense.
Some comments I must leave though:
Clustering is not the point of black and white keys. Rather it is the facility to pick an anatomically reachable, desired subset of 12 keys available per octave. As a simplified European tradition baseline, that is the white keys transposed by some number of semitones. The salient thing here is to have a row of keys which are mostly two semitones apart but have a one semitone gap at strategic locations to produce the scale.
Much of the music in the world operates roughly on the pentatonic scale which coincides with the black keys, or the complement of "European" scales in a 12 step equally tempered octave. Pentatonic scales are mostly two semitone steps with strategic 3 semitone steps.
Finally, the harmonic model in TFA does not resemble the piano very much. Would be interesting to see how different harmonic models and temperaments in various historical keyboard instruments interact with the computation. The modern piano is equally tempered. In a harpsichord, that would generate a lot of roughness for the thirds which are way out of tune. The modern piano mitigates this by having the hammers strike strings at a position that avoids exciting the 5th harmonic (which produces a justly intoned third on top of fundamental frequency).
Would be interesting to see what kind of difference to the roughness calculation it would make to omit the 5th harmonic!
Hi, author of TFA here. Thanks for your comments :)
You're right the harmonic model used is a bit of a bodge. I didn't know that about hammer position, very interesting. Let me know if you ever get around to trying different timbres in the code.
I think we are more in agreement than you think on clustering, though. Although I neglected to discuss what's anatomically reachable, and also the nuances of history, my point is that the desired subset you mention can be defined by using clustering to pick a subset that sounds good.
I think perhaps the way I would state the subset problem is that white keys are in a way the "subset that sounds good" to the culture where the piano keyboard arose. More in the vein of discussion than suggesting you'd need to change anything :)
The black keys are means of transposing that subset.
It is a very interesting but in a way unrelated insight that the black keys also form a consonant group.
And then music and musicians naturally coevolve with instruments and do whatever they please!
In a recent article about his Sixtyforgan (an organ made from a Commodore 64 and a spring reverberation tank), Linus Akesson explained how he had used the key layout of a chromatic button accordion.
This appears to differ from the Janko layout, though they both apparently share the feature that "if you know the shape of a particular chord or scale, you can automatically play the same thing in another key just by moving your hand" (so long as you have five rows of buttons, in the accordion layout.)
There are many ways to skin this cat apparently, though as with QWERTY, established convention is hard to change.
It's a bit different actually, the same keys repeat multiple times 'above' each other, in two alternating rows with half overlap. I've built a little device that you set on top of a regular piano keyboard to experiment with it but that wasn't very satisfactory. Doing a full scale conversion would be quite a job.
Actually a piano accordion, but it's been refreshing learning to play the bass buttons. They increment in 5ths along the main axis, but with an offset row also giving quick access to a major 3rd up or semitone down.
My primary instrument is piano so it's definitely involved rewiring my concept of which notes are 'close' and 'far' to match the circle of fifths. Until now I never noticed how in addition to the usual cadences, many tunes have further rising/falling sequences of 5ths in the bass.
> The modern piano mitigates this by having the hammers strike strings at a position that avoids exciting the 5th harmonic...
I think it's actually the 7th harmonic that pianos avoid, if I remember correctly. (I guess one could verify this by measuring the hammer position on a piano string and figure out if it's hitting the node at 1/5 of the string length or 1/7th).
I have a theory that pianos and guitars have become the dominant musical instruments of the last hundred years or so simply because you can get away with mis-tuning them and they still sound pretty good. (I once had the opportunity to play a 15-tone equal tempered guitar, and it still sounded good, which led me to believe that you can get away with almost anything with a guitar.) Which isn't to say that guitars don't sound better in just intonation, they do.
On the other hand organs and accordions, for instance, sound amazing in just intonation but not nearly so good in 12-tone equal temperament. The notes (especially thirds and sixths) clash with each other too much. It's tolerable, but not great.
I've been working with a group of people converting guitars to 41 tone equal temperament; they have a nicer third (off by about 5 cents instead of about 15) and a closer 4th and 5th (off by about half a cent instead of 2) and can approximate 7-limit just intonation intervals pretty closely. The trick to make it playable is to omit half the frets, so it's fretted for 20.5-tone equal temperament, and any notes not available on one string are available on the string next to it. It sounds like it shouldn't work, but it does.
> "Much of the music in the world operates roughly on the pentatonic scale which coincides with the black keys, [..]"
I think there is not so much music in the pentatonic that is formed by the black keys, or a related scale. Off the top of my head only Paul Desmond's Take Five comes to my mind. I think this mainly because it is horrible to read and write in traditional notation.
Coincidentally I played a bit of Stevie Wonder's music recently which was all in E-flat minor. I have to say it is very pleasant to play, especially if you use Steve's often unusual fingerings.
Reading it OTOH was not so pleasant. This hints to me that our music is not only influenced by the way it sounds and the way it can be played but also by what is convenient to write down and read - and sometimes it takes a blind artist to overcome this limitations.
That's what I meant when I wrote that his fingerings are pleasant to play and I meant that you don't have to be blind to benefit from the fact that they facilitate easy orientation on the keys.
For example Stevie actually plays the main melody of the main riff of Superstition distributed to both of his hands. This allows him to simultaneously play some bass notes with his left and some higher chords with his right hand while completely avoiding to move his hands away from their basic position. His hands never jump. Playing the melody with both hands is unusual and not what most Superstition tutorials show, but it is actually very pleasant and safe because everything just lies under your fingers.
The other side of the same coin is that Stevie never had to worry if his music is easy to write and read. That also facilitates playability and maybe emphasizes pleasant movements over looks on a sheet of paper.
I watched many Youtube videos to figure it out. Thankfully a good soul named Josh FunkKeyStuff Paxton had already written it down and I was glad when I found his detailed analysis. Here is a link to a PDF that another good soul made of Paxton's now lost so called "Superstition dissertation":
While similar in some ways you can't compare a computer keyboard with a piano keyboard. The piano is linear and long and sometimes you cannot avoid to jump with your hands. No matter how good you are, the further the jump the higher the risk to miss. Performance is a lot about risk reduction, so even the best of the best have a glance sometimes.
if you ever watch a professional, especially a session musician, sightread, they don't look at the keyboard, even for big strides. it's all muscle-memory. and blind concert pianists play the whole repertoire, not just stuff in Eb.
I think there is not so much music in the pentatonic that is formed by the black keys, or a related scale.
Depending how strictly you’re using “pentatonic” there, I don’t think you’re correct, unless I’m misunderstanding. In the wider classical and jazz repertoire there’s plenty of music written in Eb, Gb etc, heavy on the black notes.
For example, Debussy’s piano music often uses keys with lots of flats (or lots of sharps). Chopin supposedly played the black keys with his thumbs in some cases, against the accepted style at the time.
In jazz, Billy Strayhorn (who I think was strongly influenced by Debussy) seemed to be very fond of writing in Db.
Interesting... Do you have a source copy for this? I'm always interested to play music as the artist intended and Stevie Wonder is one of my favourites :)
No, not even remotely. But it's not trying to be. Notation is a way to communicate music to a player or conductor who applies their own interpretation to it. Notation is more like a movie script than a novel.
It's. It even well optimized for that. It's a very clunky path-dependent system that has centuries-old cruft just because it's difficult to change culture and it's mostly passable for an orchestra of different instruments to use a shared notation. Instruments with a strong culture of solo music (guitar) often use different notation.
> The modern piano mitigates this by having the hammers strike strings at a position that avoids exciting the 5th harmonic
This is fascinating! I always wondered why a piano could work so well despite in reality having slightly ill-tuned thirds. Thanks for sharing the tidbit.
But isn't everything in equal temperament "ill-tuned"? The fifths aren't real (3/2x frequency) fifths, etc. Everything except octaves - but piano tuners tell me that octaves that are "too big" (>2x) sound better![0]
Yes, but while fifths are off by 2/100, major seconds by 4/100, thirds are by 14/100 or 16/100.
The former create gentle swirling interference, the latter a rough stuttering. If you hit a fifth on a piano and listen carefully, you can hear the slow cycle in the sound.
The stretched octaves are due to high string tension causing some inharmonicity, ie. sharpening higher harmonics.
The inharmonicity is caused by the strings not being 'ideal' mathematical strings: they have stiffness, due to having some thickness - and require some additional force to bend them back and forth, beyond that of the string tension. This bending stiffness is more evident in higher harmonics than in the fundamental (tighter radius of curvature), manifesting in increased 'apparent stiffness' and therefore a higher frequency, at the higher harmonics, compared to the lower harmonics.
Not sure if I've explained it well, but vibrations of an ideal string (no bending stiffness) can be described by a second-order partial differential equation [0], whereas a real string with nonzero bending stiffness is actually more of a 'vibrating beam' problem, which is a fourth-order PDE [1].
EDIT: this is why low notes have greater inharmonicity (thicker strings).
I don't think bending modes are the usual culprit, it is more often the frequency mismatch between axial and transverse modes that cause inharmonic tones. I'd need to get a real piano and make some spectra and do some maths to show it though.
Yeah, pianos deal with both the tuning inaccuracies that are inherent to 12-tone equal temperament in addition to piano-specific oddities like having to stretch the octave.
Basically, the harmonics that rise off of piano strings aren't exact multiples of the fundamental -- they're a little bit off, because piano strings don't behave entirely like ideal strings, they behave like metal cylinders. In the mid-range they're pretty pretty close to plain equal temperament, but in the high treble the strings get shorted but the string gauge stays almost the same, which means the ratio of diameter to length increases and they act less like strings and more like cylinders. The bass has similar issues with single and double wound strings. So, the fix is to just stretch the octave enough so that the harmonics of low notes line up better with the fundamentals of higher notes, and so on. (What we perceive as "out of tune-ness" is the wobbly sound of two frequencies played together that almost but don't quite line up, creating a beat frequency.)
Old joke explained: RTFA means Read The Fine Article -- except that F doesn't really stand for fine. It's usually meant as a crabby way of saying "The article answers your question".
From that, "TFA" just means "The Article", but without any of the crabbiness, and without the F really standing for anything. That's just a shortcut, and a nice instance where a bit of Internet lore became nicer rather than meaner.
I'll sometimes read TFA as "the featured article," but I don't know whether I picked that up along the way or invented it to make internal sense of nice things said about TFA.
RTFA derives directly from RTFM ("Read The ... Manual"), which I always interpret with the full force of the embedded expletive.
Same. When I'm trying to be neutral, I usually just say "the article" / "the submitted article". TFA to me carries the same kind of connotation as "TFM" in "RTFM".
> Clustering is not the point of black and white keys. Rather it is the facility to pick an anatomically reachable, desired subset of 12 keys available per octave.
This is the essential point of critique that makes TFA an exercise in assembling loose hypotheses for publication on the InterWobbles.
I thought (read) that the distribution of black and white keys came to be like that to provide a visual pattern which allows you to easily distinguish the different octaves.
I often huff and puff at articles like TFA, but here I found myself nodding: the method is sound and makes sense.
Some comments I must leave though:
Clustering is not the point of black and white keys. Rather it is the facility to pick an anatomically reachable, desired subset of 12 keys available per octave. As a simplified European tradition baseline, that is the white keys transposed by some number of semitones. The salient thing here is to have a row of keys which are mostly two semitones apart but have a one semitone gap at strategic locations to produce the scale.
Much of the music in the world operates roughly on the pentatonic scale which coincides with the black keys, or the complement of "European" scales in a 12 step equally tempered octave. Pentatonic scales are mostly two semitone steps with strategic 3 semitone steps.
Finally, the harmonic model in TFA does not resemble the piano very much. Would be interesting to see how different harmonic models and temperaments in various historical keyboard instruments interact with the computation. The modern piano is equally tempered. In a harpsichord, that would generate a lot of roughness for the thirds which are way out of tune. The modern piano mitigates this by having the hammers strike strings at a position that avoids exciting the 5th harmonic (which produces a justly intoned third on top of fundamental frequency).
Would be interesting to see what kind of difference to the roughness calculation it would make to omit the 5th harmonic!