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The Tempered Natural Harmonic Tuning System

This article explains the issues that go into making the standard, modern tuning system that all modern music utilises. It's known as the Tempered Natural Harmonic Tuning System (TNHTS). Beginning with the occurrence of harmonics in most natural sounds, we show the relationship between these and the pitches of at least some of the notes used in musical scales of possibly the majority of cultures on earth. The Pythagorean tuning system is explored in some depth, and is noted as the foundation for the TNHTS in use today. Data sheets describing all relevant tuning systems are linked to from this document.

 

Natural Harmonics and Notes

Every musical sound produced on a tuned instrument contains harmonics. The amplitudes and relative pitches of each harmonic component determine the most important tonal characteristics of the instrument. While the range of musical timbres is enormous, the possibilities viewed in these rather dry terms seem extremely limited. This is because, as the term harmonic indicates, The frequencies that accompany – and blend with – the fundamental frequency must fit neatly with it. It is a property of the physical world that a natural vibration will tend to carry only components that harmonise with it. (This does not always occur, of course, as metallic objects will tend to produce so many inharmonics as to hide any semblance of a fundamental frequency. Some instruments, such as to be found in a gamelan orchestra, make use of near-harmonics, making a bright, garish sound). But as far as any melodically useful sound is concerned this is preponderantly true. In terms of wave theory, to harmonise, as harmonisation is known in natural systems, means that one wave must be able to fit a whole number of cycles within the other. (This implies that two coincident waves should be doing the same thing at the same time, at regular intervals. That is, at the precise moment when one reaches a peak or a trough the other one also does. This strictly must occur at regular intervals). The ratio of frequencies between two such waves will always be an integer. That is, the higher frequency is a divisor of the lower frequency. Notes pairs of non-integer ratios do harmonise, but it is rare to find such occurring entirely of their own accord in the one system.

This, then leaves us with a small number of pitches to play with while remaining within the range of easily digestible and melodically useful frequencies. But, as we said, the possibilities are nonetheless enormous. The first (fundamental frequency x 2) is one octave; the second (x3), an octave and a half. The third (x4) is two octaves, and the forth (x5) is two and a quarter octaves. The fifth (x6) is three octaves, and the sixth (x7) is three and one sixth octaves. On some instruments like guitar, techniques are available to isolate harmonics. If you play such an instrument, then you will have heard these harmonics, and possibly number of others. These, as you know, in their allotted amplitudes, are the auditory components that provide the foundation of your instrument’s timbre.

 

Building Blocks of Music

The fact that these sounds are consonant has not been lost on ancient or traditional people groups. According to Helmholz, the music of every people group known (at the time of his writing in the 1870’s) is built upon concepts of octaves, fifths and fourths. You may add to this the preponderant use of major and minor thirds. The most obvious exception to this is the traditional music of the Australian Aborigines and the people of the Kalahari region, which is much more rhythmic and non-harmonic. Nevertheless, all of the notes mentioned are based on naturally occurring harmonics whose usefulness can be learned experientially. It is certainly not so that all notes of all tuning systems have been derived this way, as not all tuning systems follow a harmonic pattern. While Helmholz claims that ancient music knew nothing whatsoever of harmony or polyphony, he does maintain that the human mind has a natural affinity for harmony, which consistently allowed for the consonance of pitches to be recognised. Thus, the notion of consonance pointed only towards the construction of melodies, and not to the actual interplay of notes. Nevertheless, when simple instruments such as the harp or lyre were created and their notes were selected, the tuning systems that were employed were built upon what Helmholz argues was a purely intuitive and arbitrary basis. Arbitration allows for the emergence of the many tuning systems that do not consistently follow natural harmonics. But whatever path the ancient artisans and bards followed, the standardisation of instrument designs made certain natural harmonics available at melodically useful pitches, along side the note that would have been their fundamental frequency in any vibrating natural object. To work purely on the basis of natural harmonics (knowingly or not) yields what is best termed, a Natural Harmonic Tuning System (NHTS).

An example of systematic design (as arbitration) in the construction of a NHTS is the Pythagorean system. Pythagoras followed a construct now known as the circle of fifths. Starting with a certain pitch, and raising the frequency by one half (or shortening a vibrating string by one third), consistently 12 times over, one arrives eventually at a pitch extremely close to the 7th octave above the original. En route, one has produced all 12 notes present in any European system. By dropping every note to the same octave as the initial note, a complete chromatic scale is created, very close to that which we know today. This last note, however, is not a proper octave, but a note known as "B sharp". Impossible, you may think, but it is a normal artifact of the Pythagorean system. The pitch of this note is in fact 24 cents higher than the proper seventh octave. Proceeding through the same pattern downwards, one obtains a mirror-image result, ending with a note called "D flat-flat", 24 cents below the 7th lower octave. (24 cents, incidentally, is a sufficient interval to produce very noticeable beating, being almost an eighth tone). In fact, every interval is microtonally inconsistent. This makes the Pythagorean system inconsistent within itself, and awkward for all but the simplest music. Having experimented a little with composing to a Pythagorean system (the upward version) I can say Pythagorean melodies sound weird as hell to an ear accustomed to the TNHTS. Key changes would produce a change in the chordal quality of a piece. History seems to show, however, that this was the dominant system in European music until the time of Gioseffo Zarlino, the creator of the modern tuning system, in the 16th century. It does, in fact, form the basis for all modern music, with the only important difference being the usage of even tempering to create consistent note intervals.

 

Tempering: The Institutionalised Hack.

Finally the last step in explaining the derivation of the TNHTS. Tempering is the practice of slightly adjusting the pitch of a sound-producing component on an instrument - such as a string. This is done in two different instances. One occurs in the act of tuning, where inconsistencies in the diameter or material of a string, or in some other instrumental property, make for improper positioning or pitching of the some notes relative to others. Thus a compromise is sought wherewith beating effects and unmusicality are minimised for the most common playing positions and note sequences or chord inversions.

The other instance of tempering is used in the standardisation of some tuning systems (or of one tuning system only…), and therefore holds influence in some instrument designs. The object here is to obtain even spacing (in frequency or in position) between all notes and thus to negate the difficulties associated with odd intervals. Tempering does mean that all note intervals, excepting only whole octaves, are slightly dissonant. Yet without this adjustment, chords sound different depending on the key in which they are played. Key changes are also effected similarly. The Pythagorean system, when crossing to the next octave above or below the frequency on which it is designed, incorporates an entirely new note which fills a gap created by what would otherwise be a very large interval. The even tempering process that makes the difference between that and the modern TNHTS allows this extra note to be dropped. All notes are spaced in such a way that they are so close to consonant intervals that beating effects are not discernible where they are not expected. If you look at the data sheet on the TNHTS as per MIDI specifications, then you will notice that even the dominant (the fifth note of the scale) is a little way out of perfect consonance.

 

Conclusion.

A tempered system, then, is much easier to work with in performance, although for purists, the drawback is that not all intervals are precisely consonant. The major and minor thirds of the TNHTS are the most pertinent examples: A major chord, for example, played in a true consonant system sounds smoother, or "more major" than in a tempered system. The equivalent is true for a minor chord. The difference is, for the unaccustomed ear, quite striking, even though it is only a matter of a few Herz at melodic frequencies.



 

Footnotes

[1] NHTS's and true consonants.

A difference exists between any systematically (or arbitrarily) derived natural harmonic tuning system and what might be referred to as a true consonant system. The former is designed to be able to accommodate, to some extent, key changes and chordal structures that overlap octave boundaries (based on the designated fundamental). What might be called a true consonant system bears no musical flexibility in mind, but features notes with a perfect harmonic relationship to the fundamental. It is not constructed by any contrived means, but adheres strictly to a range of mathematically simple intervals. Two examples of such may be found here.

[2] The Circle of Fifths.

As a former guitar teacher, I often showed my students how to construct a chord progression to back almost any kind of simple, modern song by introducing them to the chordal circle of fifths. Playing a progression in the key of C, for example, one proceeds with a pattern involving chords of C, G and F, where G is the succeeding member of the circle and F the preceding member. The complete circle incorporates one chord for all 12 notes, and works equally well with any combination of majors, minors, sevenths, major sevenths, suspensions and so forth. It conveniently happens also to be the basis for R&B. Incidentally, for beginners, another commonplace trick in building chord progressions is the use of relative minors and relative majors. These are chords that use the same scale as each other, but whose fundamental notes are 3 semitones apart. The fundamental of the major becomes the third of the minor, and the third of the major becomes fifth of the minor. Thus B minor can be a fitting substitute for D major, A minor for C major, E minor for G major, and so forth.


 

References

Helmholz, Hermann L. F., On the Sensations of Tone as a Physiological Basis for the Theory of Music, 4th edition, 1877. 2nd English translation by Alexander J. Ellis, 1885, Dover Publications, Inc, New York.

 

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