ScottOpus PRODUCTIONS
ScottOpus PRODUCTIONS
TOPIC:- Pythagorean Tuning (The Cycle of Fifths) by Kerry R. Scott
As indicated in the Topic Musical Tuning the Greek Philosopher Pythagoras (?580 - ?500 B.C.) was first credited with recognising that the Harmonic Series, especially the lower harmonics, were conducive to the tuning of string instruments. He went on to divise a system of tuning that utilized the interval of the octave and fifth (the interval between the second Harmonic and the Third Haromic) (See Fig 1. in the Topic "Musical Tuning"). Pythagoras considered the intervals of the octave and the fifth (the first second and third harmonic) as "perfect". His assumption was that if the upper harmonics of both pitches had similar or the same frequencies then they could be considered "consonent" (see the Topic Consonance and Disonance.) and it was therefore the first, second, third and (perhaps) the fourth harmonics that created the "most" consonance intervals and were to his mind therefore "Perfect". In order to develop his tuning system Pythagoras piled a series of these perfect fifths on top of each other in an ascending order and then did the same in a descending order. It should be noted that in order to do this he considered each frequency of each member of his "pile" as a fundamental for the next note. This description, to be sure, simplifies his tuning system to the extreme but does, if nothing else, describe simply the process he applied to the tuning of musical instruments. It must be remembered that Pythagoras was mostly interested in the theoretical application of his ideas rather than their practical application. His theories therefore constituted a mathematical rather than a practical approach to the tuning of musical instruments.
Although we attribute the tuning system based on fifths to Pythagoras it would be wrong to consider him the "inventor" of the system as many others were involved in the development of the system and indeed it was not until Medieval times (documented in the Ninth and Tenth Centuries) that a system evolved totally relying on Perfect Fifths. It is not the intention of this article to trace the historical development of tuning systems, particular the cycle of fifths and its role in tuning systems, but rather clarify the theoretical approach to this confusing subject.
Jumping to the Fifteenth and Sixteenth centuries, therefore, we find the system extended to encompass 12 or more members -- this was called "The Cycle of Fifths". A rudimentary understanding of the Cycle of Fifths is, I believe, necessary to fully understand the tuning systems indicated above, and therefore there follows a short discussion of the Cycle of Fifths as applicable to the Pythagorean Tuning System.
Below can be seen the cycle of fifths starting on C in musical notation:-
Fig. 1. - The Cycle of Fifths A more simple way of observing the cycle of fifths is by condensing them into one octave as shown below:-
Fig. 2. - The Cycle of Fifths Condensed In order to adequately clarify the tuning confusion though it is necessary to understand that with the development of Polyphony during the Thirteenth, Fourteenth and Fiftheenth Centuries, composers began seriously to explore the "theoretical" (rather than practical) possibilities of pitch development. That is, they wanted their music to take the listener on a more challanging emotional journey, to experience new and different sounds and to provide different kinds of intellectual stimuli. They were significanly hampered in this regard by a number factors but particularly the Cycle of Fifths. What is wrong with the Cycle of Fifths? In order to answer this question we must look at the ways the Cycle of Fifths is constructed and to do that we must return to the "Harmonic Series".
Figure 3. "the Cycle of Fifths - construction 1." (seen below) is an attempt to construct the Cycle of Fifths directly from the Harmonic Series. It is an abortive attempt and riddled with inaccuracies and bears little relationship with the tuning of, for example, the modern Piano. It is, however, a logical approach to creating a Cycle of Fifths and is included here to demonstrate that any modern day tuning system, particularly "Equal Temperamant", is an evolved system with the Musical Alphabet and music notation as its basis. From a purly Mathematical and theoretical position the Pythagorian Tuning System may be "neat and tidy" but does not relate to what we hear. Figure 3. uses the Harmonic Series seen in "Figure 1. - the Harmonic Series" in the Topic "Musical Tuning" to construct a Cycle of Fifths starting with C. It uses the frequencies of the applicable Harmonic and transposes the frequency down the required octaves to arrive at the note within the Cycle of Fifths. As can be seen this seems to work for most of the notes with the notable exceptions of the sixth and twelfth "cycle of fifths" members.
Circle of
Fifths
Number- Original
Harmonic
Series
Number -FREQ. - Octave
Transposition -FREQ. -- Note.
Equivilant --Table assumes a Fundamental (Harmonic 1.) of C -- 67.69 Hz.
-1.- 1. 67.69 Hz. None. 67.69 Hz. - C. - -2.- 3. 132.38 Hz. 1↓. 101.54 Hz. - G. - -3.- 9. 609.21 Hz. 3↓. 76.15 Hz. - D. - -4.- 13. 879.97 Hz. 3↓. 110.00 Hz. - A. - -5.- 10. 676.90 Hz. 3↓. 84.61 Hz. - E. - -6.- 59*
60**
59.5***3993.71 Hz.*
4061.40 Hz.**
4027.56 Hz.***5↓. 124.80 Hz.*
126.92 Hz. **
125.86 Hz. ***
118.46 Hz. ****- B. - -7.- 11. 744.59 Hz. 3↓. 93.07 Hz. - F#. - -8.- 17. 1150.73 Hz. 4↓. 71.92 Hz. - C#. - -9.- 25. 1692.25 Hz. 4↓. 80.38 Hz. - G#. - -10.- 38. 2572.22 Hz. 5↓. 80.38 Hz. - D#. - -11.- 27. 1827.63 Hz. 4↓. 114.23 Hz. - A#. - -12.- 41*****
42*****2775.29 Hz.*****
2842.98 Hz.*****5↓. 86.75 Hz.*****
88.84 Hz. *****- E#(E+).*****
E#(F).***** --13.- 64 4332.16 Hz. 6↓. 135.38 Hz. - B#(C). - Fig. 3. - The Cycle of Fifths - Construction I. * From Fig. 3 it can be seen that the fifty ninth harmonic produces a frequency of 3993.71 Hz. which results in what I consider a slightly flat B.
** From Fig. 3. it can be seen that the sixtieth harmonic produces a frequency of 4061.40 Hz. which results in what I consider a slightly sharp B. In other words there is no Harmonic Series Frequency equivalent, within the first 6 octaves of the Harmonic Series with a Fundamental of C, to what we know as the note "B" .
*** Neither the fifty ninth nor the sixtieth harmonic provide a frequency that is applicable for this cycle of fifths. Therefore I have taken the difference between the fifty ninth and sixtieth harmonic, made the applicable octave transposition with the resulting frequency of 125.86 Hz.
**** With regard to the indication of 118.46 Hz. applicable to member 6 of the Cycle of Fifths it should be noted that between the first and second, and third and fourth, cycle of fifth members there is a difference of 33.85 Hz. (it is also worthy of note that between the second and third, and the fourth and fifth members there is a difference of 25.39 Hz.). In order to continue this pattern the frequency of the sixth member of the Cycle of fifths should be 118.46 Hz. and not the 125.86 Hz. as calculated, albeit indirectly, from the Harmonic Series.
***** The twelfth member of this constructed cycle of fifths could be considered as a result of either the forty first or the forty second harmonic. However as they are so close -- a difference of only 67.69 Hz. -- both results have been shown here.
Music/Instrument Tuning
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A listing/portfolio of the music compositions of Kerry R. Scott
Further Information on The Tudor Rose School of Music
Further information on the Music CD -- Bubble and Squeak
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Orchestral Suite from Mass 2100 and Mass 2100 original performance edition.
Further information on the Music CD -- Smphony No. ! -- Soundscapes of a Forgotten Britian
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