Blog 8 – Pythagoras and Religious Overtones (or: When is G♯ not A♭?)

2 Oct

Ah, temperaments.  They’re enough to drive one to distraction, which is what happened to Pythagoras when he realized that, if you (effectively) start at bottom C (C1), double that frequency to get to the next octave, and keep repeating that until you get to the top C (C8), you will get a specific frequency;  but if you start at C1, multiply that frequency by 3/2 to get to the 5th above it, then keep repeating that until you get to C8, you get a different frequency.  For the same note.

Hmm.  (not “hummm”…)

It gets worse.  You would think that, if you start at C4 (middle C), and go up by three pure (i.e. beatless) major 3rds to C5, you’d get a pure octave.  However, you don’t.  If you were to mute off the two notes of an octave and tune the middle strings of the octave to be beatless, and then tune each of the three major 3rds in the octave to be beatless, but tune the upper note of the last 3rd (which is the upper note of the octave) on the right string (with the center string muted), and then compare those middle and right strings, you will find that they ain’t the same!


So, this is where we are in the process – exploring historical temperaments.  And we have discovered the wolf.  Certain temperaments dictate that the music stick to certain chords, because only those chords are acceptably in tune.  The wolf lives in the other ones.  J. S. Bach’s “Das Wohltemperierte Klavier” sounds quite different when played in the temperament for which it was written than it does when played in equal temperament.  Here’s an excerpt describing the situation:

Well v.s. Equal Temperament

Contrary to what you’ve been taught, Bach did not compose the Well Tempered Clavier to promote the equal tempered tuning system. Equal temperament actually did not come into use until the 20th century. Bach’s motivation for composing the WTC was to demonstrate the feasability of composing in well temperament and to demonstrate the varying key colors in well tempered tuning as one progresses around the circle of fifths. The various well temperaments used in Bach’s time are distinct from our equal temperament. Well temperament represented a departure from the various meantone tunings that were used in earlier music. (

(Never mind the spelling – it’s from the math department.  TD)

I have the instructions for tuning what’s purported to be the temperament that Bach used for “Das Wohltemperierte Klavier”.  If somebody wants to learn the piece, or some part of it, when I get back  it would be interesting to tune a piano to the “Bach” temperament, play the piece, then play it on a piano tuned to equal temperament, and compare the two.  (It’d be easier if someone could just find a recording…)

The way it’s been going over the last week or so is that we’ve been introduced to three different historic temperaments, then given instructions on how to tune them, then turned loose on the pianos.  If you get the tuning right, there are some chords in each of the temperaments that sound “perfect” (which doesn’t actually sound that great…), some that are interesting, and some that are real howlers.  We’re also learning a few different ways to tune pure octaves by using test intervals to even out the beats with the upper note and the lower note of the octave. It’s really interesting!

The whole experience is giving me a new way of listening, which I discovered Sunday night at a piano concert at the New England Conservatory’s Jordan Hall. The NEC is the oldest independent school of music in the US, and Jordan Hall is an absolutely gorgeous old place – leather covered doors, comfortable seats, huge proscenium, and a Steinway D that is a very nice piano.  (Here’s a link describing how they chose the piano:   )

At the concert, I found I was spending a lot of time listening to how the piano was performing, rather than how the pianist was doing  (the piano was in fine tune, although the dampers needed some adjusting), and I realized that here was Yet One More Thing to pay attention to during a concert.

And speaking of choosing pianos, here’s another link, from National Public Radio, in which Leif Ove Andsnes describes his process for choosing a piano for a concert, from the Steinway basement.

His description starts at about 2:20 in, but the whole thing is worth listening to.

And one more thing, from the BBC (so it must be true):

Fun, huh?

Oh, and before I forget, a couple of posts ago, I mentioned the (Jetsons’) Fender Rhodes Seventy-Three that lives at the NBSS up in Spinet Land, next to the refrigerator.  Here it is:


So, as one my good friends in Montana says – “Party on!”


5 Responses to “Blog 8 – Pythagoras and Religious Overtones (or: When is G♯ not A♭?)”

  1. david Swingle October 2, 2012 at 3:09 am #

    Keep ’em coming. Fascinating on many levels.

    — dave swingle, partier

  2. Shelley October 2, 2012 at 2:08 pm #

    Tim ~ I get the C1, C4 and C8 stuff, but as a visual kinda gal, you lost me with all the ‘middle’ stuff. It would be interesting to hear the differences though, in temperament styles. Sounds like you are playing and learning at many levels ~ good work. Have a great week, see you in a few days ~

    • timinboston October 3, 2012 at 1:32 am #

      I can show you with the math. That way, you’ll still be able to enjoy listening to your piano after we’re done…

  3. Suzanne October 2, 2012 at 9:59 pm #

    Ready to tune our piano at Christmas?

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