ScottOpus PRODUCTIONS
ScottOpus PRODUCTIONS
TOPIC:- Musical Tuning by Kerry R. Scott
***It is highly recommended that the Topics Pitch and "Harmonic Series" be read and understood before embarking on this Topic and its links.***
Unfortunately describing the tuning of our present day instruments and by extension the way humans perceive pitch is, to say the least, complicated. Indeed the subject's documentation is riddled with misunderstanding, myth, anomaly and the hob-goblin of all clarity, ignorance.
In Music there are four ways of producing a musical pitch. They are from human, mechanical or electronic instruments and are as follows:-
A vibrating string. A resonating column of air. A vibrating membrane. An oscillating electrical signal.
The first three (the mechanical) ways of producing a pitch automatically and naturally produce a series of pitches (frequencies) called the Harmonic Series. This series of frequencies is more extensively discussed in the Topic "Harmonic Series". It is this series of notes that all of the western systems of tuning have emanated from and indeed it is this series of notes that overall indicate the kinds of sounds (timbre) instruments make
It is, in fact, also this series of frequencies that is the basis for the confusion, misunderstanding and, as one writer put it, "incomprehensible jargon" that bedevils sensible cognizance of western tunings systems.
Before proceeding much further into the discussion of the inequalities and idiosyncrasies of western tuning systems and in order for any discussion on tuning or pitch organization to be meaningful, it is necessary to marry or join the scientific language of Acoustics to the artistic language of Music. That is, the scientific measurement of Frequency -- Hertz -- must be equivalent to the musical measurement of Pitch -- The musical Alphabet.
This is not as easy a task as it may first appear. In reality any note of the musical alphabet can be set to any frequency so long as the relationship between the notes in the musical alphabet remain the same.
In Ancient Greek times both Plato (?427 - ?347.B.C.) and Aristotle (384 - 322 B.C.), who were influenced by the Pythagorean School of mathematical tuning, believed that the tuning of instruments should be calculated whereas Aristoxenus (c. 320 B.C.) believed musical tuning and the intervals created by the musical alphabet should be determined by ear.
In the Baroque Period of music (1600 - 1750) different tuning frequencies were used for the performance of choral and instrumental compositions. Furthermore tuning frequencies varied for different organs and harpsichords even within the same city. The difference,in some cases resultated in an "A" equalling 380 Hz. on one organ to an "A" equalling 480 Hz. on another. In some cases a difference of "perfect fourth" was noted between instruments found in the home and those in a major churchs within the same town.
It was not until 1955 that the A, above "Middle C" was generally accepted as equalling 440 Hz. and thus became the international standard of tuning.
It is this standard that the following discussion and data is predicated on.
Generally the credit for discovering that the first part of the "Harmonic Series" was conducive to tuning strings on a musical instrument was the Greek Philosopher Pythagoras (?580 - ?500 B.C.). His theory was based upon the first three frequencies of the Harmonic Series, particlularly the "third harmonic", and relied on the relationship between the harmonics (ratios) rather than the frequencies themselves. Pythagoras condsidered these harmonics and the intervals between (the Octave, the Fifth and the Fourth) as "Perfect" Intervals. This "ratio" or "interval" approach to tuning continues to this day. For the musical or mathamatical layman, however, in this topic (and the topics on the individual tuning systems seen below) I am going to limit myself to explaining tuning sytems in simple arithmetical terms.
It should be noted, here, that notation systems in music developed separately from that of the tuning systems. Music notation was and is the way a composer communicates his or her wishes to the performer (see Topic What is Music) whereas tunings systems were developed to adequately, precisly and technically replicate the tones (pitches) that a musical instrument is required to play. Although separate, the development of tuning systems were, in fact, driven by the creativity and inovation of the composers requirments. Composers developed systems of notation and as such, paid little attention to the constraints of the "Harmonic Series". Their creation of the Musical Alphabet, intervals, triads, chords, modulation, keys, and scales, were all with regard to their exploration into new musical avenues and had little regard to the efforts of instrument designers and performers who vainly tried to keep up with the composer's requirments.
Amongst a mulitude of others, there are two particular examples of this juxtaposition of the composer's inovation versus instrument designers aspirations.
The first is that of "Das Wohlemperierte Klavier" by J. S. Bach (1685 - 1750). This work is generally trumpeted as the heralding of the use of the "Equal Temperament" system of tuning and it is also put forward that the work was apparently written to deomonstrate the effectivness of this system. A major part of "Das Wohlemperierte Klavier" was in fact written in 1722, well before the general acceptance of equal tempermant system and therefore, if anything, the work was the impetus for the "Equal Temperament" system to be created rather than the other way around.
The second is as outlined in the Topic "Natural Trumpet" on this site."It is thought that the Trumpet Concerto of Joseph Haydn (1732 -- 1809), completed in the year 1796 was composed for the Keyed Trumpet notwithstanding that the Keyed Trumpet only appeared in the first years of the Nineteenth Century. It is, however, true that the work could not have possibly been performed on the traditional Natural Trumpet.'
Again the composer's inovation led to the technical development of the instrument or tuning system.
The composers continued (and continue) to adjust and change notation systems to accomodate their needs and so it is no small wonder that the only "natural" or "mathematical" system "The Harmonic Series", now is far removed from the sounds played by musical instruments.
Before discussing the individual and sometimes controversial different forms of tunings, and as an example of how far the "Harmonic Series" differs from our now familar tunings, I draw your attention to the table below titled Fig 1. - The Harmonic Series. This is an extended version of the table seen in the Topic "Harmonic Series". In this edition I have extended the range of the series to the sixtyfourth Harmonic. In addition I have included the octave and made an effort to indicate the equivilant note name with the Harmonic number. It can immediatly be seen that, with the exception of the first two or three octaves there are considerable differences between the Frequencies (pitches) that are extant in the "Harmonic Series" and those that we epect to hear or find "in tune". Furthermore in the Harmonic Series starting with Octave 5 there are more than twelve notes per octave.
When there is a number of Harmonics that seem to be equivilant to a particular note value, I have indicated whether I think they are flat or sharp to the "true" note by the inclusion of "minus" and "plus" signs after the letter name. Obviously two minus signs indicate that the harmonic note is flatter than one minus sign and the same configuration relates to the plus sign.
Hm# - Freq - Oct. -- Note. -- Table assumes a Fundamental (Harmonic 1.) of C -- 67.69 Hz.*
-1.- 67.69 Hz. -- 1 -- - C. - -2.- 135.38 Hz. -- 2 -- - C. - -3.- 203.07 Hz. -- 2 -- - G. - -4.- 270.76 Hz. -- 3 -- - C. - -5.- 338.45 Hz. -- 3 -- - E. - -6.- 406.14 Hz. -- 3 -- - G. - -7.- 473.83 Hz. -- 3 -- - B(f)-. - -8.- 541.52 Hz. -- 4 -- - C. - -9.- 609.21 Hz. -- 4 -- - D. - -10.- 676.9 Hz. -- 4 -- - E. - -11.- 744.59 Hz. -- 4 -- - F#-. - -12.- 812.28 Hz. -- 4 -- - G. - -13.- 879.97 Hz. -- 4 -- - A. - -14.- 947.66 Hz. -- 4 -- - B(f)-. - -15.- 1015.35 Hz. -- 4 -- - B+. - -16.- 1083.04 Hz. -- 5 -- - C. - -17.- 1150.73 Hz. -- 5 -- - C#. - -18.- 1218.42 Hz. -- 5 -- - D. - -19.- 1286.11 Hz. -- 5 -- - D#. - -20.- 1353.80 Hz. -- 5 -- - E. - -21.- 1421.49 Hz. -- 5 -- - F. - -22.- 1489.18 Hz. -- 5 -- - F#-. - -23.- 1556.87 Hz. -- 5 -- - F#+. - -24.- 1624.56 Hz. -- 5 -- - G. - -25.- 1692.25 Hz. -- 5 -- - G#. - -26.- 1759.94 Hz. -- 5 -- - A. - -27.- 1827.63 Hz. -- 5 -- - A#. - -28.- 1895.32 Hz. -- 5 -- - B(f)--. - -29.- 1963.01 Hz. -- 5 -- - B(f)-. - -30.- 2030.70 Hz. -- 5 -- - B+. - -31.- 2098.39 Hz. -- 5 -- - B++. - -32.- 2166.08 Hz. -- 6 -- - C. - -33.- 2233.77 Hz. -- 6 -- - C#-. - -34.- 2301.46 Hz. -- 6 -- - C#. - -35.- 2369.15 Hz. -- 6 -- - C#+. - -36.- 2436.84 Hz. -- 6 -- - D. - -37.- 2504.53 Hz. -- 6 -- - D#-. - -38.- 2572.22 Hz. -- 6 -- - D#. - -39.- 2639.91 Hz. -- 6 -- - D#+. - -40.- 2707.60 Hz. -- 6 -- - E. - -41.- 2775.29 Hz. -- 6 -- - E+. - -42.- 2842.98 Hz. -- 6 -- - F. - -43.- 2910.67 Hz. -- 6 -- - F#--. - -44.- 2978.36 Hz. -- 6 -- - F#-. - -45.- 3046.05 Hz. -- 6 -- - F#. - -46.- 3113.74 Hz. -- 6 -- - F#+. - -47.- 3181.43 Hz. -- 6 -- - F#++. - -48.- 3249.12 Hz. -- 6 -- - G. - -49.- 3318.61 Hz. -- 6 -- - G#-. - -50.- 3384.50 Hz. -- 6 -- - G#. - -51.- 3452.19 Hz. -- 6 -- - G#+. - -52.- 3518.88 Hz. -- 6 -- - A. - -53.- 3587.57 Hz. -- 6 -- - A+. - -54.- 3655.26 Hz. -- 6 -- - A#. - -55.- 3722.95 Hz. -- 6 -- - A#+. - -56.- 3790.64 Hz. -- 6 -- - A#++. - -57.- 3858.33 Hz. -- 6 -- - A#+++. - -58.- 3926.02 Hz. -- 6 -- - B--. - -59.- 3993.71 Hz. -- 6 -- - B-. - -60.- 4061.40 Hz. -- 6 -- - B+. - -61.- 4129.09 Hz. -- 6 -- - B++. - -62.- 4196.78 Hz. -- 6 -- - B+++. - -63.- 4264.47 Hz. -- 6 -- - C-. - -64.- 4332.16 Hz. -- 7 -- - C. -
Fig. 1. - The Harmonic Series * This frequency is determined from the international standard of A=440 Hz. In the Harmonic Series with the Fundamental of "C", the "A" appears as the 13th Harmonic. Therefore 440 Hz./13 = 67.69 Hz. Fundamental.
Most, if not all, of the tuning systems that exist today that encompass up to twelve notes to an octave, are based in some way on the work of Pythagoras and the tuning system known as "Pythyagorean Tuning".
The following is a list of the more well known tuning systems that have as a basis the first harmonics of the Harmonic Series and the observations of Pythagoras.Pythagorean Tuning System.(Cycle of Fifths)
Just Intonation.
Equal Temperament.
Mean Tone System. Each of the tunings systems indicated above have individual Topic articles associated with each one. For a detailed description of each use the links below or click on the titles above.
Pythagorean Tuning System (Cycle of Fifths)
Further information on Kerry R. Scott's life and Compositions
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
Further information on the Music CD -- Rattle and Rhyme
Further information on the Music CD -- Beyond the Virtual Creation
Further information on the Music CD -- Brandy Butter, Brass and Bells
Further information on the Music CD -- The Old, The New, and an Eclectic Medley
Further information on the Music CD -- Mass 2100
Orchestral Suite from Mass 2100 and Mass 2100 original performance edition.
Further information on the Music CD -- Smphony No. ! -- Soundscapes of a Forgotten Britian
Further information on the live recording of the first performance of Mass 2100
Further information on compositional and composition commissions.
Further information on the stories and writings of Kerry R. Scott.
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