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posted by hubie on Friday March 01, @02:50PM   Printer-friendly

https://phys.org/news/2024-02-pythagoras-wrong-universal-musical-harmonies.html

The tone and tuning of musical instruments has the power to manipulate our appreciation of harmony, new research shows. The findings challenge centuries of Western music theory and encourage greater experimentation with instruments from different cultures.

According to the Ancient Greek philosopher Pythagoras, 'consonance'—a pleasant-sounding combination of notes—is produced by special relationships between simple numbers such as 3 and 4. More recently, scholars have tried to find psychological explanations, but these 'integer ratios' are still credited with making a chord sound beautiful, and deviation from them is thought to make music 'dissonant,' unpleasant sounding.

But researchers from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics, have now discovered two key ways in which Pythagoras was wrong.

Their study, published in Nature Communications, shows that in normal listening contexts, we do not actually prefer chords to be perfectly in these mathematical ratios.

"We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us," said co-author, Dr. Peter Harrison, from Cambridge's Faculty of Music and Director of its Center for Music and Science.

The researchers also found that the role played by these mathematical relationships disappears when you consider certain musical instruments that are less familiar to Western musicians, audiences and scholars. These instruments tend to be bells, gongs, types of xylophones and other kinds of pitched percussion instruments. In particular, they studied the 'bonang,' an instrument from the Javanese gamelan built from a collection of small gongs.

"When we use instruments like the bonang, Pythagoras's special numbers go out the window and we encounter entirely new patterns of consonance and dissonance," Dr. Harrison said.

[...] "Quite a lot of pop music now tries to marry Western harmony with local melodies from the Middle East, India, and other parts of the world. That can be more or less successful, but one problem is that notes can sound dissonant if you play them with Western instruments."

"Musicians and producers might be able to make that marriage work better if they took account of our findings and considered changing the 'timbre,' the tone quality, by using specially chosen real or synthesized instruments. Then they really might get the best of both worlds: harmony and local scale systems."

Journal Reference:
Raja Marjieh et al, Timbral effects on consonance disentangle psychoacoustic mechanisms and suggest perceptual origins for musical scales, Nature Communications (2024). DOI: 10.1038/s41467-024-45812-z


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  • (Score: 1, Interesting) by Anonymous Coward on Friday March 01, @03:44PM (8 children)

    by Anonymous Coward on Friday March 01, @03:44PM (#1346977)

    Then there is John Lennon [reddit.com]

    • (Score: 3, Interesting) by RamiK on Friday March 01, @05:36PM (6 children)

      by RamiK (1813) on Friday March 01, @05:36PM (#1346988)

      I wouldn't be surprised if his guitar's voicing (the actual body's resonant tuning) was a slightly flat A so flatting the D made the secondary harmonic resonate better.

      Regardless, the guitar stands out for having strings with fundamental frequencies that have less frequency than the harmonics so you're starting out with an exception to the rule...

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      compiling...
      • (Score: 3, Informative) by RamiK on Friday March 01, @05:40PM

        by RamiK (1813) on Friday March 01, @05:40PM (#1346989)

        FIX: "fundamental frequencies that have less frequency" -> "fundamental frequencies that have less amplitude"

        And see here for some details: https://music.stackexchange.com/questions/6942/music-theory-frequencies-related-to-playing-a-note-on-guitar [stackexchange.com]

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        compiling...
      • (Score: 4, Interesting) by nostyle on Saturday March 02, @06:30PM (4 children)

        by nostyle (11497) on Saturday March 02, @06:30PM (#1347103) Journal

        Some fifty years ago, I read a book about building a classical guitar from scratch. It included extensive sections on proper bracing of the sound board and instructions on how to tune the unstrung body to resonate at A. Ultimately I was daunted by the plethora of jigs one needed to fabricate but a single guitar. It seemed only to make sense to do this if one intended to fabricate dozens of them.

        Some years later, I purchased one in Pisa for $350 - an hand-built small-name Spanish guitar whose sound pleased me. When I got home, I pulled off all the strings and measured its native resonance with an electronic tuner, and discover it resonated in G. I still use this as my "beater" guitar when composing. It features a cedar top and mahogany back and sides. I have nearly worn a hole in it like the one Willie Nelson has worn in his.

        Later when wanting to record some compositions, I found a second-hand big-name spanish-made "student" guitar on ebay for around $1200 - I think the "student" classification indicated that some parts of it were machine made and imported into Spain from southeast Asia. It features a cedar top and rosewood back and sides and has a much louder voice. I measured its resonance at A-flat. After making some recordings with it, I lent it to a friend whose guitar got smashed.

        Ultimately, I traveled to Madrid and bought a signed #1-A guitar from Manuel Contreras II [guitarsalon.com], for more money than I am willing to admit. It is something of a cross between a "classical" and "flamenco", featuring a spruce top and rosewood back and sides. It does indeed resonate in A, and it is the instrument I use exclusively when recording these days. It produces the most balanced sound of all the instruments I have played.

        I deduce that there is something to this "master-crafted" stuff, and that no two things are ever quite the same.
        --
        "You're a shining star - No matter who you are" -Earth, Wind & Fire

        • (Score: 1, Informative) by Anonymous Coward on Saturday March 02, @10:27PM (3 children)

          by Anonymous Coward on Saturday March 02, @10:27PM (#1347142)

          And here [youtube.com] is what the Contreras looks and sounds like - if you can find anyone who can play like this.

          • (Score: 3, Funny) by The Vocal Minority on Sunday March 03, @03:23AM (2 children)

            by The Vocal Minority (2765) on Sunday March 03, @03:23AM (#1347164) Journal

            I was a little disappointed to find that this wasn't a classical guitar version of "never gonna give you up".

            • (Score: 3, Interesting) by nostyle on Saturday March 09, @05:25AM (1 child)

              by nostyle (11497) on Saturday March 09, @05:25AM (#1347981) Journal

              I finally got around to looking that one up. It seems to work better on a steel-string guitar. Here ya go [youtube.com].

              No less amazing than the other piece. Humans are phenomenal.

              --

              If I could play the guitar
              I'd play Hawaiian songs

              -Loeka Longakit, The One They Call Hawaii

    • (Score: 3, Interesting) by istartedi on Friday March 01, @07:09PM

      by istartedi (123) on Friday March 01, @07:09PM (#1346999) Journal

      TFL says part of the reason was to make their guitars distinguishable on mono recordings. With stereo, you can separate the musicians by having different levels from each speaker. Kind of funny to think about it, but people *did* care about a certain amount of fidelity even if there was only one channel, and they had to mix so it would even sound good on AM radio.

      This is why remasters seem worthwhile. We may not have the full band any more, but we've still got people who have a clue about what they really wanted us to hear.

      --
      Appended to the end of comments you post. Max: 120 chars.
  • (Score: 5, Insightful) by Thexalon on Friday March 01, @06:00PM (6 children)

    by Thexalon (636) on Friday March 01, @06:00PM (#1346990)

    People who were paying attention realized there's a problem centuries ago.

    * What European-based music theorists call an "octave" is generally seen as the most consonant sound, and is a 2:1 frequency ratio. So given a frequency x and an integer n, the pitches that are consonant with x because they're octaves is 2^n * x.
    * The next most consonant sound, based on this concept, is the 3:2 ratio, what European-based musicians would consider a "perfect fifth", which is kinda part of most chords that you'll hear in anything other than really weird modern classical stuff (I'll get to the "kinda" in a moment). So 1.5 x. And then you can generate another 5th on top of that, so 1.5^2 x, and another, and another and so on.

    So combine the two, and you should have a whole bunch of pleasant sounds you can combine in all sorts of interesting ways, right? Wrong! Because there are no non-zero integers n and m such that 2^n = 1.5^m. So you can get octaves, or you can get perfect fifths, but not both.

    This little math problem has been vexxing musicians for centuries. The solutions have generally been:
    - Tune your instrument to only sound in tune in a particular key and octave. That's what ancient Greek musicians typically did, so that's what Pythagoras would have heard.
    - Fudge it a bit, so that it's close enough. That's what led to the 12-note European-style-music chromatic scale: 1.5^12 is ~129.75, 2^7=128, so they "temper" the scale in one of a few ways to make them line up. If you're wondering, this is what J.S. Bach was talking about when he titled the Well-Tempered Klavier.
    - Don't limit yourself to specific notes, allowing the musicians to shift things slightly to match what they're trying to do. This is standard in Indian classical music: While they use a 7-note scale similar to the Europeans, exactly which 7 notes they use shifts based on what raga they're playing in. If you look at a sitar, you'll see all those pegs along the fingerboard, which allow them to adjust the tuning slightly of all of the frets, which is also why it will sound a bit out of tune when you combine one of those with European-style instruments.

    And really, once you're used to hearing tuning a certain way, you'll think everything else is a bit "wrong". For example, my alma mater had an organ with an unusual tuning and split keys (so that C# and Db are different notes, for example) that allowed the organist to get perfect fifths, and it sounds really weird to people who are used to hearing the 12-note well-tempered keyboards that most people hear all the time.

    --
    The only thing that stops a bad guy with a compiler is a good guy with a compiler.
    • (Score: 2) by darkfeline on Saturday March 02, @10:15AM (1 child)

      by darkfeline (1030) on Saturday March 02, @10:15AM (#1347060) Homepage

      Is this really a problem though? Music is at best a "suggestion". Notes (and rhythms) can be smeared quite a bit and still be the "same song", and people do so all the time according to their preferences and proclivities.

      --
      Join the SDF Public Access UNIX System today!
      • (Score: 2) by Thexalon on Saturday March 02, @11:48AM

        by Thexalon (636) on Saturday March 02, @11:48AM (#1347063)

        It's not an unsolvable problem, and you're right that musicians fudge things in interesting ways all the time, but it does interfere with the goal some people have of mathematical perfection in music. It also means that intonation is a bit weird at times, because to get it really right you have to know not only what note you're playing but what note you're trying to match up nicely too and adjust your pitch accordingly. For example, if you watch a professional orchestra with a tuba in it, and you watch that tuba player, you'll notice they have a slide to adjust their tuning easily, which they do almost every note, because especially deep in the bass range a little bit off sounds a lot off.

        --
        The only thing that stops a bad guy with a compiler is a good guy with a compiler.
    • (Score: 2, Interesting) by khallow on Saturday March 02, @01:26PM (2 children)

      by khallow (3766) Subscriber Badge on Saturday March 02, @01:26PM (#1347066) Journal

      The next most consonant sound, based on this concept, is the 3:2 ratio

      You're not in the right scales. For pure periodic waveforms there is a fundamental frequency and the harmonics in question would be all integer multiples of that harmonic. So you have 1f, 2f, 3f, etc. In your base 2 logarithmic scale that would be exponents of 0, 1, ~1.58, 2, ~2.32, ~2.58, ~2.81, 3,... (corresponding to 1f, 2f, 3f, 4f, 5f, 6f, 7f, 8f, ...) That's typical sound from sources that have a dominant single dimension like rigid horns (not trombones), vibrating strings, tuning forks, etc. You can quickly go beyond that with odd shaped drums, bells, instruments that can change their dimensions (trombones, guitars especially fretless ones), etc which can generate more than that.

      And then people found that tuning the scale up or down resulted in even more interesting combinations.

      • (Score: 1, Interesting) by Anonymous Coward on Saturday March 02, @01:57PM

        by Anonymous Coward on Saturday March 02, @01:57PM (#1347072)

        If you haven't seen it, you might find John Baez's blog on this topic [wordpress.com] interesting.

      • (Score: 3, Informative) by Thexalon on Saturday March 02, @03:18PM

        by Thexalon (636) on Saturday March 02, @03:18PM (#1347076)

        Among the instruments I've played on well enough to not offend people in public are 3 different brass instruments, so I'm very familiar with harmonic series, their benefits, and limitations.

        First off, there are notes that are in the commonly used scales that are not part of the harmonic series, at all. For example, if your fundamental is C, your first note that could be even approximately called A is the 27th harmonic, and it's wildly out of tune for other uses of A. No A = no C pentatonic scale, no C major scale, and a bunch of other less common scales and modes. So now there's a problem.

        So where does A really come from if you use C0 as your fundamental? And the answer is to switch fundamentals and try to pick an f' where n*f' = m*f for some whole-number ratios of n and m. Here's a couple versions of the math you can do that would get you to a frequency for A:
        1. 3*f' = 5*f: So the 3rd harmonic with a fundamental of A should be an E, and the fifth harmonic with a fundamental of C should also be E.
        2. f' = 27f: The 27th harmonic with the fundamental of C is A.

        The problem is that there's no way to make the harmonic series from #1 and #2 the same pitches. They're always a little bit different.

        And yes, you can do weird things with microtonality and the full spectrum of notes that are possible to produce. Classical electronic musicians do that sort of thing all the time, which is why you get things like the old Doctor Who theme not sounding quite right to a lot of western ears even though it's perfectly in tune with itself.

        --
        The only thing that stops a bad guy with a compiler is a good guy with a compiler.
    • (Score: 2) by VLM on Saturday March 02, @09:00PM

      by VLM (445) on Saturday March 02, @09:00PM (#1347131)

      It's always been enjoyable reading about this from a music perspective, and a math perspective, but there's also a physics perspective where the natural and unavoidable harmonics of real instruments will also mix and appear in various ratios. So it's actually even worse than proposed, because the 3rd harmonics of two notes will sound even weirder than 2^n = 1.5^m because the absolute distance will increase as the harmonic number increases.

      Using made up numbers you'd like to think the 12-note is only off by 1.75 or so parts per 128, but its an even bigger gap at harmonic frequencies.

      Sometimes it seems amazing to me anything sounds "good".

  • (Score: 4, Informative) by hendrikboom on Friday March 01, @06:42PM (3 children)

    by hendrikboom (1125) Subscriber Badge on Friday March 01, @06:42PM (#1346995) Homepage Journal

    This argumentation will be a bit dry without experiencing some Gamelan music. Here's a sample o YouTube: Gamelan Bali (Balinese Gamelan) - Traditional Music [youtube.com].

    See Composite Theory of Dissonance [youtube.com] for some serious theory.

    And Tuning of Gamelan and Sensory Dissonance [youtube.com] for its application to Gamelan.

  • (Score: 0) by Anonymous Coward on Saturday March 02, @03:21AM

    by Anonymous Coward on Saturday March 02, @03:21AM (#1347043)

    Notes and Neurons: In Search of the Common Chorus:
    https://www.youtube.com/watch?v=S0kCUss0g9Q#t=58m52s [youtube.com]

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