Music and Math: The Genius of Beethoven

Laurence Glazier comments:

Very nice, I would add that Bach was the engineer who enabled Beethoven and everyone else to write in lots of different keys. 1.5^12 and 2^7, in music 12 fifths and 7 octaves, are almost but not quite the same. Bach fixed this with a tuning system which averages out the difference.

Peter Saint-Andre adds:

Indeed, there were a lot of tuning systems developed around then: Neidhardt (seemingly Bach's preferred system), Werckmeister (he developed several), etc. Just last night I read all about them in The Esoteric Keyboard Temperaments of J. S. Bach. These folks were the quants of their day!

Peter Grieve comments:

Yes, the problem with getting good fifths and good octaves in the same scale is find a power of 3 that is equal to a power of 2. This is because a fifth is a ratio of 3/2, and an octave is a ratio of two.

Of course, there is no power of 3 that is exactly equal to a power of 2. There is a fairly good match at 3^5=243, and 2^8=256. The power of 5 on the 3 means that this corresponds to a pentatonic scale. And 3^12=531,441 while 2^19=524,288, (proportionately a better match) which as Laurence says is the basis of a diatonic scale.

Because the matches aren't exact, something's gotta give, and this is what Bach's temperment ideas addressed (as Laurence said).

There are other near matches at larger powers, but a scale with dozens or hundreds of notes has limited appeal.

Laurence Glazier writes:

Excellent attachment on the tunings, esoteric is the right word. The fact that this is being rediscovered after hundreds of years, is of special interest to me.

Adam Grimes adds:

I have built and played harpsichords for many years. When you play harpsichords, you also tune them. A lesser-known fact is how quickly this instrument goes out of tune… you can have it in tune for a concert and then it will need a touch up at intermission.

So, harpsichord players quickly become very familiar with these tunings. Some are much more useful than others, but it also explains what composers meant when they talked about affects or emotions associated with certain keys. This was a very real thing, in some of the older tuning systems, but has been completely lost (for better or worse) with modern equal temperament.

Another interesting aside is that I find these historical tunings don't work that well on the modern piano. Completely aside from the temperament issues, there's also the issue of inharmonicity (the deviation of a physical string from the theoretical ideal). All strings have this, but the piano has A LOT because of the thickness of the strings. (Certain types of harpischords (Italian) have scalings that are much closer to the theoretical ideals.) A piano is tuned ever-sharper in higher octaves so that it is in tune with its own overtones rather than the actual pitches. It's subtle, but it's real and important… and it also obliterates the precision of these historical tunings. (Another interesting aside is that once your ear learns to hear in these historical tunings, moving back to ET is a kick in the gut. You'll sit down at a piano, play a chord, and think "wow. everything really IS out of tune." which is the compromise of ET. (For the record, ET is a beautiful and useful thing, as well.)

What I don't see much value in are the microtonal modern experiments, but I understand what drives that line of thought.

For any musicians, if you haven't had the experience of singing pure-tempered intervals against a drone I'd highly encourage it. You can spend hours or even weeks exploring the beauty and power of these resonances… and you'll know musical materials as an EXPERIENCE of resonance rather than a sound or a theoretical construct.

One might imagine that it was these experiences of resonance that encouraged early humans to sing, to seek sound, and maybe even to seek language… maybe in those caves where they left us paintings of mystery and power… somewhere a very long time ago.

But, seriously, go get a bass drone sound and sing some pure octaves, fifths, and thirds against it. You'll never hear the same way again.

A reader adds:

Each open tuning has a special resonance that is different than the same notes played in concert. Similarly chord inversions carry different overtones from base fingering.

Jeff Watson adds:

I love Fripp’s New Standard Tuning, CGDAEG. The mnemonic for recalling it is “California guitarists drop acid every gig.”

Adam Grimes responds:

yeah but slightly different. Fretted instruments are ET. You could potentially bend some notes, but you're still working in an ET world. (Scordatura certainly changes the timbre of instrument, and resonance of open strings, etc., but is a substantially different thing from temperaments.)

Laurence Glazier writes:

Thanks Adam, fascinating thoughts.

When transcribing from inspiration, I am sometimes unable to use the note I hear in my mind, which lies somewhere between a pair of adjacent semitones. As my software uses ET tuning, I have on occasion resorted to using MIDI control instructions to nudge the pitch into place, but in the light of your post, I now see that the issue may be with the tuning system. On one of the historical keyboard instruments, the note I require might simply be there.

I have enjoyed writing music in the past for clavichord, because of the pressure sensitivity, but am now writing mainly for orchestra.

As you say, experience trumps academic construct. I personally consider music to be an elemental force of nature, and species evolve to sense it along with every other aspect of reality. It's also interesting that lunar and planetary orbits often lock into similar ratios. The Pythagorean Comma has a counterpart in the slight divergence between the lunar and solar calendars. The term live music, in my opinion, is literally true.

Adam Grimes responds:

Clavichord is a beautiful and intensely problematic (at least in my experience!) instrument.

I own one. The intimacy of it is incredible… it puts the player's finger in almost direct, expressive contact with the vibrating string… but that brings up so many issues of control and it's such a different technique than any other keyboard instrument. To say nothing of the whisper-soft sound level (that defies amplification, which might seem to be the obvious answer.)

And you're right… all those "in between" notes exist as a possibility on that instrument. Not hard to imagine someone playing in a remote key and instinctively bending the out of tune notes into an acceptable range.

Zubin comments:

Guitar players always bend notes giving infinite micro tones. Squeezing the string to approach the note can give great feeling. Of course singers all do it too.

Vic is reminded of a Beethoven story:

During a performance of one of his piano concertos Beethoven was the soloist, and he got so carried away with conducting that at one point he forgot to play the piano. He flung his arms wide and knocked the candlesticks off each side of the piano. The audience burst out laughing, and Beethoven got so mad that he ordered the orchestra to start over again.

Two choirboys were enlisted to hold the candlesticks out of harm's way. One of them got increasingly intrigued by the piano score and came in closer and closer just as a loud passage broke forth. Out went Beethoven's arm, knocking the choirboy in the mouth so that he dropped his candlestick. The other choirboy, having followed Beethoven's motions more cautiously, ducked, to the complete delight of the audience.

Beethoven fell into such a rage that on the first chord of his solo he pounded the piano so forcefully that he broke half a dozen strings. Die-hard music lovers in the audience tried to restore order, but failed. After that debacle Beethoven became increasingly reluctant to give concerts.

From Wisconsin Public Radio: The Catastrophic Conductor





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