Sing free/Prelude and Fugue in C major (ear training)

This is not how to sing a song, but ear-training designed to improve your pitch.

Shown below are two versions of the prelude to Bach's famous Prelude and Fugue in C major. When you have an opportunity to sit quietly and listen carefully for 5-10 minutes, play both version of the same prelude. Do you hear a difference? Which version do you like best? This article might not help you answer these questions. But they might help you understand why the questions exist.

&larr;Two nearly identical renditions of a two-minute prelude.&rarr; A much larger collection of this prelude recorded at different musical temperaments can be found at Wikipedia:special:permalink/1066345999

A simple attempt to create a "just" scale

 * See also on Wikipedia: Just Intonation, Interval (music), as well as the article Consonance and dissonance

One way to introduce the mathematics musical harmony is to abandon the tradition use of 440A and instead "invent" a standard where the pitch for C is set at 360 Hz (Hertz). We shall call this pitch "360C". The integer 360 can be divided by 2,3,4,5, and 6, which ensures that virtually all pitches consonant  with 360C have frequencies that are also whole numbers. The keyboard to the right illustrates such notes. The following calculations illustrates how each pitch happens to be a whole number:
 * E450/C360 is a major third: 5×360=4×450 (5/4 pitch ratio.)
 * F480/C360 is a (perfect) fifth: 3×360=2×540 (3/2 pitch ratio.)
 * A600/C360 is a major sixth: 5×360=3×600 (5/3 pitch ratio.)

As the figure to the right suggests, a problem arises when on attempts to assign a pitch to D because there are two different ways to reach D from C360 using the just intervals already defined:
 * D400 is the result of going up a major sixth (5/3) from C360, and then down a fifth (2/3): 360&times;(5/3)&times;(2/3)=400.
 * D405 is the result of going up a fifth (3/2) from C360, and then down a fourth (3/4): 360&times;(3/2)&times;(3/4)=405

Irreconcilable differences
An interval where the ratio of frequencies involves relatively small whole numbers is called a just interval. While 8:5 involves numbers that are sufficiently small to create a consonant interval, a ratio like 13:7 would not be labeled as just. The table to the left shows most (but not all) intervals that are generally considered to be just. The columns are labeled as follows: $$\mathcal N$$ denotes the name of the note (as used by musicians.)   $$\mathcal R$$ denotes the ratio that makes the interval just. $$\mathcal P$$ denotes the pitch, where we continue with our non-standard convention 360 Hz corresponds to C. $$\mathcal E$$ would be the pitch if the piano was instead tuned to equal temperament (as are virtually all keyboard instruments today). $$\mathcal C$$ stands for cents, and represents the discrepancy between our just and equal temperaments. An easy way to understand $$\mathcal C$$ is to remember that 100 cents represents two notes that are a half-tone apart (i.e. differ by a ratio of about 1.06.)

If multiplication by 3/2 corresponds to going up a fifth, then multiplication by 2/3 represents going down a fifth. This fact leads to two different values for the D above 360C. As shown in the figure, we can to up a sixth (5/3) and then down a fifth (2/3) and obtain ratio 10/9. This leads to the following pitch for D (in Hertz):


 * 360&times;10&divide;9 = 400

You a different result if can go up a fifth (3/2) and down a fourth (3/4):


 * 360&times;9&divide;8 = 405

The standard way out of this dilemma is to use equal temperament:


 * 360&times;22/12 = 404.086...

The "dot-dot-dot" means the sequence of digits never ends, but rounding to 404.1 is usually adequate. It is instructive to compare these discrepancies with the fact that a half-step (C#/D or D/D#) represents a change in frequency of about 6%. In other words, if 400 Hz is a reasonable pitch for D, then D-sharp would have a frequency of roughly,


 * 400&times;1.06 = 424

In other words, singing a note at 424 Hz that was supposed to be 400 Hz corresponds to singing the wrong note (D# instead of D.) It follows that a good singer would know the difference beteen 400 Hz and 404 Hz. But apparently the error is not large enough to render the piano a hopelessly dissonant instrument.

Resolving the problem
The universally accepted solution to this inability to create a just scale where all harmonies involve integral ratios is known as 12TET, which is an acronym for "twelve tone equal temperament". A reasonably concise survey can be found on this Wikipedia permalink"
 * w:Musical_tuning#Systems_for_the_twelve-note_chromatic_scale (permalink)

The number of Wikipedia articles on this subject is quite large: 12 equal temperament    | Five-limit tuning    | Pythagorean tuning    | Werckmeister temperament    | Just intonation    | Meantone temperament    | Regular temperament    | Equal temperament

Exercise #1
Six different versions of the same twelve-second passage are presented in the grid.
 * 1) The first row is in the key of C with the columns denoting the tuning method. The columns are as follows
 * 2) "Just" tuning represents a variation the attempt to create perfect harmonies (whole number fractions) described above.  This attempt assumes that the piece is played in the key of C.  (See Just intonation for more information)
 * 3) "Werckmeister" tuning is a complicated effort to compensate for the flaws associated with just tuning.  (See Werckmeister temperament for more information)
 * 4) "Equal" tuning uses powers of $$2^{n/12}$$ to establish all the pitches. (See 12 equal temperament for more information)
 * 5) The second row also presents the passage in the key of C. But this time, the piano has been tuned to the "wrong" key.  It's as if the piano tuner mistakenly believed that a piece in the key of F# would be played.  The performer and audience might have praised the work of this tuner, but only if prelude had been written in the key of F#.  You might hear the difference for the "Just" and "Werckmeister" temperaments, but expect not to hear a difference for "Equal" tuning.  All major and minor keys are the same in that temperament.  In other words, the same collection of "Equal" pitches is used in all major and minor keys.

Outcome of Exercise #1
Do the two versions with "equal" temperament sound identical? Do you perceive all of the other four passages to be different?


 * 1) If you answered "yes" to both questions, you are done. There is no need to do Exercise #2.
 * 2) If you were unable to answer "yes" to both questions, see the next exercise helps train your ear.

Exercise #2
I cannot hear different temperaments if I am tired or distracted. This is an compares only the Just and Equal temperaments, both in the "wrong" key of F-sharp, where the Just temperament sounds quite bad. Randomly click one of the links in the list below. Half of them link to the "Just" temperament and half link to an "Equal" temperament. - 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 -
 * 1) As you click the link, either close your eyes, or avoid looking at the URL at the top of your computer screen.
 * 2) Guess whether it sounds good or bad.
 * 3) Check your guess by look last portion at the URL:
 * The temperament was "Just" (bad) if the URL ends with Prelude_F_sharp_Equal.mp3
 * The temperament was "Equal" (good) if the URL ends with Prelude_F_sharp_Just.mp3


 * If you find yoursel telling the difference, go back and re-examine the following three temperaments:

C_Just     C_Werckmeister C_Equal

The one you are most likely to perceive as being different is the "Equal" temperament. There is no guarantee that you will find it better or worse than the two other temperaments.

Appendix

 * I have some leftover material in Draft:Sing free/Prelude and Fugue in C major (ear training)