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Discussion on microtones (and why I love a fretless.).


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What is microtonality? Put simply, microtonality relates to those intervals "between the cracks" of the keyboard or any other instrument. Since the days of Pythagoras, Western music has evolved into the highly formalised system we have today with twelve equally spaced tones within the octave. These twelve notes are so arranged that any intervals played in one key sound exactly the same in another - this is good news for our predominantly harmonic music but bad news for tuning because, with the exception of the octave, not one interval is in tune. Hang on a minute, I hear you say, when I play chords on my keyboard they sound fine to me. That's because your ears have become so accustomed to these intervals that you don't notice the errors. These "errors" arise due to compromises that have to be made in tuning any fixed pitch instrument like a piano or synth.


Unlike a violin player, who has the freedom to play an almost infinite variety of different notes by virtue of the fact that he has extremely fine control over the pitch of the vibrating string and thus use his ears to select the most appropriate one to blend in with the other instruments, keyboard players have no such facility. The pitch of their instrument is fixed - the string player can play hundreds of different intervals within the span of a single octave, the poor piano player only has twelve and these must serve him in all weathers. This is where temperament comes in; tempering seeks to make small adjustments in the tuning of intervals, such as Major 3rds and 7ths, so that when these intervals are played in different keys, they sound to our ears pretty much the same or, to put it another way, so that they retain the same level of concordancy (or even discordancy!). Why bother to do this? Well if we tuned a piano to pure intervals, we'd soon find out that while a C Major 7 chord sounds fine, playing an F# Major 7 would set our teeth on edge - in fact, in a pure temperament tuned on C, a C Major 7 chord would sound much "sweeter" than it would on a modern equally tempered keyboard.


The most important thing to bear in mind is that we judge whether an interval is consonant or dissonant solely by its effect on the ear; that and a hell of a lot of neural processing. Consider this: if you give someone two tone generators and ask him or her to adjust the relative pitch of one of them to a number of different intervals, such as a fifth, Major 3rd etc., and then you measured the interval ratios (i.e. the ratio of their respective vibration rates) of these intervals, you'd probably find that their ears had led them to produce intervals in simple whole number ratios; 2:1 - an octave, 3:2 - a fifth, 5:4 - a Major 3rd and so on. I should perhaps point something out here: although, in a broad sense, pure intervals are formed of simple whole-number ratios, there is no natural or physical law that suggests that consonant intervals cannot be formed from other, more complex ratios. In other words, there's no real reason for a scale to have only twelve notes in its span or even for the octaves to be in tune in order for it to produce consonant intervals.


Our ears are exquisitely sensitive devices - a pin dropped at our feet in a quiet room produces about one quadrillionth (0.00000000000001) of a watt of acoustic energy which moves the eardrum less than the diameter of a hydrogen molecule. An orchestra going full bore pumps out about 70 watts of which 40 watts is generated by the bass drum alone! In terms of amplitude, the ear has a useful sensitivity range of 20,000,000 to 1 - our sense of pitch discrimination is almost as good. One can even go so far as to say that our intervallic sense is a direct result of the very physiology of the ear and its associated processing devices[1].


In the history of music, it has been noted that technological advance has usually preceded developments and innovations in artistic style and musical direction - the invention of the double escapement mechanism in piano action, which allows the hammer to fall back off the strings even with the key still pressed home, permitted Listz his fiery virtuosity and allowed Chopin to write some of the most extraordinary music for the instrument. Arguably the most influential technical advance has been temperament. The earliest music we know of was homophonic, that is, consisting of single melody lines; the earliest temperament was probably Pythagorean (see below) which only really allows four truly concordant intervals thus restricting the form of the music (you could of course use discordant intervals for variety - little if any authenticated material exists from those times so who knows?).


The comparative lack of concordant interval leaps also limits modulation and duophony; no one in ancient Greece would have dreamed of either moving to the Major 3rd or placing it in parallel with the tonic as it was dissonant, but today we consider the Major 3rd consonant - weird! Let's take a quick look at some numbers and try and get a handle on all this - don't worry, you won't need a calculator just yet! We've seen how the interval between any two notes can be described in terms of their frequency ratios; for example, the octave has a ratio of 2:1 - that is to say, the frequency of one component is twice that of the other. This allows us to compare the magnitude of the intervals under scrutiny without reference to their frequencies. We can express other pure intervals in similar fashion: the fifth has a ratio of 3:2, the major third 5:4, the minor seventh 16:9 and so on. By adding ratios together (actually, being fractions we have to multiply them) we can form new intervals. Add a perfect fourth to a perfect fifth and we get an octave (4/3 x 3/2 = 12/6 or 2/1). If, when adding two ratios together, we exceed by an octave or more, multiply by 1/2 for each octave raised to get the interval back within the confines of an octave (two perfect fifths: 3/2 x 3/2 = 9/4 x 1/2 = 9/8 or Major Second)


If we examine the interval ratios of all the diatonic intervals in the puretone temperament, two curious facts emerge:













Pure Minor Second






Perfect Fifth



Pure Major Second






Pure Minor Sixth



Pure Minor Third






Pure Major Sixth



Pure Major Third






Pure Minor Seventh



Perfect Fourth






Pure Major Seventh



Pure Augmented Fourth









First, intervals which we consider concordant (pleasant-sounding) have simpler ratios than discordant (harsh-sounding) ones; generally, the higher the number, the more discordant the interval and second, all these ratios are numbers which are multiples of the prime numbers two, three and five: you may also notice that these ratios are identical to certain members of the harmonic series. Right, we've got nice simple numbers describing these intervals so what's the problem? The problem is that the interval ratios between successive degrees of the scale are not equal. This means that if you play an interval in one key, it'll sound different in another. Take a fifth, for example. Play the interval C-G in this puretone temperament and it'll sound perfectly OK but play F#-C# and the interval ratio, instead of being 3:2 is now 1024:675 or almost a fifth of a semitone too sharp (divide 45:32, the augmented fourth from C, by 32:15, the Minor Second raised one octave). Now since the ear can just about hear a difference of one hundredth of a semitone, playing this interval is going to sound distinctly odd.


So, although the puretone temperament has the benefit of greater overall concordancy, because its primary interval ratios (3:2, 4:3 etc.) lie alongside those some of the harmonic series, it will only work in one key. This limits the possibilities of harmony, an essential component of Western music of the last 400 years, thus many attempts were made by various people over the years to "re-tune" or temper the interval ratios such that as many keys as possible sounded approximately the same.





1. An interesting consideration is the phenomenon of the octave. Why is it, when we consider the audible frequency range from 20Hz to 20 KHz, we perceive a series of points along this scale that we can consider as having the same "quality" while patently being a different note? Part of the explanation may be that if we take a bi-lateral cross-section through the cochlea, that part of the ear's mechanism responsible for converting acoustic energy into electrical impulses, it reveals a spiral shape which can be described mathematically by a Fibonacci Series; the same maths govern the principles of the harmonic series. Neuro-pathology of the ear shows that octaves are decoded at the same point in each layer of the spiral. Some experts maintain that if the cochlea was a straight cone, rather than a tightly-wound spiral, we would have no perception of the octave at all; all we would hear would be a series of successively rising tones.



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Notation for microtonal scales, part 1


© 1997, John S. Allen


One of the great virtues of traditional musical notation is its consistent, logical and easily readable representation of pitch. Before discussing microtonal notation, it is useful for us to examine some of the features of conventional notation. This article briefly describes the features of standard notation and begins an exploration of how it may be expanded to include microtones.


Spatial and symbolic representation of musical pitch


A system of notation may represent musical pitch spatially, by the location of symbols on a surface -- or symbolically, by the appearance of the symbols.


An entirely spatial representation of pitch, as in a piano roll, assigns a different spatial location to every defined pitch. Some experimental systems of notation have represented pitch in this way, but they have not been widely accepted. They are not as compact as conventional notation, and the larger staff they require makes reading difficult. The only place where the piano roll representation is commonly used today is in music software for computers, where it is necessary to provide a mathematically accurate representation of pitch and timing on a computer screen.


Pitch may be represented non-spatially, purely symbolically, with letter symbols such as A3 or c#'''. Such symbols are commonly used in printed text such as this article. While symbolic notation is satisfactory to identify single musical pitches, it becomes difficult to read when applied to entire compositions.


Conventional music notation


Conventional musical notation uses both spatial and symbolic representation of pitch, -- and the result is greater than the sum of the parts. The symbolic representation came first, in the Middle Ages. With the addition of a single staff line, and then additional lines, the spatial reference was added to represent increasingly complex musical structures while keeping the notation readable. The great staff, with 11 lines, represented the high point of this evolution -- but the great staff was difficult to read. The system of treble and bass clef used today for keyboard music is, however, simply a great staff split apart in the middle, and covers exactly the same range. Besides its easier readability, the system with two five-line staffs allows for clearer notation of overlapping figures in the two hands.


The several different symbols -- naturals, sharps and flats -- at each degree of the staff make possible a compact and easily readable presentation. The compactness of the presentation is assisted further by placing notes in the spaces as well as on the lines of the staff. Conventional notation uses a similarly two-pronged approach in representing timing. The start of a note is represented spatially, but its duration is represented symbolically.


Notational Spelling


The sharp and flat signs let conventional western notation distinguish 35 pitches per octave, if we allow alterations to sharp, flat, double sharp and double flat. Such distinctions have existed in fact as well as in theory in western music. When mean-tone tuning was standard for keyboard instruments, the distinction between pitches such as G# and Ab was not just one of correct musical "spelling." It reflected an actual -- and not subtle -- difference in musical pitch.


The rules of musical spelling continue to be observed today, even in music written for today's equal-tempered keyboard instruments, on which different spellings no longer represent different pitches. Spelling continues to be important because it clarifies the role of pitches in tonal structures and makes the notation easier to read.


For example, we spell an E major triad E G# B, not E Ab B. The G# is a musical third above the E, as required in a triad. The Ab appears in notation as a flatted fourth; not only would this spelling lead to confusion as to whether the structure represented is a major triad, but more importantly, the Ab would occupy the position on the musical staff already occupied by the A natural of the E major scale. Could we solve this problem by calling the A a Bb? No, because our scale already includes a B. If we spell incorrectly, we are playing a game of musical chairs with musical notes. Two degrees of the scale are going to end up sitting in one line or space of the musical staff. This happens much less often, at least in tonal music, when spelling is correct.


Expanding the notational system to include microtones


The microtonal system which I favor is based on sequences of musical fifths. Such a system may use nearly conventional system of notation. In fact, we may conveniently use conventional notation unchanged to represent the pitches of a series of 17 fifths from Gb through A# -- the seven naturals, plus the five sharps and five flats.





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However, if we carry this system onward into the double sharps and double flats, it becomes confusing. For example, a Cb is the same or nearly the same as a B, not a C; an Ebb is the same or nearly the same as a F, not an E. We avoid such difficulties as much as possible by using enharmonic transpositions. In twelve-tone equal temperament, the five sharps and five flats duplicate each other. There is enough overlap that double sharps and double flats are rarely needed.


Since this duplication does not occur in a microtonal system, we can not use enharmonic transpositions in the same way. Furthermore, even though 35 distinct pitch classes are possible using double sharps and double flats, more than 35 may be needed: some interesting microtonal systems are based on cycles of 41, 43, 50 and 53 fifths. And, as already stated, excessive use of double sharps and double flats makes notation clumsy and confusing. For all of these reasons, we do well to look for another approach.


Dr. Adriaan Fokker's microtonal notation


Dr. Adriaan Fokker, of Leyden, Holland, took up the study of music theory when prevented from working as a physicist during the Nazi occupation. He developed a theory of 31-tone music and built instruments to play in 31-tone equal-temperament. Dr. Fokker's work is a good point of departure in exploring our options for microtonal notation.


Other than that it continues on past 12 pitch classes, the 31-tone scale which Dr. Fokker used is almost identical to traditional mean-tone tuning. Its major thirds are nearly just and its fifths are slightly flatter than those of 12-tone equal temperament. As a result, a Db, for example, is somewhat sharper than a C#. If we continue into the double sharps and double flats, a Dbb is, in the same way, flatter than a Db and a Cx is sharper than a C#, and we get the sequence shown here.

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All of the naturals in this 31-tone scale are approximately the same as in the 12-tone system (and almost exactly as in mean-tone tuning). Fokker's "half-sharps" and "half-flats," replacing double sharps and double flats, make it possible for a musician to read Fokker's notation with almost no training. About all the musician needs to know is: "flats are sharper than the enharmonically equivalent sharps; half-sharps are a shade sharper than the corresponding naturals, and half-flats, a shade flatter." The purer harmonies of the 31-tone system provide a further guide to correct intonation in this system


Through the sequence of 17 musical fourths and fifths from Gb through A#, Fokker's notation is conventional. Following the A#, an additional series of fifths through seven "half-flats" from F1/2b to B1/2b is followed by seven "half-sharps" from F1/2# to B1/2# and the return to the conventional sequence at Gb. The 17 conventionally notated pitches, the seven "half-flats" and seven "half-sharps" conveniently add to a total of 31. As already described in the article on the Fibonacci series, this is not a mere mathematical accident: it results from the requirement that a whole number of octaves and a whole number of fifths nearly coincide.


Fokker designed and built 31-tone keyboards. In the terminology I use in this series of articles, they were horizontal row, sharps-forward general keyboards with alternating, interlaced columns of 6 and 5 keys and 65 keys per octave. Fokker's keyboard layout is shown in the two-octave section here:

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The Italian Renaissance composer and theorist Nicola Vicentino (1511-1576) [1] experimented with microintervals and built for example a keyboard with 36 keys to the octave, known as the archicembalo. However Vicentino's experiments were primarily motivated by his research (as he saw it) on the ancient Greek genera, and by his desire to have acoustically pure intervals available within chromatic compositions.


While experimenting with his violin in 1895, Julian Carrillo (1875-1965)[2] discovered the sixteenths of tone, i.e., sixteen clearly different sounds between the pitches of G and A emitted by the fourth violin string. He named his discovery Sonido 13(13th sound). (The resultant 1/8 semitone is used today in the tuning of MIDI instruments, as is called a Triamu, written as 3mu [3]). Julian Carrillo reformed theories of music and physics of music. He invented a surprisingly easy musical notation based on numbers that can represent scales based on any musical interval within the octave, like thirds, fourths, quarters, fifths, sixths, sevenths, and so on (even if Carrillo wrote, most of the time, for quarters, eights, and sixteenths combined, the notation is able to represent any imaginable subdivision). He invented, adapted, and made new musical instruments that can produce microintervals. He composed a large amount of microtonal music and recorded about 30 of his compositions. Carrillo was nominated for the Nobel Prize in Physics in 1950 because of his work about the node law.


Some Western composers have embraced the use of microtonal scales, dividing an octave into 19, 24, 31, 53, 72, 88, and other numbers of pitches, rather than the more common 12. The intervals between pitches can be equal, creating an equal temperament, or unequal, such as in just intonation or linear temperament.


Microtonalism in rock music


The American hardcore punk band Black Flag (1976-86) made interesting vernacular use of microtonal intervals, via guitarist Greg Ginn, a free jazz aficionado also familiar with modern classical. (During their peak in the late '70s and early '80s, long before American punk was mainstream, the band was considered, not unwarrantedly, a thuggish and hostile street unit, although time has given their work a considerable measure of musical acclaim.) A worthwhile song is "Damaged II," from 1981's Damaged LP a live-in-studio recording in which intentional (and surprisingly scale-aware) use of quarter- and eighth-steps suggests a guitar in danger of detonation. Another is "Police Story," most versions of which end in a cadence played a quarter-tone sharp, to similar effect.


Other rock artists using microtonality in their work include Glenn Branca (who has created a number of symphonic works for ensembles of microtonally tuned electric guitars) and Jon and Brad Catler (who play microtonal electric guitar and electric bass guitar).


The American band Zia founded by composer Elaine Walker has released several partially microtonal rock albums since the early 1990s. Their works include use of the Bohlen-Pierce scale. http://www.ziaspace.com/ZIA/sections/music.html

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Ghent Belgium... :)


Well, it's news to me. I didn't even know that the website was referring to Ghent. I developped an ear for microtones when I started to play fretless, and it's taking me to uncharted waters, which is where I find adventure. Have you ever been there and heard what they do? So far what I hear is simply theory, not musical pieces. Various tones and chords using microtonal notes. Quite fascinating. I've been experimenting with some microtone sounds when I play. I play the blues using two microtones, on the Four I drag back from the One about 1/4 tone, and the Five I play over the One about 1/4 tone. And it sounds fine with me....

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Not exactly the same, but you can produce microtones on a fretted instrument with bends (bending the strings). It takes some practice to make automatic the logistics of fret choice and direction of the bend, though.


On a bass, it's probably more physically demanding to bend than to just play a fretless, although string choice will be influential.


Also, just because you're playing a fretless doesn't mean you can't bend, too. I've heard some tastey bends come out of (fretless) URB with some very flexible strings.


Bends are not quite the same, though, because generally you're bending up (or down) to the desired pitch (glissando) instead of just playing the desired pitch straight out.

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