7 ms·

(disclaimer: music degree) 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 m

(disclaimer: music degree) 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!

tomsmeding 5y ago
> 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.
- yesenadam 5y ago
  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] [0] https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and_inharmonicity https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and...
  - vnorilo 5y ago
    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.
    - HPsquared 5y ago
      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). [0] https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.pdf https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.... [1] http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation.pdf http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation...
      - salty_biscuits 5y ago
        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.
        vnorilo 5y ago
        In the case of the piano, it is definitely stiffness.
- weinzierl 5y ago
  Piano tuning is a science in its own. Google for Railsback Curve if you want to fall down a rabbit hole.
magicalhippo 5y ago
Your post reminded me of this video[1] by Adam Neely, where he tries various tuning systems around A = 432 Hz. As a non-musician it really made me appreciate how little I know about the technical side of music. [1]: https://www.youtube.com/watch?v=ghUs-84NAAU https://www.youtube.com/watch?v=ghUs-84NAAU
weinzierl 5y ago
> "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.
- vnorilo 5y ago
  Surely not related to black keys, that is what I tried to say. Most pentatonic music traditiona predate keyboards.
- jmrm 5y ago
  AFAIK Stevie's songs uses a lot of black keys due to being easier to him to locate them using touch instead of sight.
  - carlob 5y ago
    I highly doubt that anyone with that level of musical training really needs to look at the keyboard...just like you probably don't while you type.
    - weinzierl 5y ago
      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.
pantulis 5y ago
You mention "TFA" a couple of times, what is it?
- vnorilo 5y ago
  The featured article.
  - pantulis 5y ago
    Thanks!
  - mxmilkiib 5y ago
    Read The Featured Manual
  - meowface 5y ago
    I always considered the middle word something slightly different...
    - pertymcpert 5y ago
      Yeah. I thought it was odd to use the term TFA when you're not trying to attack the piece.
      - meowface 5y ago
        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".
    - mannykannot 5y ago
      It's a context-sensitive grammar (OK, not strictly...)
- jfengel 5y ago
  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.
heresie-dabord 5y ago
> 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.
sideshowb 5y ago
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.
- vnorilo 5y ago
  Thank you for writing it! 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!
- jacquesm 5y ago
  Are you aware of the Janko keyboard?
  - recursive 5y ago
    I had not heard of it. It looks similarly motivated the harmonic table layout. https://en.wikipedia.org/wiki/Harmonic_table_note_layout https://en.wikipedia.org/wiki/Harmonic_table_note_layout
    - jacquesm 5y ago
      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.
moralestapia 5y ago
Disclaimer: I do not have a music degree. 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.
elihu 5y ago
> 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). This random google result agrees with me: https://pages.mtu.edu/~suits/badnote.html https://pages.mtu.edu/~suits/badnote.html 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. https://kiteguitar.com/ https://kiteguitar.com/

tomsmeding 5y ago

> 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.

yesenadam 5y ago
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] [0] https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and_inharmonicity https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and...
- vnorilo 5y ago
  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.
  - HPsquared 5y ago
    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). [0] https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.pdf https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.... [1] http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation.pdf http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation...
    - salty_biscuits 5y ago
      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.
      - vnorilo 5y ago
        In the case of the piano, it is definitely stiffness.
weinzierl 5y ago
Piano tuning is a science in its own. Google for Railsback Curve if you want to fall down a rabbit hole.

magicalhippo 5y ago

Your post reminded me of this video[1] by Adam Neely, where he tries various tuning systems around A = 432 Hz. As a non-musician it really made me appreciate how little I know about the technical side of music. [1]: https://www.youtube.com/watch?v=ghUs-84NAAU https://www.youtube.com/watch?v=ghUs-84NAAU

weinzierl 5y ago

> "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.

vnorilo 5y ago
Surely not related to black keys, that is what I tried to say. Most pentatonic music traditiona predate keyboards.
jmrm 5y ago
AFAIK Stevie's songs uses a lot of black keys due to being easier to him to locate them using touch instead of sight.
- carlob 5y ago
  I highly doubt that anyone with that level of musical training really needs to look at the keyboard...just like you probably don't while you type.
  - weinzierl 5y ago
    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.

pantulis 5y ago

You mention "TFA" a couple of times, what is it?

vnorilo 5y ago
The featured article.
- pantulis 5y ago
  Thanks!
- mxmilkiib 5y ago
  Read The Featured Manual
- meowface 5y ago
  I always considered the middle word something slightly different...
  - pertymcpert 5y ago
    Yeah. I thought it was odd to use the term TFA when you're not trying to attack the piece.
    - meowface 5y ago
      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".
  - mannykannot 5y ago
    It's a context-sensitive grammar (OK, not strictly...)
jfengel 5y ago
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.

heresie-dabord 5y ago

> 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.

sideshowb 5y ago

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.

vnorilo 5y ago
Thank you for writing it! 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!
jacquesm 5y ago
Are you aware of the Janko keyboard?
- recursive 5y ago
  I had not heard of it. It looks similarly motivated the harmonic table layout. https://en.wikipedia.org/wiki/Harmonic_table_note_layout https://en.wikipedia.org/wiki/Harmonic_table_note_layout
  - jacquesm 5y ago
    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.

moralestapia 5y ago

Disclaimer: I do not have a music degree. 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.

elihu 5y ago

> 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). This random google result agrees with me: https://pages.mtu.edu/~suits/badnote.html https://pages.mtu.edu/~suits/badnote.html 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. https://kiteguitar.com/ https://kiteguitar.com/