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

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{{center|'''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 [[w:Prelude and Fugue in C major|''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.

[[File:Bach C Major Prelude Werckmeister.ogg|thumb|left|200px|[[w:Werckmeister_temperament|Werckmeister tuning.]]]][[File:Bach C Major Prelude Equal.ogg|thumb|right|200px|Equal tuning]] {{center|&larr;Two nearly identical renditions of a two-minute prelude.&rarr;}}
{{center|A much larger collection of [[w:Prelude and Fugue in C major, BWV 846|this prelude]] recorded at different musical temperaments can be found at [[Wikipedia:special:permalink/1066345999#Further_media]]}}

==A simple attempt to create a "just" scale==
[[File:Natural piano scale.svg|thumb|275px|A keyboard illustrating the ratios associated with musical intervals.]]
:''See also on Wikipedia: [[w:Just Intonation|Just Intonation]], [[w:Interval (music)|Interval (music)]], as well as the article [[w:Consonance and dissonance|Consonance and dissonance]]''

One way to introduce the mathematics musical harmony is to abandon the tradition use of [[w:Concert pitch|440A]] and instead "invent" a standard where the pitch for C is set at 360 Hz ([[w:simple:Hertz|Hertz]]).<ref>"Hertz" or Hz is a technical term for "cycles per second"</ref>  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 [[w:Consonance and dissonance|consonant ]] with 360C have frequencies that are also [[w:Whole number|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 ===
{| class="wikitable floatright"
! <math>\mathcal N</math> 
! <math>\mathcal R</math> 
! <math>\mathcal P</math> 
! <math>\mathcal E</math> 
! <math>\mathcal C</math> 
|-
| C || ***|| 360 || 360 || 0
|-
| E&flat; || 6/5 || 432 || 428.1 || -16
|-
| E || 5/4 || 450 || 453.6 || 14
|-
| F || 4/3 || 480 || 480.5 || 2
|-
| G || 3/2 || 540 || 539.4 || -2
|-
| A&flat; || 8/5 || 576 || 571.5 || -14
|-
| A || 5/3 || 600 || 605.4 || 16
|-
| C || 2/1 || 720 || 720 || 0
|}
[[File:Harpsichord.9023840.jpg|200px|thumb|This [[w:special:permalink/1032835651|split sharp]] is designed to allow the instrument to play a note at one of two slightly different pitches.]][[File:Syntonic comma on C.mid|thumb|These [[w:Syntonic comma|two notes]] are about 1.25% apart.]] 
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:  <math>\mathcal N</math> denotes the name of the '''note''' (as used by musicians.) &nbsp; <math>\mathcal R</math> denotes the ratio that makes the interval just. 
&nbsp; <math>\mathcal P</math> denotes the '''pitch''', where we continue with our non-standard convention 360 Hz corresponds to 
C .&nbsp; <math>\mathcal E</math> would be the pitch if the piano was instead tuned to equal temperament (as are virtually all keyboard instruments today). &nbsp; <math>\mathcal C</math> stands for '''cents''', and represents the discrepancy between our just and equal temperaments.  An easy way to understand <math>\mathcal C</math> 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):

:<big>''360&times;10&divide;9 = 400''</big>

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

:<big>''360&times;9&divide;8 = 405''</big>

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

:<big>''360&times;2<sup>2/12</sup> = 404.086...''</big>

The "dot-dot-dot" means the sequence of digits never ends, but [[w:simple:Rounding|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%.<ref>The percent increase in frequency for any pitch raised by a half tone is 5.9463094...</ref>  In other words, if 400 Hz is a reasonable pitch for D, then D-sharp would have a frequency of roughly,

:<big>''400&times;1.06 = 424''</big>

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 [[w:permalink|permalink]]"
*[[w:special:permalink/1056806428#Systems_for_the_twelve-note_chromatic_scale|w:Musical_tuning#Systems_for_the_twelve-note_chromatic_scale (permalink)]]

The number of  Wikipedia articles on this subject is quite large:<br>
[[w:12 equal temperament|12 equal temperament]] {{spaces|2}}'''|'''{{spaces|2}}
[[w:Five-limit tuning|Five-limit tuning]] {{spaces|2}}'''|'''{{spaces|2}}
[[w:Pythagorean tuning|Pythagorean tuning]] {{spaces|2}}'''|'''{{spaces|2}}
[[w: Werckmeister temperament| Werckmeister temperament]] {{spaces|2}}'''|'''{{spaces|2}}
[[w:Just intonation|Just intonation]] {{spaces|2}}'''|'''{{spaces|2}}
[[w:Meantone temperament|Meantone temperament]] {{spaces|2}}'''|'''{{spaces|2}}
[[w:Regular temperament|Regular temperament]] {{spaces|2}}'''|'''{{spaces|2}}
[[w:Equal temperament|Equal temperament]]

==Ear training==
{| class="wikitable floatright" style=text-align:center float="right"
|+ ''Three temperaments in different keys''<ref>http://www.yuvalnov.org/temperament</ref>
|-
|   *  
| Just
| Werckmeister
| Equal
|-
| C
|[https://www.yuvalnov.org/temperament/Prelude_C_Just.mp3 C_Just]
|[https://www.yuvalnov.org/temperament/Prelude_C_Werckmeister.mp3 C_Werckmeister]
|[https://www.yuvalnov.org/temperament/Prelude_C_Equal.mp3 C_Equal]
|-
| F#
|[https://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 F#_Just] 
|[https://www.yuvalnov.org/temperament/Prelude_F_sharp_Werckmeister.mp3 F# Werckmeister]
|[https://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 F#_Equal]
|-
| colspan="4" | A much more musical collection is shown below:
|}

{{Listen|type=music|pos=right|header=[[w:special:permalink/1066345999#Further_media|Lifted from Wikipedia]]
|filename=Bach C Major Prelude Equal.ogg|title=equal
|filename2=Prelude 1, Kirnberger temperament.ogg|title2=Kirnberger
|filename3=Bach C Major Prelude Werckmeister.ogg|title3=Werckmeister III 
|filename4=Prelude 1, Werckmeister temperament.ogg|title4=Werkmeister}}

===Exercise #1===
Six different versions of the same twelve-second passage are presented in the grid. 
# The first row is in the key of C with the columns denoting the tuning method. The columns are as follows
## "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 [[w:Just intonation|Just intonation]] for more information)''
## "Werckmeister" tuning is a complicated effort to compensate for the flaws associated with just tuning.  ''(See [[w: Werckmeister temperament|Werckmeister temperament]] for more information)''
## "Equal" tuning uses powers of <math>2^{n/12}</math> to establish all the pitches. ''(See [[w:12 equal temperament|12 equal temperament]] for more information)''
#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?  

#If you answered "yes" to both questions, you are done.  '''There is no need to do Exercise #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.<ref>You might wonder why a tired or distracted person would want to listen to Bach.  I was able to teach a bit about the physics of music in certain courses.  Sometimes I was a bit worn near the end of the day.</ref>  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.  
-----
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 1]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 2]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 3]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 4]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 5]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 6]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 7]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 8]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 9]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 10]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 11]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 12]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 13]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 14]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 15]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 16]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 17]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 18]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 19]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 20]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 21]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 22]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 23]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 24]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 25]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 26]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 27]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 28]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 29]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 30]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 31]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 32]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 33]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 34]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 35]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 36]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 37]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 38]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 39]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 40]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 41]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 42]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 43]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 44]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 45]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 46]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 47]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 48]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Equal.mp3 49]{{spaces|3}}
[http://www.yuvalnov.org/temperament/Prelude_F_sharp_Just.mp3 50]{{spaces|3}}
-----
#As you click the link, either close your eyes, or avoid looking at the [[w:URL|URL]] at the top of your computer screen.
#Guess whether it sounds good or bad.
#Check your guess by look last portion at the URL:
:*The temperament was "Just" (bad) if the URL ends with <pre>Prelude_F_sharp_Equal.mp3</pre>
:*The temperament was "Equal" (good) if the URL ends with <pre>Prelude_F_sharp_Just.mp3</pre>
* If you find yoursel telling the difference, go back and re-examine the following three temperaments:

[https://www.yuvalnov.org/temperament/Prelude_C_Just.mp3 C_Just]  &nbsp; &nbsp; [https://www.yuvalnov.org/temperament/Prelude_C_Werckmeister.mp3 C_Werckmeister] &nbsp; &nbsp;
[https://www.yuvalnov.org/temperament/Prelude_C_Equal.mp3 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.

==See also==

''See also [https://www.kylegann.com/tuning.html kylegann.com/tuning.html]''

==Appendix==
{{subpages}}  
*I have some leftover material in [[Draft:Sing free/Prelude and Fugue in C major (ear training)]]