With a little patient study, if you have a bit of scientific temperament, Equal Temperament is not hard to understand. In this first post on the topic, I made a chart (below) that shows the difference between the Naturally Occurring Overtones Series and the same notes generated by the Equal-Tempered Scale. The Overtone Series is the collection of pitches that result when you play a SINGLE note on a piano or any other instrument. The pitches are all "integral multiples" of the original pitch that sound simultaneously ( 1,2,3,4 etc. times the frequency of the fundamental pitch ) As you see on the chart, the F overtone series, we naturally hear a number of F pitches at 2, 4, 8, 16,and 32 times the fundamental pitch. This is not an abstract concept - the overtones ARE audible if you pay attention - it's easy to not notice them and take them for granted! Aural piano tuners will listen directly to selected overtones to count beats and compare beat-rates and it's a skill that can be learned.
There is a wealth of information just in the left side of the chart before I even touch on the "Tempered Scale". Here are some things to observe about the Natural Overtone Series:
1. All sounds that are an octave apart have a frequency ratio of 2:1 ( this applies to the A's and C's as well as the F's)
2. All sounds that are a 5th apart have a frequency ratio of 3:2
3. All sounds that are a 4th apart have a frequency ratio of 4:3
4. All sounds that are a Maj-3rd apart have a frequency ratio of 5:4
5. All sounds that are a Min-3rd apart have a frequency ratio of 6:5
6. All sounds that are a Maj-6th apart have a frequency ratio of 5:3
7. All sounds that are a Min-6th apart have a frequency ratio of 8:5
The reason we recognize this intervals as "musical" is because our ears are sensitive to these simple frequency relationships. The other overtones that occur naturally are so divergent from the scale that we normally use ( the tempered scale ) that I see no reason to include them now.
THE TEMPERED SCALE
Some time ago it was decided to cram 12 notes into an octave in such a way as to be able to play in all 12 keys without some keys sounding dissonant and some sounding smooth. The notes were distributed in the octave "equally" ( actually "geometrically", see previous post "The 2 Most Important Musical Numbers"). All resultant equal intervals are equally out of tune because, if you try to tune a series of pure intervals you will end up with an octave that is EXTREMELY wide and out of tune and you will also end up with some resultant intervals that are seriously out of tune. Basically, you can't do it! ( It's fun to try this if you have an old piano and a tuning hammer). So all 3rds are equally out of tune, all 4ths are equally out of tune, etc etc. Fourths and Fifths end up being only slightly out of pure, but all of the other intervals are considerably out! We currently have the Equal Temperament compromise that dominates almost all of music. With this system we can do endless 1/2- step modulations of "Mack The Knife" and it continues to sound like "Mack the Knife". Didn't used to do that.
The right side of the chart shows the pitches that would result from an Equal Temperament Tuning. There is also a wealth of information on THIS side of the chart. Here are some things to observe about Equal Temperament Tuning:
1. The octaves of the fundamental ( F in this case ) are the same as the naturally occurring Octaves - equal temperament maintains pure octaves. ( Note that "octave stretching" and "inharmonicity" are different concepts altogether - I will post on that later and should not be confused with "tempering")
2. Tempered 5ths ( The F to C intervals ) are narrower than the naturally occurring pure 5th ( a slight amount ) ( Red up to blue on the chart).
3. Tempered Maj-3rds are wider than naturally occurring pure maj-thirds ( quite a bit, by the way) ( Red up to green on the chart. )
4. Tempered 4ths are wider than naturally occurring pure 4ths ( a slight amount ) ( Blue up to red on the chart)
5. Tempered Minor-6ths are narrower than naturally occurring m6ths ( quite a bit) ( green up to red on the chart)
6. Tempered Major 6ths are wider than naturally occurring Maj 6ths ( quite a bit) ( blue up to green )
7. Tempered Min-3rds are narrower than naturally occurring min-3rds ( quite a bit ) ( green up to blue)
Whenever an interval is wider or narrower that the pure inteval, it will create beats ( slow vibrato effect). Every interval on the piano produces beats except the octave ( though even that may generate some beats for other reasons that I will leave out here). There are no pure intervals in Equal Temperament.
This is why unaccompanied choirs sound so wonderful - singers will naturally gravitate to pure intervals because they are so natural and they will make that adjustment at every chordal opportunity. It also explains why choirs tend to pull flat - if the third of a chord is on the top, the singers will naturally sing it considerably flat to achieve a pure interval against the lower root. This new audible pitch reference becomes the new running reference for the piece and this can happen multiple times - a not-so-practiced choir piece can drop a quarter step or more by the end!
CHECKING OVERTONE PITCH ON BASS OR GUITAR
An interesting thing that happens when musicians are checking their overtones against an electronic tuning machines reference. Those machines are normally preset to Equal Temperament. If you play string overtones ( by gently touching the strings at the various node-points) and check against the machine, here is what you will naturally find because of the difference between Natural Overtones ( the pitches from the instrument unfretted harmonics) and Equal Temperament ( the pitch reference built into the tuning device) :
Octave overtones will be right on.
Third and Tenths will appear to be considerably FLAT.
Fifths and Twelfths will appear to be slightly SHARP.
Those results would be expected and is independent of fret location. If you get something else, there might be string damage or who-knows-what.
Hope that clears everything up.......
Thanks, Nice explanation.
ReplyDelete