Sep. 26, 2015

Temperaments & Tuning, Part I

In the 6th century B.C. the Greek mathematician Pythagoras is thought to have made some acoustical experiments using a device called a monochord, with which he demonstrated the ratios of vibration frequencies for some of the musical intervals found within the octave.  In his monochord a single, long vibrating string fixed at both ends, when plucked or bowed, produced higher pitches by means of progressive shortening of the vibrating portion of the string.  To shorten the string Pythagoras used a movable bridge to divide the monochord string, first in half, then into smaller divisions or segments.

If it's supposed that the entire length of the monochord used by Pythagoras yielded the note tenor C3, then its half length in a pure ratio of 2:1 would have yielded middle C4, the octave.

Similarly, 8/15 of its length in a pure ratio of 15:8 would have yielded tenor B3, the major seventh.

3/5 of its length in a pure ratio of 5:3 would have yielded the note tenor A3, the major sixth.

2/3 of its length in a pure ratio of 3:2 would have yielded the note tenor G3, the perfect fifth.

3/4 of its length in a pure ratio of 4:3 would have yielded the note tenor F3, the perfect fourth.

4/5 of its length in a pure ratio of 5:4 would have yielded the note tenor E3, the major third.

5/6 of its length in a pure ratio of 6:5 would have yielded the note tenor Eb3, the minor third.

8/9 of its length in a pure ratio of 9:8 would have yielded tenor D3, the major second.

Musical intervals are thus expressible as fractions which are pure ratios comparing the vibration frequency of a named pitch with the vibration frequency of a constant reference pitch.  To tell what kind of interval a fraction designates today, scientists have devised a standard unit for measuring the size of perceived intervals resulting from 2 frequencies vibrating in a pure ratio.  It's a logarithmic system which mathematically defines the distance of an octave as 1200 cents.  The unit called Cent was first put forth by A.J. Ellis and was defined as 1/100th part of an equal tempered semitone.  From 1 equal tempered semitone to another the distance is always 100 cents no matter what frequencies may be involved, and the octave distance is always 1200 cents.

In any organ tuned in equal temperament the half steps within the octave would be divided into cents as follows:

Equal tempered semitone       =    100 cents

Equal tempered whole tone    =    200 cents  

Equal tempered Minor 3rd      =    300 cents

Equal tempered Major 3rd      =    400 cents

Equal tempered Perfect 4th   =    500 cents

Equal tempered Tritone         =    600 cents

Equal tempered Perfect 5th   =    700 cents

Equal tempered Minor 6th      =    800 cents

Equal tempered Major 6th      =    900 cents

Equal tempered MInor 7th      =  1000 cents

Equal tempered Major 7th      =  1100 cents

Octave                                   =  1200 cents

For any pure ratio n / p, the number of cents in the interval is always:

Cents = log ( n / p ) X 1200 / log 2

           = log ( n / p ) X 3986.3137 ...

Using this formula, the following important pure interval sizes within the octave, rounded to the nearest cent, are obtained:

Diminished 2nd                    =       41 cents  (128:125)

Minor (chromatic semitone)  =       71 cents  (25:24) 

Major (diatonic semitone)     =     112 cents  (16:15)

Minor 2nd                             =     182 cents  (10:9)

Major 2nd                             =     204 cents  (9:8)

Septimal whole tone             =     231 cents  (8:7)

Diminished 3rd                      =     245 cents  (144:125)

Augmented 2nd                     =     275 cents  (75:64)

Pythagorean Major 3rd          =     408 cents  (81:64)

Minor 3rd                              =     316 cents  (6:5)

Major 3rd                              =     386 cents  (5:4)

Diminished 4th                      =     427 cents  (32:25)

Augmented 3rd                      =     457 cents  (125:96)

Perfect 4th                            =     498 cents  (4:3)

Augmented 4th                      =     569 cents  (25:18)

Tritone                                  =     590 cents  (45:32)

Diminished 5th                       =     631 cents  (36:25)

Perfect 5th                            =     702 cents  (3:2)

Diminished 6th                      =     743 cents  (192:145)

Augmented 5th                      =     773 cents  (25:16)

Minor 6th                               =     814 cents  (8:5)

Major 6th                               =     884 cents  (5:3)

Diminished 7th                       =     925 cents  (128:75)

Augmented 6th                      =     955 cents  (125:72)

Minor 7th                               =  1018 cents  (9:5)

Major 7th                               =  1088 cents  (15:8)

Augmented 7th                       =  1159 cents  (125:64)

Octave                                   =  1200 cents  (2:1)

From this listing it can be readily determined which binary combinations of pure intervals combine arithmetically (in cents) to make up an octave.  Those which do are as follows:

Diminished 2nd (41) + Augmented 7th (1159)  =  Octave (1200)

Major semitone (112) + Major 7th (1088)        =  Octave (1200)

MInor 2nd (182) + Minor 7th (1018)                 =  Octave (1200)

Diminished 3rd (245) + Augmented 6th (955)   =  Octave (1200)

Augmented 2nd (275) + Diminished 7th (925)   =  Octave (1200)

Minor 3rd (316) + Major 6th (884)                    =  Octave (1200)

Major 3rd (386) + Minor 6th (814)                    =  Octave (1200)

Diminished 4th (427) + Augmented 5th (773)    =  Octave (1200)

Augmented 3rd (457) + Diminished 6th (743)    =  Octave (1200)

Perfect 4th (498) + Perfect 5th (702)               =  Octave (1200)

Augmented 4th (569) + Diminished 5th (631)    =  Octave (1200)

(continued in Part II)