I wanna be well tempered

If someone could direct me to a place where I could be enlightened with the concept of 'equal temperament' I would be forever in their debt, I understand a little, like equal temperament is not quite 'in tune' compared to the eastern way of thinking missing out the 13th note et al but I need a basic grounding on the maths behind it ? or do I need worry?

I never understood Pythagorean theorems in maths and I understand he is the main culprit behind this.

Should I bother or should I just play piano?

Kind regards

You don't need to worry, unless you are planning a career as a piano tuner, or plan to moonlight playing Indian classical music. However, to satisfy your curiosity, here's one possible starting point:

Hi Drumey:

I have done extensive research and work with different tuning systems. Here's a little primer on scales and temperament that I wrote as an answer to another poster a couple of months ago. Sorry if the little o's don't line up properly in your browser:

Tuning is mostly about intervals. An interval is the musical 'space' between two notes. If you play two musical notes at different frequencies, some intervals sound more pleasing than others. The human brain prefers two sounds where the vibrational waves match up more often. For example, if one string vibrates 440 times per second and another at 880 times per second, every other vibration matches up and your brain is happy.

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This is called 'consonance' and is produced when frequencies are small ratios of each other (2:1 in the last case). The interval of a fifth (C to G) has the ratio 3/2, so every third vibration of one string matches up with every second vibration of the other.

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The fewer stray, unmatched vibrations in between, the more consonant the string sounds. 2:1 is an octave and has only one stray vibration in between on one string, and none (all match up) on the other, so it sounds the most consonant of all. The fifth has one stray vibration on one string and two on the other. A fourth (C to F) is the next most consonant interval down the line with a ratio of 4/3.

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Now there are two unmatched waves for one string and three for the other.

The set of interval ratios for a diatonic scale is

1/1 do 9/8 re 5/4 mi 4/3 fa 3/2 sol 5/3 la 15/8 si 2/1 do

As the number of stray, unmatched waves increases, the intervals sound increasingly dissonant and less pleasing to the brain. Note that all of the most consonant fractions possible, viz. 2/1, 3/2, 4/3, 5/3 and 5/4, have been used in the scale. Now if I want to play in another key (play the scale starting on a different note) I will need more notes.

Here's why: If I start on the fifth note of the scale shown above and go up another fifth it puts me at a place 9/8ths above the higher 'do'. That's where the 're' comes from. In turn, to have a note a major third above 're' requires a whole new fraction (accidental, black note), 7/5. Fortunately, you eventually only need 12 fractions to satisfy all the keys. Note that 're' and 'si' are very dissonant fractions when played with 'do', but may sound perfectly fine when played with another note in a different key.

Since the interval of an octave is the most consonant interval possible and since it doubles the frequency with each iteration, we can set up these same intervals every twelve notes and each degree will still be exactly twice its counterpart one octave below (C is twice the C below, F# is twice the F# below, etc.). There is nothing magical about the octave. It's all mathematics. The fact that we can keep repeating the same tunes in each octave as we go up and still sound good with the octave below is a matter of pure convenience. Thus, since the human brain likes ratios, not just finite intervals in terms of an even number of vibrations, the musical gamut is an exponential one. In other words the octaves go 2x, 4x, 8x 16x, and not 2x, 3x, 4x, 5x. The modern set of notes in Western music simply came from setting up the nicest-sounding intervals (simplest fractions), then adding accidentals so that you can play these intervals universally in any key. With such an ingenious system a singer with any vocal range can sing any tune along with any instrument(s) (with their own various ranges) in any key desired, as high or as low as desired.

So everything is fine now, right? Well, there is a slight problem. There is no practical group of fractions that can give you all the same perfect intervals in all keys. For example, from 'do' to 'sol' in the above scale has a ratio of exactly 3/2 or 1.500 and sounds great. From 're' to 'la' should also be a fifth, but 5/3 divided by 9/8 is 1.481. Close, but not exact. For this reason the 'just' scale that we set up is only perfect for the key we set it up in and not quite right for music in other keys played on the same instrument.

Well, it turns out that to get this universal, nice-sounding set of notes that repeat as double frequencies every twelve notes, we unknowingly set up a set of ratios that are approximately logarithmic. Have you ever seen logarithmic graph paper? It's the kind that has lines that get increasingly close together as you go along. This is how the frequencies of the modern scale are. Every twelve notes you have doubled the frequencies, so the graph scale must be scrunched up through each octave to fit the frequencies that are twice as large. Now, some clever fellow figured out that if you describe the frequencies of the scale with a certain mathematical equation (an exponential one), you can get the same exact intervals between any two notes no matter which note you start with! The formula looks like this:

f = 27.5 * 2^(N/12)

Where f is the frequency (vibrations per second) and N is the note number on a piano (the lowest A is 1, A# = 2, etc.)

Don't let the math scare you. It's really quite simple. The 27.5 is simply the frequency of the lowest note (A0) on a piano. 27.5 is chosen because it will ensure that 440 will be the frequency of A above middle-C. This is a universal standard and I won't even get into who decides such things or why. The exponential base 2 is because the scale is based upon octaves and repeats after each doubling (2:1). The 12 is there because there are 12 different notes in our system. If you wanted a new scale system based on the interval of a fifth (3/2) You could replace the 2 with 3/2 or 1.5. If you wanted a scale with only ten even degrees, you could replace the 12 with 10 in the equation. But these imaginary scale systems wouldn't allow you to include all the nicest-sounding intervals and have a system that repeats as you go up.

This arrangement of frequencies is called Equal Temperament and is now universally adopted for all musical instruments. Octaves are preserved as a perfect 2:1 ratio, but all other intervals are an approximation. A fifth, for example (in any key), is about 1.498 instead of 1.500. So what we are accustomed to hearing is slightly off, but not bad. We get to play with any instrument in any key, at will, at the cost of slightly less (though uniformly) consonant sound. Virtually every piece you have ever heard was played/recorded in Equal Temperament and you are accustomed to it.

Don

Visit Ed Foote's website for explicit information on this topic.

[...]

A brilliant choirmaster with whom I worked in a German school would not permit a piano in his rehearsal room. He gave a note on his pitchpipe and the rehearsal proceeded without accompaniment. His belief was that the youngsters' intonation would never be secure if they had to accommodate it to ET, or to any other fixed temperament. Their repertoire was varied, with great emphasis on Renaissance polyphony. When they sang with orchestra (and they annually sang Bach's St Matthew Passion, uncut, at the parish church) they naturally, effortlessly, matched the instruments by, as Ed says, moving notes hither and yon. Unaccompanied, the chording was gorgeous, the thirds ringing out splendidly - just like a good string quartet's - with constant flexibility of intonation. Alas, that can't ever be achieved on a standard piano. It is in principle possible on a clavichord, though, and I seem to have read that the clavichord was Bach's preferred keyboard instrument. He was also a fine violinist, and I can't think he would have been happy with ET.

Alan Jones

Dear Ed:

Thank you for your elegant, polite and genteel critique on my posting. Here are a few answers.

I'm referring to consonance. Listen to the interval of an octave and then listen to the interval of a half-step or a major seventh.

Sorry, I meant permanently intoned instruments, not voice or non-fretted stringed instruments. The holes in wind instruments, tubes and valves of brass instruments and frets of guitars are all arranged for equal temperament. Other adjustable, but still permanently intoned, instruments, like the piano, are usually also tuned to this system for obvious accompaniment reasons.

I am a trombonist as well as a pianist and am quite familiar with the bending of tones to produce more pleasing intervals, but this is only disirable (or possible) when playing along with other instruments which can do the same. Just intervals can only be used in an 'active' sense by the performer since it is different in each key. Only certain instruments are capable of this, viz. strings, voice, etc. If you want to play along with the ET instruments, you'd better be thinking in ET.

Nicely done, thanks.

Go to
http://members.aol.com/chang8828/contents.htm

and click on Section 2, Chapter Two C. C. Chang