Make a mixture of 5 minute epoxy and wood dust and push into the fret slots with a business card or razor blade.
Each of these methods requires wood dust lighter than the wood you are using. The mixture of dust and glue is always much darker than the wood the dust came from. If you are a perfectionist, and you should be if you want to be satisfied with your guitar, you should make some tests on wood dust to glue ratio in order to match the color of your fingerboard. Compare color of a sanded patch of fingerboard, as some manufacturers dye their wood, and your fingerboard is probably darker as the result of use.
****** If you have a light wood fingerboard, I recommend the veneer method, although recently I was able to do a pretty good match on a maple fingerboard with acetone-diluted Plastic Wood. It took a lot of testing, but it worked pretty well. It does need a top coat of super glue or urethane, as Plastic Wood is not that flexible.
FRETLESS INSTRUCTIONS
If you are making a fretless guitar, you are almost done. Once you have the fret slots filled and level sanded, you must sand with increasingly fine grits. Sand with 220, 320, 400 and 600 grit, then brush off with an old toothbrush. Inspect the side of the neck and carefully trim any protruding veneer or glue with a single edge razor bleade or x-acto knife. If you have been extremely careful, you won't need any touch-up. Wax the fingerboard, remove the tape and polish the entire guitar with a soft cloth.
You will need to sand the nut down since the distance between the strings and the neck is now less by the height of the frets. Mark off a little less than the height of a fret along the bottom of the nut with a sharp pencil. Glue a piece of 80 grit sandpaper to a flat piece of wood and sand the nut by rubbing the bottom of the nut over the sanding block, sanding almost to the line. Put the strings on and try the guitar. Play it for a while and adjust the height of the nut until it is just right.
Calculating the fret positions
Now you have a smooth, filled fingerboard and are ready to mark the positions for fret slots. To calculate the length of your fret scale, measure the distance from the nut to the octave fret and multiply by two. Measuring from the nut to the bridge is not reliable. The general formula for the distance from the nut to the kth fret is
f(k)=length(1-1/r)
where f(k) is the distance to the kth fret from the nut, length is the total scale length and r is the ratio you want from the fret. Let's say, for example, you measured a 12" from the nut to the 12th fret and that you want frets placed so as to give a scale of
1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1.
This is an example of a just major scale. The nut is at 1/1, the first fret is calculated by
f(1)=24(1-1/(9/8))=24(1-8/9)=24(1/9)=2.67",
the second fret is calculated by
f(2)=24(1-1/(5/4))=24(1-4/5)=24(1/5)=4.8".
The remaining calculations are similar. The octave is 2/1, so the fret calculation is
f(8)=24(1-1/(2/1))=24(1-1/2)=24(1/2)=12",
just as it should be.
The formula for equal tempered fret spacing is the same, but now instead of ratios, we use powers of two (unless you are using a non-octave scale).
The formula
f(k)=length(1-1/r)
is better written as
f(k)=length(1-2^(-k/n))
where the carat is the power sign and n is the number of equal tempered steps to the octave. As an example, let's calculate a few fret measurements for 19tet with a 35" bass guitar scale. The 0th fret is the nut, and the distance to the first fret is given by
f(1)=35(1-2^(-1/19))=35(1-.9642)=35(.0358)=1.25".
The distance to the 8th fret is
f(8)=35(1-2^(-8/19))=35(1-.7469)=35(.2531)=8.86".
Keep your figures accurate to four places and your final measurement should be accurate to 1/100". Notice that .2531 is very close to 1/4. This is as it should be, since the 8th fret in 19 approximates the perfect fourth (4/3) very well, but is a little sharp. Plug 4/3 into the equation for just frets and you will get
length(1-3/4)=length(1/4)=length(.25).
It is a good idea to test the algorithm a couple of times in this way before you cut to make sure you are using it correctly.
The general procedure for measuring for fret slots is the same whether you are using just intonation or equal temperament. Draw a pencil line down the centerline of the neck. Tape your ruler against this line with two sided tape with the end exactly against the end of the fretboard, and make a mark where the first fret will be. Be sure you do this very precisely. Use a very sharp pencil or a mechanical pencil with .5mm lead. Mark each fret position until you reach the octave. The last mark should line up exactly with the former 12th fret. Retape the ruler at the octave, and mark at 1/2 the distance calculated for the first octave. Now, depending on the type of ruler you have, either flip it over and tape it on the other side of the center mark, or slide it over and tape it parallel to the center mark. In either case you want a new set of marks parallel to the first and on the other side of the neck. Repeat the process until you have two sets of parallel marks. Connect the marks and you have a set of lines marking where all the frets will be placed. It is very important to do the measuring and cutting precisely. Check your work and redo the process if you have any doubt about the accuracy.
Cutting the new fret slots
It is time to cut the fret slots. If you are cutting the slots by hand, you will need a guide to clamp or tape on the neck to assure that the saw cuts the slots accurately. I use a straight piece of wood that has a little cup cut out so as to fit snug against the face of the fingerboard. It is possible to cut fret slots by hand without a guide, but you increase the possibility that the saw will slip and mar the fingerboard. Tape or clamp the guide to the neck using a leather or cork jawed clamp, being careful to line it up about 1/2 the width of the fret saw behind the pencilled in fret mark. Making sure the saw is parallel to the neck, take one light swipe over the neck from side to side to make sure the block is properly placed. If the saw has cut out the line you are OK. Realign the guide if necessary. Cut into the fingerboard to a depth of about 1/40th of an inch deeper than the fret tang. You don't have to be anal about this--you just don't want to cut the slot too deep or too shallow. The same depth as the old slots is usually about right. Continue up the neck until you have all the slots cut. You may want to cut the top fret slots first so that your technique improves as you get closer to the lower frets, which, in general, will be the most used. If you nick the fingerboard during this process, wait until after the position markers are laid in to fix the nicks.
Laying in the position markers
We will now lay in the position markers. Decide where you want the position dots (if any) and mark them by connecting with pencil the left and right ends of the fret slots between which the markers will be placed. You now have an "X" whose center marks the dot's position. Punch the center and drill the hole to the depth of the marker. Place the position dot in the hole and wick in super glue. When the dots are set, sand the fingerboard with the long sanding block using 120, 220, 320 and 400 grit paper until the neck is smooth and blemish free. If there is any touch up you need to do, now is the time to do it, as once the frets are laid in this chore will be extremely difficult.
Placing the frets
Re-wax the fingerboard, being careful not to get any in the slots. If you do, drag the fret saw lightly through the slot a couple times to remove it. The wax will help keep the super glue leaks from sticking to the fingerboard. To lay in the frets, cut off a piece of fret wire with the end nippers (or fret puller or toenail clipper), making sure there is enough straight tang to span the entire slot. The fret should be a little longer than the slot so that the deformed end where is was cut does not have to be pounded into the slot. The protruding ends will be cut off later with the end nipper. Lay the neck in a padded cradle made from a 4"x4"x4" piece of wood with a cup cut out about the same radius as the back of the neck. I use a piece of sheepskin to line the cradle. This cradle insures that the back of the neck is protected while the fret is being hammered in and also that there is a good coupling from the table to the neck to transfer the energy of the hammer into the fret. For a classical guitar, the cradle may have to be taller. Plase the fret over the slot and tap the fret in lightly on one side of the fingerboard using the fret hammer. Once the fret is seated on one side, tap across the fret to seat it all the way across. Check to make sure it is seated against the fingerboard by eye and by trying to run you fingernail under it. If it needs to be tapped in harder, tap from the center to the outside in both directions. Be especially careful when doing this on a classical guitar. I prefer not to fret a classical above the 12th fret using a hammer since the risk of cracking the top or loosening braces is so great with this method. If you want to get fancy, there are a number of products available to press in frets. Again, you might want to practice on the higher frets first until you get a hang of the method.
Take the end nipper and start from the lower frets, nipping off the protruding frets flush with the edge of the fingerboard. With care, you will not mar the surface. Carefully apply a coat of wax with a Q-tip to the edge of the fingerboard, avoiding the fret slots. Apply one drop of super glue at a time to the fret ends, allowing it to wick in. About 3 drops is enough for each side is enough. Make sure glue is not oozing out the other side of the fret slot. If there are any frets whose ends just wouldn't stay down when you tapped them in, clamp them down before you apply the glue. A little baking soda on the glue will help it set up faster (though super glue accelerator is better). When the frets are all glued in, you may find that a little glue has seeped out between the fret and the fingerboard. Carefully cut the glue off with a single edge razor blade. You can score the glue next to the fret so it will come off easily. Here is where the fingerboard waxing is so important. With enough wax, the glue actually just peels off.
Dressing the frets
Your frets are laid and it is time to dress them. Polish the wax on the fingerboard until it shines, then lay masking tape between all the frets to protect the fingerboard. This time, you may need to use painter's masking tape so it will stick better to the waxed surface. Run a magic marker across the top of each fret so you can tell what has been cut. Run the flat file lightly over the surface of the frets along the length of the neck, taking care to avoid making the same stroke twice. You should make a sort of criss-cross pattern along the length of the neck, trying to cover the length of the neck from several different angles. This way the file will tend to average the height of the frets over the whole surface.
As you file, the marker is cut from the highest portion of the frets. You are finished when there is at least a thin line cut through the magic marker all the way across every fret. Some of the lines of visible fret material will be wider than others, and that is all right. Brush off the filings and re-ink the frets. If you have a fret crowning file, run it across the top of each fret until there is a thin continuous line of fret material showing.
If you don't have a fret crowning file, run the long sanding block lightly over the fret surfaces using 320 grit sandpaper until a thin line of fret material shows through every inked fret. Now is a good time to lay in side position dots if you want them. The procedure for removal of the old and laying in the new is the same as that for dots on the surface of the fingerboard. Use the flat file to smooth the sides of the fingerboard where the frets end and the dots have been laid in. With care, you can do this without marring the finish much.
You may want to refinish the fingerboard edges, or you may want to wax them after a light touchup sanding. Never use silicone wax, as it will make refinishing impossible should you decide to do so in the future. With a jeweler's file or a file with a flat, noncutting edge (place the non cutting edge in contact with the fingerboard), using one or two light strokes only, file the tips of the fret at the edge of the fingerboard where the fret meets the fingerboard surface. The file should start at 45 degree angles to both the plane of the fingerboad and a parallel to the length of the fret. The file remains at a 45 degree angle with the plane of the fingerboard, but during the stroke its angle relative to the length of the fret changes from 45 degrees to 90 degrees. This action takes a little practice, but the end result is a fret with a smooth edge where the hand is likely to contact it.
To finish the frets, start with 220 paper (or 320 if you used a crowning file) and vigorously hand sand (no block, fingers gripping a small piece of sandpaper) each fret along its length to smooth the crown and remove nicks. When you are done with one grit, move up to the next finer, taking care to brush the fingerboard free of grit from the previous coarser sandpaper. This insures that a finer sandpaper will not drag a piece of detached coarse grit across the fret and mar the surface. Following the 600 grit paper, use 00 or 000 steel wool for the final finish. Brush off the neck, remove the tape and admire your handiwork.
Final steps
Apply a fresh coat of wax or polish, re-attach the neck and replace the nut. Since you have new frets you may have to adjust the height of the nut. You can remove material from the bottom of the nut or from the slots, or if you need to raise the height of the nut (more likely) you can shim it with slips of brass strip (to be found at hobby stores), or with drops of super glue on the bottom of the nut hardened and filled by dropping baking soda on it. File the baking-soda-super-glue filler with a jeweler's file of an X-acto saw to get the right string height at the nut. You will probably have to adjust the bridge height too. Open a new set of strings, string it up and play!!
Where to get Supplies
A fret saw, fret puller, weighted plastic or brass faced hammer, ruler marked off in hundredths of an inch, and fretwire can be obtained from Luthier's Mercantile, Stewart McDonald, GHS, or any number of luthiers's supply companies. A hand fret saw may be made by purchasing a good quality miter saw of approximately 0.023" blade width and flattening the kerf (the waviness of the set of the teeth) carefully with a light hammer on an anvil or other flat surface.
The ruler in hundredths of an inch can usually be obtained at an art supply store or sometimes at a college book store (drafting classes!).
Jewelers files can be obtained from jewelry supply stores or good hardware stores.
Instead of a fret puller, you can grind the face of an end nipper (availableat a good hardware store) until the bevel of the cutting edges is ground away and the face is smooth and rounded. I have also heard of people using toenail clippers. The main thing is to have a flat or convex clipping face.
....................................................................... 4. Newbie Questions 1 ....................................................................... 4.1. What is microtonality? - Margo Schulter ../margoschulter/what_is_microtonality.html ------------------------------------------ What is microtonality? What is paucitonality? ------------------------------------------ ---------------------------------------------------- 4.1.1. Microtonality and paucitonality: a short answer ---------------------------------------------------- If asked "What is microtonality," members of the Tuning List might offer definitions taking at least three general approaches, all of them with delicate and often controversial cultural implications: (1) MICROTONALITY AS THE USE OF "SMALL" INTERVALS. In the most obvious definition, microtonality (from Greek _mikro_, "small") is the use of intervals smaller than the usual whole-tones and semitones of the best-known Western European compositional traditions, although the use of such intervals is a routine feature of many world musics. (2) MICROTONALITY AS THE USE OF "UNUSUAL" INTERVALS OR TUNINGS. In a second and related definition broadening the first, microtonality is the use of any interval or tuning system deemed "unusual" or "different" in a given cultural setting -- in many 20th-21st century settings, for example, just about any tuning for keyboard or guitar other than a division of the octave into 12 equal semitones (12-tone equal temperament, or 12-tET). The composer Ivor Darreg's concept of _xenharmonics_, which it is tempting to describe in a paraphrase of the Latin poet Terence as the conviction that "nothing intonational is alien to me," seems synonymous with this sense of "microtonal." (3) MICROTONALITY AS A MUSICAL CONTINUUM OR DIMENSION. In a third definition, microtonality is simply the dimension or continuum of variation among intervals and tuning systems, embracing _all_ musics. Seen from another perspective, the first two definitions treating "microtonality" as a special and often suspect category of music imply an unspoken norm of what we might term _paucitonality_, literally "few tones" -- or better, "scarce tones." This concept -- not a tuning system or musical style, but a confining state of mind -- also invites at least three definitions: (1) PAUCITONALITY AS CULTURAL MYOPIA. In its root definition, paucitonality or "scarce-tonedness" is a state of musical and cultural myopia in which the use of intervals and intonational nuances routinely occurring in many world musical traditions -- including some European ones -- must be relegated to a special "microtonal" category. (2) PAUCITONALITY AS INTONATIONAL MONOMANIA. In its more aggravated forms, paucitonality could be defined as an ideology (often unspoken) restricting musicians (at least in theory) to a single tuning system, and viewing talented musicians even of one's own historical or cultural tradition who dare to propose something "different" as "straying far afield from the mainstream." (3) PAUCITONALITY AS THE ABSENCE OF CHOICE. In its third definition, corresponding to the definition of "microtonality" as a universal property of all music, paucitonality means tuning by default, or by a decision not to decide, rather than by informed and aware choice. As the first two definitions of "microtonality" or "paucitonality" may suggest, defining certain intervals as "unusual" can have witting or unwitting cultural implications, especially when the definition of "usual" is based on one subset of European composed music. The less pleasant overtones -- to use a musical figure of speech -- sometimes get articulated all too plainly. In his "A Brief History of Microtonality in the Twentieth Century"[1], microtonal composer and historian Brian McLaren tells how a German theorist named Willi von Moellendorf experimented with one of the popular early 20th-century tuning systems: 24-tone equal temperament (24-tET). Taking as a starting point the 12-tET system which had then just recently become the norm in Europe for keyboards, 24-tET divides each of the 12-tET semitones into two "quartertones." Often very mistakenly equated with "microtonality" in general, the 24-tET system is simply one equal division of the octave among a myriad of equal and unequal divisions, not to speak of nonoctave tunings. In fact, many self-declared microtonalists today might regard it as a rather conservative and overworked choice, although others find it charming and exciting, especially when applied to certain less familiar styles. In 1917, however, von Moellendorf had to confront not only the usual objections to any musical innovation, but also the charge that intervals such as quartertones represent a "primitive, even barbaric condition of a lower cultural level." Given subsequent history in Germany and elsewhere in Europe during the years 1933-1945, such possible cultural implications may have an interest more than musical.[2] Indeed the word "barbaric," another Greek-derived term, may say much about the concept of "microtonal" as applied by "antimicrotonalists" or "paucitonalists." To the ancient Greeks, the speech of outsiders sounded like a meaningless "bar-bar," thus the term "barbarian," one outside the "civilized" world. To "antimicrotonalists" (some of whom might define themselves simply as "lovers of _normal_ music"), similarly, musics based on unfamiliar intervals or tuning systems sound like "mistunings" or "random dissonance." Such judgments would relegate not only self-consciously experimental or avant-garde composed musics, but age-old musics of a vast range of world traditions, to an "inferior" (or at best "exotic") status. It is against this backdrop of "intonational politics" that champions of intonational pluralism -- "microtonality" in definition (3) -- consider such issues as "Should I call myself a microtonalist, a xenharmonicist, an alternative tuning advocate, or simply a musician?" People hesitating to embrace the "microtonal" label, or even actively resisting it, often take the principled stand: "My music is an integral whole, with intonation simply one aspect of this whole: I leave others to make categorizations." Other musicians, ranging from Nicola Vicentino (1511-1576) to many members of this Tuning List, eagerly embrace terms such as "microtonal" (or earlier historical equivalents such as "enharmonic," see Section 4.1.2.1 below) because they see new intervals and tuning systems as a central theme of their music. Also, musicians may embrace the term "microtonal" as an act of affirmation and solidarity in the face of intonational oppression: "If that's what the 'mainstream' scene wants to call us and our music, let's make the most of it." Our third definition of microtonal as a universal but not universally recognized aspect of all music may be one way of reconciling these viewpoints. The decision (active, passive, or unknowing) to use _any_ tuning system places a musician somewhere on the microtonal continuum, or within the microtonal multiverse. From this perspective, to declare that one is a "microtonalist" is simply to acknowledge this reality, and to invite others to join in this musical act of self-awareness and mutual celebration. ------------------------------------------------ 4.1.2. A longer answer: perspectives and paradoxes ------------------------------------------------ Since both microtonality and paucitonality may be concepts of most relevance to composed European traditions, questions of how best to define them may be implicitly ethnocentric, distracting us from a more balanced survey of the use of intervals both large and small in a plethora of world cultures through the millennia. Nevertheless, examining a few historical examples and patterns may suggest the perennial nature of some European and related issues of microtonality/paucitonality, and also bring out some of the paradoxes inherent in definitions sometimes more often cited (on this Tuning List and elsewhere) than carefully examined. One 20th-21st century factor which may transform the debate for better or for worse is technology. On the positive side, electronic means both of recording and of synthesizing musical sounds have created a basis for free cultural interchange and mutual knowledge, and also for realizing just about any known or not-yet-conceived tuning system, and for quickly switching a keyboard or similar instrument from one system to another. On the negative side, mass production can mean mass standardization of the most confining kind, threatening to produce what composer and historian Douglas Leedy has described as "an unsavory echo of imperialism" subjecting the "musical cultures of the world" to a process of "control and appropriation."[3] Despite these vital technological changes, a journey to the Europe of five centuries ago may reveal how many microtonal/paucitonal issues are not so new in substance. -------------------------------------------------------- 4.1.2.1. Microtonality/paucitonality and "small" intervals -------------------------------------------------------- In 1482, the often iconoclastic Spanish musician and theorist Bartolome Ramos published a treatise noted then and now both for its daring innovations, intonational and otherwise, and for its biting satire against the musical "establishment" of the time. Both aspects of his work led to heated controversies. Discussing the art of finding regular intervals on a keyboard likely tuned in the new meantone fashion then coming into vogue, Ramos considered an interesting question: might a 12-note tuning more usefully include G# or Ab, two distinct notes often differing in Renaissance meantone temperaments by around a fifth of a tone. After considering the arguments and opting for Ab, Ramos added that some people prefer to satisfy both sides of the question by designing a keyboard with both accidentals -- typically by splitting the key for an accidental so that pressing the front portion would sound G#, for example, while the back portion would sound Ab. This approach was followed, for example, in the organ at Lucca in Italy with such split keys for G#/Ab and Eb/D#, providing 14 notes in each octave, and enjoyed widespread favor in 16th-17th century Europe. Ramos, however, raises the objection that having both Ab and G# would introduce an interval not part of the diatonic order -- the interval smaller than a semitone between these two notes. Showing the Renaissance love for Classic allusions and precedents, Ramos cited the case of a musician banished from Sparta for the "crime" of adding extra strings to his instrument beyond those accepted by tradition, thus upsetting the musical and political order. Although Ramos adds that nevertheless there are good arguments for having both G# and Ab which he reserves for another discussion (this discussion is not known in his writings which have come down to us), his allusion to Spartan paucitonalism may reflect a cultural theme still quite relevant in the year 2001, as artists facing the professional consequences of "microtonality" can attest. Moving ahead in time to the year 1555, we find composer and theorist Nicola Vicentino publishing his great work on _Ancient Music Adapted to Modern Practice_. By a happy coincidence, in one popular flavor of meantone tuning with pure or near-pure major thirds, the diesis or small interval by which accidentals such as G# or Ab differ happened to be around the same size as the enharmonic diesis of ancient Greek theory. Vicentino made the most of this by seeking, as his title suggested, to combine the expressive enharmonic genus of the Greeks with 16th-century techniques of polyphony and counterpoint. Masterfully analyzing the "common practice" of the time as well as documenting his own experimental music, Vicentino described his _archicembalo_ or "superharpsichord" dividing the octave into 31 dieses, each equal to 1/5-tone. In 1561, he advertised his similar _arciorgano_ or "superorgan," sounding a notable cross-cultural theme. With this instrument, he announced, one could perform "all manner of songs and airs according to the idiom which all the nations of the world sing" -- including, for example, the Spanish, French, Polish, English, Turkish, and Hebrew manners.[4] Given the special interest of the Renaissance in fitting music to a text elegantly and expressively, this passage could be read to suggest an interest both in the melodic intervals and nuances favored by other cultures, and in the most flexible choice of intervals by a composer in setting poetic or other texts in a range of world languages. Along with 16th-century enharmonicists -- or xenharmonicists, to use a more recent term -- there were also the "anti-enharmonicists," among them the otherwise often outspokenly radical musician Vincenzo Galilei, father of the astronomer Galileo. In Galileo's view, Vicentino's enharmonic dieses were "contrary to the nature of singing and disproportionate to our sense of hearing."[5] As his comments on Vicentino and his students revealed, such matters could be issues of economics as well as theoretical psychoacoustics. According to Galilei, Vicentino's disciples found it necessary to abandon the enharmonic style in order to succeed in the marketplace. Many xenharmonicists of today may also hear a familiar if not so pleasant ring in another view of Galilei, an accomplished lute player as well as composer and polemicist: his ridicule of lutenists favoring the use of _tastini_ or "little frets" added to the instrument in order to obtain purer thirds. In 1584, while meantone tunings with pure or near-pure thirds were standard on keyboards, 12-tET was standard on fretted instruments such as the lute; but then as now, some players favored alternative or modified frettings. Galilei seems to be describing a kind of just intonation scheme or the like adding special frets to the usual ones. Poking fun at such players when they strive for pure thirds but find that their fingers stumble into impure fifths or fourths, showing that their ears are not as fastidious as one might have guessed, Galilei adds that a truly expert performer (such as himself!) needs no such special gadgets to impress an audience, thus assuming the 16th-century role of the author as self-promoter. For Galilei, both Vicentino's enharmonic music and the just intonation lute frettings represented the use of intervals smaller than a diatonic or chromatic semitone -- a concept closely coinciding with the most familiar definition of "microtonalism." His reaction was not favorable. Galilei's views as a musician who felt free to question authority on many points, and to challenge the conventional rules of counterpoint and dissonance treatment, are of special interest. To advocate Nicola Vicentino's enharmonicism, or Ivor Darreg's xenharmonicism -- a difference of only one letter -- is to take a stand which may carry an appreciable professional price. ------------------------------ 4.1.2.1.1. How small is "small"? ------------------------------ Accepting for the sake the argument the "small interval" definition of microtonality, a fine issue arises: "How small is 'small'?" The standard interpretation of "smaller than a semitone" leaves considerable room for debate and ambiguity, because diatonic or chromatic semitones of historical European tuning systems can vary in size from around 1/3-tone (say 63 cents) to around 2/3-tone (say 126 cents). Semitones of these specific sizes occur in the equal 19-note division of the octave advocated by the French composer Guillaume Costeley in 1570, or the almost identical meantone temperament with pure minor thirds described by the Spanish theorist Francisco Salinas in 1577. Another interpretation might take as "microtonal" an interval too small to be perceived in a given musical context as a usual semitone. Under this definition, the enharmonic diesis or fifthtone of Nicola Vicentino at around 41 cents clearly qualifies, certainly in the setting of his music: it has a radically different effect than the diatonic semitone of his system at around 117 cents (or 3/5-tone), or the smaller chromatic semitone at around 76 cents (or 2/5-tone). In contrast, a single step of 22-tET at 1/22 octave or about 55 cents makes a very convincing semitone in connection with a whole-tone of four steps, 4/22 octave or about 218 cents. This is an example of how in the right setting a literal "quartertone" can serve as a regular diatonic semitone, thus arguably falling outside the category of "microtonal" in the sense of "too small to be heard as a usual scale step." This 22-tET interval, by the way, illustrates how a "quartertone" can take on various sizes or shapes; while a step of 1/24-octave or 50 cents may be the most familiar example because of the popularity of 24-tET in the 20th century, it's only one possible form. One curious conclusion is that an interval of around 50-55 cents might be "microtonal" in one musical setting, and a regular "semitone" in another. If we regard the drawing of such a blurred line as an engaging but rather parochial study in the ethnomusicology of one regional compositional tradition among the many musics of the world, then the exercise can be at once edifying and harmless. ------------------------------------------------------------- 4.1.2.2. Microtonalism as "unusual" intervals or tuning systems ------------------------------------------------------------- When introduced into a European style of composition where diatonic or chromatic semitones are the smallest recognized intervals, smaller intervals such as Vicentino's diesis at around 41 cents can have a strikingly "unusual" effect, in 1555 or 2001. A broadening of the "microtonal" concept focuses on this perception of the "unusual," whether induced by intervals of small size, or simply of _unaccustomed_ size. Consider, for example, a division of the octave into five equal parts, each an interval of 240 cents, which a listener accustomed to historical European tuning systems might hear as either a very large major second or whole-tone, or a very small minor third. This creative ambiguity, by the way, lends a special charm to tunings such as 20-tET when used for "Western European-like" styles. This interval of 1/5-octave or 4/20-octave is much larger than a semitone, and therefore not "microtonal" in the narrow sense, but it arguably has a strikingly "different" quality for the uninitiated listener analogous to that of Vicentino's diesis. One should hasten to add that for members of many world musical cultures, an interval of around 240 cents dividing the octave into five roughly equal parts is not "unusual" at all, but the routine norm, known simply as "everyday musical practice." Again, we find that "unusualness" is in the ear of the listener, and that paradoxically defining something like 240 cents as "microtonal" in order to celebrate its "unusual" qualities may be tacitly accepting a definition of "usualness" with definite ethnocentric implications. ---------------------------------- 4.1.2.2.1. How unusual is "unusual"? ---------------------------------- Some intonational activists will argue that "everything is microtonal except for 12-tET," including historical European tuning systems such as medieval Pythagorean intonation, Renaissance meantone, and 17th-19th century unequal well-temperaments. One curious paradox of this interpretation is that it leads to the conclusion that "Renaissance and Manneristic music of the 16th century in Europe may generally be considered microtonal -- ranging from the most conventional settings to the most radical enharmonic styles of Vicentino and his colleagues -- except for anything which happened to be written for or performed on a 12-tET lute or similar instrument." Applying our narrow definition of "small intervals" would lead to a distinction more like that which might be drawn and evidently was drawn, both by Vicentino and his critics such as Vincenzo Galilei, between "usual" music and "enharmonic" styles using the diesis or fifthtone. If we do apply the broader "microtonal-as-unusual" concept in this 16th-century setting, we may well reach the conclusion that while meantone on a harpsichord is everyday reality, 12-tET on such an instrument is "xenharmonic" or even "microtonal." Galilei, much enamored of this "perfect" temperament on the lute, tried it on a harpsichord -- and found the thirds unsatisfactory, a not unsurprising result given the nature of 16th-century style with its restful thirds, and of the harpsichord with its prominent fifth partial. As a "strange" keyboard tuning, this "spherical" temperament with its perfect symmetry and easy circumnavigability nevertheless had an attraction for Galilei and for an abbot named Girolamo Roselli who also celebrated these qualities. In the 1630's, the great composer Girolamo Frescobaldi reportedly advocated that a new organ be tuned in 12-tET -- and was roundly ridiculed by one theorist of the time for allegedly being ignorant of the difference between a large and small semitone. Others remarked that 12-tET might be more palatable if it were less unfamiliar -- a comment sometimes offered concerning "microtonal" music in more recent times. Although a definition of "microtonal" including all tunings except one may lead to such paradoxes, the "anything but 12-tET" approach does have a certain political logic. This would seem to be the logic of "uniting against a common adversary." A group of xenharmonicists who cannot agree among themselves as to whether a major third should ideally be tuned at 386 cents, 418 cents, or 435 cents can nevertheless say "We're all non-400-cents!" The next step, however, might be to realize, "Hey, 400 cents is interestingly different, somewhere `between the cracks' of the smaller or larger interval sizes (maybe both) we typically use -- and that makes it `microtonal!'" The adversary, then, is not 12-tET or any other tuning, but intonational monopoly, in other words, paucitonality. The problem is not how to distinguish between 12-tET and the "microtonal" universe, but rather how to put 12-tET squarely within that universe, somewhere on one arm of a certain spiral galaxy of equal tunings. ----------------------------------------- 4.1.2.3. Microtonality as an open continuum ----------------------------------------- Defining "microtonality" not as a property of a specific interval or tuning, but as a dimension of variation encompassing all intervals and tuning systems, may avoid some of the pitfalls and paradoxes of the "small interval" and "unusual interval/tuning" concepts, while affirming the ideal of intonational pluralism often embodied by the "small" or "unusual" in a European-related setting. More specifically, the "microtonality as universal continuum" concept may suggest a kinder and gentler strategy for convincing innocent bystanders or even decided opponents that 12-tET is only _one_ of the myriad of possibilities. Rather than arguing why 12-tET is "bad" or "dissonant" or "out-of-tune," all judgments based on stylistically specific assumptions which when generalized become just as questionable as the categorical assertion that "12-tET is ideal," we can argue that "12-tET is a fine tuning -- but only one." From that starting point, as musician and mathematician Dan Stearns has said, there are fertile fields in all directions. An enthusiast of 12-tET, or someone who has simply used it by default, might try a regular tuning making the fifths pure, or tempering them somewhat more heavily in the narrow direction, or tempering them a bit unevenly. A different strategy involving at once a small step and quantum leap is to tune _two_ 12-tET chains in 24-tET or 24-out-of-36-tET, discovering whole new families of intervals. These approaches, and others, can help curious and inquiring minds and ears recognize the xenharmonic universe they already inhabit, with 12-tET simply as one place (more attractive to some than to others) in a very rich musical cosmos. ----- Notes ----- 1. Brian McLaren, "A Brief History of Microtonality in the Twentieth Century," _Xenharmonikon_ 17:57-110 (Spring 1998), at pp. 61. 2. Ibid. On the prejudice against 1/24 octave as representing a "lower cultural level," McLaren remarks, "Shades of Himmler and Heydrich." While students of this period must address the question of any direct political lineage between Moellendorf's critics in 1917 and the Third Reich, the affinity between the concept of a "primitive ... condition of a lower cultural level" and the Nazi category of "subhuman" are all too unpleasantly evident. It is sobering to reflect that such cultural prejudices are not neatly confined to one narrow range of the political spectrum, and that today the 500-year struggle for the survival of Indigenous peoples, cultures, musics, and intonational systems continues. 3. Douglas Leedy, "Review of Martin Vogel's _On the Relations of Tone_," ibid., pp. 120-123 at 123. 4. Henry W. Kaufmann, "Vicentino's Arciorgano: An Annotated Translation," _Journal of Music Theory_ 5:32-53 (1961) at 36-37. 5. Karol Berger, _Theories of Chromatic and Enharmonic Music in Late Sixteenth Century Italy_ (Ann Arbor, MI: UMI Research Press, 1980), ISBN 0835710653, p. 73. ....................................................................... 4.2. Why study tuning? - Alison Monteith ../alisonmonteith/why_study_tuning.html Q. I am a musician (composer, teacher, performer) curious to know more about tuning as a resource for creativity. Why should I devote time and energy to the study of tuning; in otherwords, what's in it for me? A. If you are a conventionally trained musician (or even self taught) performing or composing in the Western world it is most likely that your experience of tuning systems is limited to the 12 tone equal temperament (12 tet ), such as found on the conventionally tuned piano or the fretting of a guitar. You might have come across Asian and other musics which use different systems of tuning - these will probably have struck you as exotic or unusual. In otherwords the largest part of your musical experience is one based on one sole tuning system. For the best part of two centuries ( there is some debate on this ) 12 tet has reigned supreme in Western "classical" and popular music and has indeed produced some beautiful music. A close look at modern music of all genres will reveal much innovation and originality. These developments are usually in the field of new technologies, perhaps original orchestrations and occasionally new timbres and mixes of sound. But virtually all in 12 tet. It is said more and more that we are continually recycling old resources and that it's all been done before. Not so with tuning systems. Apart from the riches of "non-Western" musics there exists an almost infinite palette of creative colours in new tunings that have been and are currently under exploration by new "classical" composers, electronic musicians, jazz musicians and composers and performers of various "neo" musics, such as neo-Gothic and neo-Ethnic. The tuning field is probably the most fertile of all musical resources at the moment and represents a challenge for musicians of any ilk who have the courage to explore new territory. It can be a lifelong study involving pure mathematics, physics, computer programming and instrument design or with some imagination a little knowledge can go a long way. Either way, you learn something new and your music has new life in it. ....................................................................... 4.3. Why twelve notes as _one_ attractive arrangement? - Margo Schulter ../margoschulter/Why_12_notes_as_one_attractive_arrangement.html Q. Why are there 12 notes per octave on typical keyboards? A. The form of the typical European keyboard instrument seems to have evolved during the era from around 757, when the court of King Peppin of the Franks received an organ as a gift from the Byzantine Empire, to around the middle of the 14th century, when 12-note keyboards in the famous arrangement with seven diatonic or "white" keys and five accidental or "black keys" were becoming standard. Here our focus is on the question: how did these keyboards come to have their familiar 14th-century arrangement, still standard in the 12-key instruments of the early 21st century? An equally important and revealing question, however, is "Why stop at 12 keys?" As other portions of this FAQ discuss from various viewpoints, the best answer may be that a range of musicians over the last six centuries or a bit more _haven't_ stopped at 12, but have designed and often have actually built instruments with anything from 13 to 31 or more notes per octave. Let's also note that while the history of the familiar 12-note keyboard is largely an adventure in Western European musical styles and tastes, various musical cultures have leaned toward larger tuning systems. For example, some medieval Arabic or Persian traditions favor a 17-note system of a kind in some resembling that advocated in early 15th-century Italy. Traditional Chinese theory recognizes sets of 53 or more notes per octave. With this bit of perspective, let's return to medieval Europe around the time of Charlemagne and his successors in the 9th century, a period celebrated by one poet as a kind of "rebirth" of Roman culture (scholars sometimes call this the Carolingean Renaissance). A vital element in the cultural mix was music, with the treatise of the revered philosopher Boethius (c. 480-524) the basis for the learned study of this art. Boethius took a great interest in the theory of consonance and dissonance, and also in the ancient Greek authors and systems, especially those of Pythagoras (as recorded by his followers) and Ptolemy. Like many world musics, the best-known Western European music of this time was based on a system of tuning in pure fifths and fourths, known in the West as Pythagorean tuning after the Greek philosopher Pythagoras. (Pythagoras, like many of the pre-Socratics, is known to us mainly by repute and by reported quotations or teachings written down by later authors). Scholars have suggested that the ancient Greeks may have borrowed it from a Babylonian tradition. While both sacred and secular music were practiced in 9th-10th century Europe, and some writers such as Hucbald tell us a bit about popular instruments such as lyres or harps, we know mainly about music for the Church: both traditional chant, and a newly documented technique of _organum_ or "organized" part-music involving the concord of different notes sounding at the same time. Medieval liturgical chant or plainsong uses a system of eight standard notes: the seven diatonic or "white" keys on a familiar keyboard, plus Bb. In other words, there are _two_ versions of the step Bb/B, and both may occur in certain chants. In the Pythagorean tuning described by Boethius, and followed as standard practice, we can derive this usual set of eight notes for chant as a chain of seven pure fifths or fourths: Bb F C G D A E B Early organs of this era seem often to have had these eight notes per octave, although the term "keyboard" might be misleading: these instruments had devices such as large sliders for opening or closing the wind supply to a pipe, so that two or more players might be needed, and the action was likely _slow_. If we applied a layout like the modern ones, we might arrange the notes like this, with the "black key" Bb set apart from the others: Bb C D E F G A B C However, we will recall, both Bb and B were regarded as regular forms of the same scale step, so a layout with all eight notes on the same row was common, and still in use in some instruments of the 14th century: C D E F G A Bb B C In medieval Western Europe, as in many other world musics, fifths and fourths were favorite consonances: Boethius described their concordant effect, and during the period of around 850-1200, musicians developed more and more complex styles contrasting these stable intervals with a wide range of unstable ones having various degrees of concord or discord. Pythagorean tuning made the stable fifths and fourths pure, or ideally smooth and concordant, and produced an intriguing continuum of tension among the other intervals. By around 1200, the great composer Perotin and his colleagues were writing pieces in the high Gothic style for three and four voices, using not only the traditional eight notes of most chant but other accidentals: Eb, F#, C#. These notes could also be added to an organ by extending the chain of fifths in either direction, for example in this ten-note chain: Eb Bb F C G D A E B F# C# At about this same epoch, there was apparently a major technological breakthrough: at least some organs acquired agile keyboards of the modern kind, which allowed them to play the flowing and often ornamented melodic lines favored in the music of the time. A 13th-century poem, the _Roman de la Rose_, tells us that small or portative organs could play either the supporting lower part or the florid upper melody of the sophisticated motets then in fashion, pieces artfully combining voices singing different texts. Just how quickly and frequently some or all of the extra accidentals became standard on the limber keyboards of the 13th century is uncertain, but by 1325, the theorist Jacobus of Liege tells us that their diatonic whole-steps or major seconds were "almost everywhere" divided into two semitones. This comment become clearer if we look at a possible keyboard layout around 1300, using the 11 notes which had sometimes been in use for at least around a century: C# Eb F# Bb C D E F G A B C Here the seven diatonic notes form an octave with five whole-tones (C-D, D-E, F-G, G-A, A-B) and two semitones (E-F, B-C). Four of the five whole-tones have added accidentals dividing them into semitones: C-C#-D, D-Eb-E, F-F#-G, and A-Bb-B. Given the special status of Bb as a "regular" note, it might also have been placed in the same row as the diatonic notes, giving us an arrangement of our eight "regular" notes plus three extra accidentals: C# Eb F# C D E F G A Bb B C By around this same epoch of 1300, a new and compelling argument for these extra keyboard accidentals was at hand: the preference for "closest approach" at cadences or in other directed progressions where an unstable sonority moved to a stable one. The basic rule, as stated by various 14th-century writers, is that a third expanding to a fifth, or a sixth to an octave, should be major; a third contracting to a unison should be minor. If they are not so naturally, then they should be altered by using accidentals. For example, a typical "closest approach" cadence on D might take two forms; here I use a notation showing middle C as "C4," with higher numbers showing higher octaves: C#4 D4 C4 D4 G#3 A3 G3 A3 E3 D3 Eb3 D3 Either form features a major third between the lower two voices expanding to a fifth, and a major sixth between the outer two voices expanding to an octave. In the first solution, the two upper voices each ascend by a semitone; in the second, the lower voice descends by a semitone. While the second form was available using the 11-note set known in 13th-century compositions, the first form called for a 12th note, namely G#. This first form was in fact much in demand, because in 14th-century style a cadence with ascending semitones was usually considered more conclusive than one with descending semitones, the latter form usually signalling a kind of musical "halfway" point rather than a final cadence. Since pieces centered on the octave-type or mode of D-D were very common, final cadences with the major third E-G# expanding to the fifth D-A were routine in theory and practice. Therefore, as a modern software developer might say, G# had found a "compelling application" -- in this case, an application calling for a modest hardware upgrade, the addition of a G# key. Around 1325, the time Jacobus wrote about the semitones being divided "almost everyone," this 12th note may have already been added on some instruments. The earliest known European compositions for keyboard, preserved in the Robertsbridge Codex with proposed dates anywhere from 1325 to 1365, calls for all 12 notes of such a keyboard, with a chain of fifths very likely tuned like this, from Eb to G#: Eb Bb F C G D A E B F# C# G# However, as we might say, the "user interface" wasn't quite yet standardized. The scholar Mark Lindley shows how the 14th-century Noordlanda organ had an arrangement with the eight "regular" notes (including both B and Bb) on one row, and the complement of four "extra" notes on another: C# Eb F# G# C D E F G A Bb B C At Halberstadt in 1361, however, it appears that the 12-note organ placed Bb in the row with the other accidentals, possibly with earlier unrecorded precedents, and this quickly prevailed as the "standard" arrangement: C# Eb F# G# Bb C D E F G A B C Q. We know that there's no need to stop at 12, but why it is _one_ attractive size for a keyboard tuning? A. Here there may be a quick answer and a more involved answer. Let's take the quick one first. In his encyclopedic treatise of 1325, Jacobus of Liege mentioned that keyboards "almost everywhere" divided diatonic whole-tones into two semitones by extra accidentals. With a 12-note instrument, as we can see on either the Noordlanda keyboard or the more familiar Halberstadt keyboard, _every_ whole-tone is divided into two semitones, including G-A (by the new 12th note G#). In all we have five such whole-tones, and five accidentals (counting Bb and the four "extra" ones) to do this dividing. There's a certain cozy symmetry in this arrangement, not to mention that a Pythagorean chain from Eb to G# nicely covers the accidentals typically used in the great preponderance of 14th-century pieces for ensembles or keyboards, including the music of composers such as Guillaume de Machaut (1300-1377) and Francesco Landini (1325-1397). The more involved answer looks at this feeling of "coziness" or "balance" more closely, and brings into the play an important concept of modern theorist Ervin Wilson: the idea of a Moment of Symmetry (MOS). Such an "MOS" occurs when a tuning system has only two sizes of _adjacent_ intervals, that is, intervals between the pairs of notes immediately adjacent to each other. If we look at our 12-note Pythagorean tuning with a chain of fifths from Eb to G# -- we can call this an "Eb-G#" tuning for short -- we find that there are in fact only two such intervals sizes, formed by two varieties of semitones known as "diatonic" and "chromatic," and here marked as "D" and "C," whose distinct sizes we're about to consider: C D D C D C D C D D C D C - C# - D - Eb - E - F - F# - G - G# - A - Bb - B - C In Pythagorean tuning, as it happens, the diatonic semitones are _smaller_ than the chromatic semitones. To see why they differ in size, we might look again at our chain of pure fifths for this tuning: C = 7 fifths up |---------------------| Eb Bb F C G D A E B F# C# G# |--------------| D = 5 fifths down A diatonic semitone such as E-F is formed from a chain of _five_ fifths; if we start at the lower note of this interval E, we must move five fifths _down_ the chain to reach the upper note F. Note that this relationship also holds for B-C, D-Eb, A-Bb, F#-G, C#-D, F#-G, and G#-A. It is semitones of this kind which are the usual melodic semitones of medieval European music, for example in our sample 14th-century cadences above involving accidental steps (G#-A and C#-D, or Eb-D). Chromatic semitones such as F-F#, however, are formed from chains of seven fifths _up_. Other chromatic semitones are Bb-B, Eb-E, C-C#, and G-G#, each illustrating this same relationship on the chain. The use of these intervals as direct melodic steps is rather unusual in this era, but Marchettus of Padua and some other Italian composers of the 14th century do it boldly and beautifully. How large is each of these semitones? One way to compare their size uses the modern yardstick of _cents_, with a pure octave divided into 1200 equal parts or cents. Keeping the math simple, we can take the size for a pure fifth (a ratio of 3:2) as a rounded 702 cents, and of a pure fourth (at 4:3) as a rounded 498 cents. These two intervals add up to a pure 2:1 octave, and their sizes add up to 1200 cents. A diatonic semitone such as E-F is five fifths down; another way of looking at this is to say that it's five fourths up, starting again at the lower note E and moving to the upper note F: 498 498 498 498 498 E3 - A3 - D4 - G4 - C5 - F5 Adding up these five fourths, we get an interval of (498 x 5) cents, or 2490 cents. Actually, in moving up five fourths, we've moved up a semitone plus two extra octaves (E3-F5), as the octave numbers for the notes in our chain show. However, by now moving down two octaves or 2400 cents from F5, we arrive at the upper note of our desired diatonic semitone E3-F3. This interval has a size of (2490 - 2400) cents, or 90 cents. For a chromatic semitone, let's say F-F#, we similarly move up by seven fifths of 702 cents each, and then down by four octaves: 702 702 702 702 702 702 702 F3 - C4 - G4 - D5 - A5 - E6 - B6 - F#7 Our seven fifths up (F3-F#7) give us an interval of (702 x 7) cents, or 4914 cents; moving back down four octaves to the upper note of our desired chromatic semitone F3-F#3, we have a size of (4914 - 4800) or 114 cents. Thus our 12-note Pythagorean keyboard gives us an MOS with two and only two sizes of adjacent intervals: diatonic semitones at a rounded 90 cents, and chromatic semitones at a rounded 114 cents. A diatonic plus a chromatic semitone forms a regular whole-tone (e.g. E-F# from E-F plus F-F#) at around 204 cents: C D D C D C D C D D C D C - C# - D - Eb - E - F - F# - G - G# - A - Bb - B - C 114 90 90 114 90 114 90 114 90 90 114 90 There is an elegant poise and balance here which makes 12 an attractive number not only in a 14th-century Pythagorean tuning, but in various other historical European tuning systems fitting various eras and styles. However, one person's ideal "Moment of Symmetry" can be another person's overworn rut. By the earlier 15th century, only decades after the Halberstadt 12-note design had won out, European theorists were proposing tunings and keyboards based on the next Pythagorean MOS: 17 notes (also found in medieval Arabic and Persian systems). Other such larger MOS systems feature 29, 41, or 53 notes -- Chinese theorists, interestingly, being aware of a special property of 53. Other approaches to tuning can produce different MOS sizes: in early modern Europe of the 16th and 17th centuries, for example, the prevailing meantone temperaments for keyboards offered such sizes at 12, 19, or 31 notes, and instruments of all three sizes were designed and built, and music written taking advantage of the larger systems. (It's worth noting, as the earlier medieval instruments with from 8 to 11 notes show, that there's no law requiring that a keyboard size must match some MOS: for example, instruments of 13-16 notes were quite common in 15th-17th century Europe.) In sum, for 14th-century European musicians and organ-builders, 12 notes was an attractive point of repose; and its symmetrical qualities, as well as familiar keyboard ergonomics, still have their appeal. However, as musicians have recognized at various times and places in the six centuries and a bit more since, 12 is not the _only_ place to stop, and there are enticing if not so widely recognized reasons to explore larger tunings and instruments. ....................................................................... 5. Newbie Questions 2 ....................................................................... 5.1. What are harmonics? - Robert Walker../robertwalker/harmonics.html
What are harmonics?
Harmonics are whole number multiples of the basic pitch of a note. Many timbres are made up entirely of harmonics of various strengths.
Sounds of a flute, recorder, or ocarina can be simulated with just two or three harmonics. Other instruments will need a fair number, especially, string instruments have many harmonics.
A string player can selectively sound harmonics of a note by playing a note, and lightly touching on the string in various places.
It's also possible to train to hear harmonics.
One can also try singing the partials to a drone, and this is a lot of fun, and anyone can do it. Just play a drone, e.g. on a cello, then try to sing an octave above, an octave plus a fifth and so on - one can use a clip such as this one to give one guide tones to find the notes easily.
Here is the 'cello drone without the guide tones:
Even if you can't hear the harmonics in the timbre as such, you will very likely find you can sing them to the timbre as a drone, perhaps using a guide tone to find them the first few times. This is also a great way to learn to sing pure (just intonation) intervals.
One field where the recognition of partials in a timbre has been developed to a high degree is the craft of bell making. The more general term for a constituent frequency of a timbre is "partial", and bells have inharmonic partials - frequencies that aren't constrained to simple multiples or near multiples of the basic frequency.
The 17th century carillioneur (and recorder player) Jacob Van Eyck pioneered modern methods of tuning bells. This is done by ear, in combination with measuring instruments.
Interestingly, bell tuners also use ratios to target the partials. They make some use of the so called sub-harmonic series (the inverses of the harmonics) for the first few partials of a bell, favouring the ratios 1/12, 1/6 1/5 1/4 1/3.
http://www.oakcroft13.fsnet.co.uk/lehr.htm
This gives the characteristic minor chord type sound of a church bell.
This is because the physics of the bell makes it hard to target a major third. Major third bells do exist. Can be by serendipity as for this bell: but one can also nowadays make major third bells to order.
Another pattern of partials that concern bell founders are doublets - closely spaced pairs of partials caused by a small deviation of the bell from perfect symmetry.
Here is a fascinating web site about bell partials, which also has an excellent user - friendly program, designed for bells,that one can also use to find the partials in any instrument.
http://www.oakcroft13.fsnet.co.uk/index.htm
The numbers such as 9/8, 6/5 and 5/4 that one sees so often in definitions of scales often come straight from the harmonic series. So for instance, you get the 5/4 - major third, from the ratio of the fifth and fourth multiple of the asic note. The 6/5 is a minor third, 3/2 is a major fifth and 9'8 is a whole tone.
10/9 is another form of the whole tone. If you play harmonics 8 9 10 12, you get four of the five notes of the just intonation pentatonic scale. 8/8 = 1/1, 9/8, 10/8 = 5/4, and 12/8 = 3/2.
It is perfectly possible to play this fragment scale on the harmonics of a string instrument (and on other instruemnts that can be played in this way).
The missing note is the major sixth which is a third below the octave, so is at a ratio of 8/5 (i.e. 4/5 of 2/1)
The missing note is the major sixth which is a minor third below the octave, so is at a ratio of 5/3 (i.e. 5/6 of 2/1). Note that this is alsoa 10/9 whole tone above the 3/2.
You won't find this one in the same fragment of the harmonic series as it would be at a ratio of 40/3 to the fundamental.
However, if you multiply all the numbers by 3, you can then find the entire pentatonic scale in the harmonic series as
(8 9 10 12 40/3) times 3 = 24 27 30 36 40
These harmonics are a bit high even for a stringed instrument (can a skilled player play these? anyone know).
Some instruments use notes from the harmonic series when played normally, notably the natural trumpet of course. It is also possible for a flute player to play the harmonic series on a hosepipe by blowing across the end, playing it as an end blown flute.
There is quite a degree of interest also in scales based on inharmonic partials - i.e. using the constituent frequencies of a timbre such as a bell sound to make a scale.
There is also some speculation that the Indonesian gamelan scales may have originally been inspired partly in this way. (Ok to say this?)
....................................................................... 5.2. What is a Moment of Symmetry Scale (MOS)? - Robert Walker../robertwalker/mos.html
What is a Moment of Symmetry Scale (MOS)? - Draft
Answer:
The most consonant interval is the octave.
Ex.. a to a', 440 Hz to 880 Hz = double the frequency. As a ratio: 2/1.
Next most consonant interval is 3/2, e.g 440 Hz to 660 Hz.
All cultures recognise the octave as a consonance, and the 3/2 is also universally recognised, in fact, often when a group of people sing a song, some of those who are musically untrained may sing at an interval of 3/2 instead of 2/1, mistaking it for an octave, giving motion in parallel 3/2s.
Now, to build a MOS scale, start with two notes at a ratio of 3/2, such as our 440 Hz and 660 Hz.
Keep adding new notes at an interval of 3/2 above the previous note, and reduce by an octave (halve the frequency) if it goes above 880 Hz.
If you carry through this construction, you will find that usually there are three step sizes in the scale, but there is a "moment of symmetry" at five notes, when the scale has only two step sizes.
| 5 notes | 3 minor thirds, 2 whole tones | pentatonic |
| 7 notes | 5 tones, 2 (small) semitones | diatonic |
| 12 notes | 7 large and 5 small semitones | twelve tone |
| 17 notes | 12 large semitones, 5 Pythagorean commas | seventeen tone (arabic) |
| 29 notes | 17 Pythagorean commas, twelve other steps | twenty nine tone |
There are more moments of symmetry at 41, then 53 notes, ...
Scales built in this way from 3/2s are known as "Pythagorean".
If you go up two notes in any of the scales, again one has two interval sizes. E.g. in the diatonic scale, the interval will either be a major or a minor third (A to C = minor third, C to E = major third in C major).
In fact, you have two sizes of intervals for all the numbers of steps. A scale with this property is known as a Myhill scale. All MOS scales are Myhill, e.g. in the pentatonic scale the two step interval is either a major third (C to E say) or a fourth (D to G say), and so on.
In the same way, the Pythagorean twelve tone has two sizes of semitone, two sizes of tone, two sizes of minor third, two sizes of major third, and so on.
The Pythagorean twelve tone major thirds are 81/64 and 8192/6561.
Actually the sweetest major third you can play is a 5/4. So, in early times (especially 17th century and earlier), it was common to temper the scales by reducing the size of the 3/2 to get pure 5/4s.
Tempering to give a pure 5/4 gives the scale known as the quarter comma meantone. Tempering to give a pure minor third at 6/5 gives the third comma meantone scale.
Comma here = syntonic comma = the ratio between the pythagorean 81/64 major third and the sweet "just intontation" 5/4. It is 81/80.
In the construction, four 3/2s get you from 1/1 to 81/64, so to get a pure 5/4, you need to temper each one a little flat, by a quarter of the syntonic comma.
Quarter here means a quarter of the value of the comma in cents, not a quarter of its value as a ratio.
Three 3/2s get you to 27/16, which is a pythagorean minor third below the octave at 2/1. To get pure minor thirds, one wants it to be at 5/3, i.e. a pure minor third below the octave.
You will find that you ned to divide by 81/80 again to do this. Since there are three 3/2s this time, you need to temper each one by a third of the syntonic comma.
This makes sense since 1/1 5/4 3/2 gives a pure triad with the minor third between the 5/4 and the 3/2. So if you need to flatten the pythagorean major third by a comma to get the sweet j.i. triad, then you need to sharpen the minor third by the same amount.
These scales all have sharp fifths (flat by 5.38 cents for the quarter comma meantone 3/2). A musician will be able to hear this, but they are still reasonably consonant and perfectly usable.
However, at a moment of symmetry (MOS), all the intervals come in two sizes, so corresponding to the pure 3/2 is a wolf fifth, which for the Pythagorean scales is 678.495 cents instead of the pure 3/2 at 701.955 cents - this is very noticeably flat. For the quarter comma meantone, the wolf fifth is very sharp rather than flat, 737.647 cents.
Luckily there is only one wolf fifth. However, one in three of the major thirds is in the alternative interval size, and one in four of the minor thirds is. Unfortunately, in quarter comma meantone, the wolf fifth also makes a triad with one of the impure major thirds. This impure triad has wild beating, and is completely unusable for a chord intended to sound as a moment of consonance and rest. So the effect is that in quarter comma meantone, one of the twelve scales can't be used in the normal way at all. At the time, this was thought a small price to pay, as music didn't tend to modulate to distant keys anyway.
Quarter comma meantone gradually fell out of fashion from about the time of Bach onwards. Musicians came to relish the ability to modulate to distant keys more and more. Quarter comma meantone continued to be used for church organs for a long time.
In its place, various scales were used that temper the 3/2 in the opposite direction, flat rather than sharp. This time the aim is to facilitate modulation, and to keep the 3/2s as pure as one can in most of the scales. Such scales are known as well tempered scales.
The sweet 5/4 was no longer targetted - by this time musicians had got used to having sharp major thirds, and came to accept them as a consonance, and even to like them.
The modern tempering in which all semitones are equal in size is actually, surprisingly, a very early development, and was in use for lutes. Equal temperament major thirds sounds much sweeter on a lute than on a harpsichord, because of the harpsichord's prominent fifth partial. Also the lute is easier to tune to equal temperament by geometrical positioning of the frets. Tuning a keyboard to equal temperament is far harder, and in fact at the time, the best method to do so would perhaps be to tune a lute, then tune the keyboard to it.
When one tempers the 3/2, then the positions of the moments of symmetry change. So, for some temperings, one will get a MOS at 19 notes and 31 notes instead of 17 and 21.
If you temper the 3/2 to 696.774 cents then you get 31 tone equal temperament as a MOS - with all thirty one notes equal in size. This is a particularly popular scale amongst microtonalists, because it has relatively sweet major and minor thirds.
19 tone equal temperament is also popular, and has especially nice approximations to the septimal (or bluesy) minor third of 7/6.
In 19 tone scales of this type, one makes a distinction between sharps and flats, with Eb > D#. There is a sharp key for E and B too, with E# = Fb and B# = Cb.
You also get 17 tone scales - these have a distinction between sharps and flats, but E# = F and B# = C.
In 31 note scales, one uses half sharps and flats, giving six notes between C and D : C, C half sharp, C sharp, D flat, D half flat and D.
The pythagorean 17 note scale can be notated using sharps and flats too, - it can be presented as a twelve note scale with each of the accidentals doubled. This time, the C, again one has a distinction between sharps and flats, but this time, the Db is flatter than the C#. It is tuned so that the interval Db to F is a Pythagorean major third at 81/64, and the interval A to C# is also an 81/64, which places the C# above the Db.
So, some musicians think of C# as sharper than Db, while others think of Db as sharper than C#. Perhaps it is more common to use the 19 or 17 tone type convention where the order of the notes is C C# Db D. But, if playing in Pythagorean intonation, one would make the order of the notes C Db C# D.
Pythagorean intonation is one of the natural tunings that can be used on strings - if the open strings of a quartet are tuned to perfect 3/2s, then the high E of the violin will be a Pythagorean major third (plus some octaves) above the low C of the 'cello.
The fifth harmonic of the 'cello C (which one can easily sound by touching the string lightly one fifth of the way across while playing the open string) will then be flatter than the E string of the violin, by 21.5063 cents, quite a large amount really, and one then hears the 81/80 syntonic comma in action!
So, it would be quite natural for a string quartet to fall into Pythagorean intonation, and one would then find that the order of the notes was C Db C# D.
....................................................................... 5.3. What is an Euler-Fokker genus? - Manuel op de Coul../manuelopdecoul/efg-e.html
What is an Euler-Fokker genus?
In ancient times the Greeks made a distinction between different genera. The fourth was the interval between the two extreme tones of the tetrachord. Two other tones divided this distance into three intervals, in various different ways. This multiformity gave rise to a system of classification into genera: the enharmonic, the chromatic and the diatonic genus.The Euler-Fokker genera (plural of genus) are a different kind of genera, but it's the same sense of the word. In the Western music of today, not the fourth but the octave is the basic interval of tone relations. Therefore all tones are shifted as many octaves as necessary to bring them within the range of one octave. The relations between the intervals themselves form the distinguishing characteristic of the genus. An Euler-Fokker genus originates when one repeatedly adds certain pure intervals to a fundamental tone to form a network of tones. The pure intervals that are applied are the basic intervals of our tone system: the pure fifth (frequency ratio 2:3, therefore symbolised by the number '3'), the major third (frequency ratio 4:5, or '5') and the harmonic seventh (4:7 or '7', clearly smaller than the normal minor seventh and thus absent in the common 12-tone system). The construction of an Euler-Fokker genus is indicated by means of a kind of formula, for instance [33377] {D+,G-}. D+ (a small step higher than D) is the fundamental of the network. The genus is formed by adding three fifths and two harmonic sevenths. G- (a small step lower than G) is diametrically opposite to D+ and is being called the guide tone. The formula leads to the following network of tones, in which the horizontal connections represent fifths, and the vertical connections harmonic sevenths:
Bb- 3 F- 3 C- 3 G-
7 7 7 7
C 3 G 3 D 3 A
7 7 7 7
D+ 3 A+ 3 E+ 3 B+
[55777] {C,C#} can be sketched as follows:
F 5 A 5 C#
7 7 7
G+ 5 B+ 5 D#+
7 7 7
Bb- 5 D- 5 F+
7 7 7
C 5 E 5 G#
In this network the horizontal connections represent major thirds, and the
vertical ones harmonic sevenths. After stacking the intervals, the tones are
transposed down as many octaves as necessary to bring them inside the range of
one octave. One can choose the name of the fundamental at will, like D+ in the
first example. These genera have been employed by Alan Ridout.
We can clarify this first example by giving the frequency ratios:0: 1/1 C 0.000 cents 1: 9/8 D 203.910 cents 2: 8/7 D+ 231.174 cents 3: 9/7 E+ 435.084 cents 4: 21/16 F- 470.781 cents 5: 189/128 G- 674.691 cents 6: 3/2 G 701.955 cents 7: 27/16 A 905.865 cents 8: 12/7 A+ 933.129 cents 9: 7/4 Bb- 968.826 cents 10: 27/14 B+ 1137.039 cents 11: 63/32 C- 1172.736 cents 12: 2/1 C 1200.000 cents
The number of intervals which form the basis of an Euler-Fokker genus is called the degree. The genera [33377] and [55777] are genera of the fifth degree. Genera of a particular degree do not all have to have the same number of tones. The number of tones also depends on the exponents of the basic intervals. ([33355] can also be notated as [3³.5²].) The degree is thus the sum of the exponents. One obtains the number of tones by increasing all exponents with one and taking the product of that. So [3³.5²] has (3+1)×(2+1) = 12 tones. It's written sometimes also as [2m.3³.5²] to indicate the arbitrariness of the number of octaves, as in the following figure:
Fig.1: Spectra of the genera of the third degree, illustration from
Rekenkundige bespiegeling der muziek.
Each line herein depicts one tone of the frequency spectrum. The regular and
sometimes irregular character of the genera becomes visible this way.
They are of the third degree, so with three, not necessarily different,
generating factors. The tone lattice of the first three is 1-dimensional
(they contain one factor), the next ones 2-dimensional and only the last one is
3-dimensional because of the three different generating intervals
3, 5 and 7 - ignoring the octave with factor 2.
Euler considered in his Examination of a new
music theory (1739) only genera (named genus musicum) with
generating factors 3 and 5. Fokker
added the factor 7 to them and made an inventory of the different
genera (1942-1946, 1966). Euler had already noted the possibility of a factor 7
however, but left it at that. One doesn't have to leave it with these three
factors, a genus can be made with an arbitrary combination of two or more prime
factors.
An Euler-Fokker genus can also be defined as a complete
contracted chord. This term is Fokker's. The term complete chord
is coming from Euler. A complete chord has a fundamental and a guide tone,
with in between all tones that are both a whole multiple of the fundamental,
and a divisor of the guide tone. The guide tone is also a whole
multiple of the fundamental. For example if we take the number 1 for the
fundamental, and the number 12 for the guide tone, then the
complete chord consists of the tones 1, 2, 3, 4, 6 and 12.
We can also reverse this and say that no tones can be added to a complete chord
without altering the ratio of the fundamental to the guide tone.
This quotient of guide tone and fundamental is called the tension number
or exponens (by Euler: Exponens consonantiae).
Let's take as another example the following complete chord: 1:3:5:15. This is
the genus [35] {C,B} of the second degree. Bringing these tones within the
range of one octave, they become 1/1, 3/2, 5/4 and 15/8 (C, G, E and B), and
this is then a complete contracted chord. The 1/1 (C) now becomes the
substitute fundamental, and 15/8 (B) is called the substitute guide tone of the
complete contracted chord, identical to the fundamental and guide tone of the
Euler-Fokker genus. The shape of the tone lattice of an Euler-Fokker genus or a
complete chord is always a rectangle, or a rectangular parallelopiped
in the 3-dimensional case.
The mathematical definition of the fundamental is the greatest common
denominator of the tone frequencies. The guide tone is the lowest common
multiple of them.
The genera of the third degree are preprogrammed among others on the
12-tone manual of the Fokker-organ.
Although the Euler-Fokker genera are strictly speaking tuned pure (just), they
can also be played in the 31-tone equal tempered system because the three
basic intervals are well approximated in it.
The beatings caused by the slight impurity make the sound of them more lively.
The major third in 31-tone equal temperament is formed by 10 steps, the fifth
by 18 steps and the harmonic seventh by 25 steps.
With the above notation of + (1 step higher) and - (one step lower), a
step of 1/31 octave or a fifth tone is meant.
In Fokker's notation system this is indicated with the
and 
signs respectively. Bb- (3
steps lower than B) is B
en C#+ is
C
.
There are 10 genera of the third degree:
| [333] {C,A} | C - D - G - A |
| [335] {F,B} | F - G - A - B - C - E |
[355] {A ,B} | A - B - C - E - E - G
|
| [357] {C,A} | C - D - E - F - G - A -
B - B
|
| [377] {C,D} | C - D - F - G - G -
B
|
[555] {C,B } | C - E - G - B
|
| [557] {A,D} | A - B - C - D -
E - G
|
| [337] {C,C} | C - D - F - G - B - C
|
| [577] {C,B} | C - D - E - G - B -
B
|
| [777] {C,F} | C - F - G - B
|
See also the article Expériences musicales avec les genres
musicaux de Leonhard Euler contenant la septième harmonique, which contains
some scores of compositions of Adriaan Fokker and Jan van Dijk.
Go to the music page to listen to these
compositions via MIDI-files.
Euler and music is an article by Patrice Bailhache that also goes into the genera. The original French version is called La musique traduite en mathématiques: Leonhard Euler.
With the computer program Scala it's possible to calculate Euler-Fokker genera with arbitrary factors and of any degree and analyse them.
Manuel Op de Coul, 2000
....................................................................... 6. Notations ....................................................................... 6.1. 19-tET notation - Joseph Pehrson ../josephpehrson/19-tET_notation.html Q. How does one notate music correctly in 19-tET? A. 19-tET is actually an HISTORICAL tuning. It is one of a series of meantones which work on the "traditional" five lined staff! Our "traditional" staff originated with an early "Pythagorean" tuning of pure fifths and has been used for several historical notations since that time. This is one of them. If one is to notate 19-tET as a MEANTONE, in the traditional historical usage, one uses NON-EQUIVALENT enharmonics. In other words, C# and Db become two different notes! The correct 19-tone pitches for this scale would be as follows: C, C#, Db, D natural, D#, Eb, E natural, E#, F, F#, Gb, G natural, G#, Ab, A natural, A#, Bb, B natural, B# 19 pitches... ------------- ....................................................................... 7. Test entries ....................................................................... 7.1. Original text for the test links - Robert Walker ../robertwalker/test_no_auto_links.html Original text - no auto substitution of the links because of this text before the $.$.$, which has to be the first non blank line on the page. $.$.$ For the original text that got converted into this page, see ../robertwalker/test_no_auto_links.txt ==================================================== $.1 Using relative urls to make links ==================================================== See ../johnstarrett/guitar See ../alisonmonteith/why_study_tuning See ../jdl/ad Note that the file extensions can be left out, and the urls can be truncated. ---------------------------------------------------- $.1.1 Changing the text for the link ---------------------------------------------------- See ../margoschulter/why_12_notes -> See ../margoschulter/why_12_notes|Margo Schulter's entry: Why 12 Notes...| ---------------------------------------------------- $.1.2 Adding a link to any page on the WWW ---------------------------------------------------- http://groups.yahoo.com/group/tuning/messages| Tuning group messages | ---------------------------------------------------- $.1.3 Add a link to a section in another entry: ---------------------------------------------------- See ../margoschulter/what_is_microtonality#2.2.1 ---------------------------------------------------- $.1.4 link to another section on the same page: ---------------------------------------------------- See section $$.2 Switching off the automatic links. See $$.1 Using relative urls to make links See $$.1.1 Changing the text for the link ==================================================== $.2 How to switch off the automatic links ==================================================== Perhaps you need the $ sign in an ascii diagram: $$off $ $.$ $.1.$ $.1.1.£ $.1.2.1.$ $.1.3.3.1.$ $.1.4.6.4.1.$ This link will remain in its original form: see $$.1.2 $$on Another link to show that it is working again: See $$.1.2 Adding a link to any page on the WWW. ....................................................................... 7.2. Test links from ascii original drafts - Robert Walker ../robertwalker/test.html For the original text that got converted into this page, see 7.1. Original text for the test links - Robert Walker ==================================================== 7.2.1 Using relative urls to make links ==================================================== See 3.1. Guitar refretting - John Starrett See 4.2. Why study tuning? - Alison Monteith See 1.1. John deLaubenfels adaptive system - JdL Note that the file extensions can be left out, and the urls can be truncated. ---------------------------------------------------- 7.2.1.1 Changing the text for the link ---------------------------------------------------- See 4.3. Why twelve notes as _one_ attractive arrangement? - Margo Schulter -> See 4.3.Margo Schulter's entry: Why 12 Notes... ---------------------------------------------------- 7.2.1.2 Adding a link to any page on the WWW ---------------------------------------------------- Tuning group messages ---------------------------------------------------- 7.2.1.3 Add a link to a section in another entry: ---------------------------------------------------- See 4.1.#2.2.1 What is microtonality? - Margo Schulter ---------------------------------------------------- 7.2.1.4 link to another section on the same page: ---------------------------------------------------- See section 7.2.2 Switching off the automatic links. See 7.2.1 Using relative urls to make links See 7.2.1.1 Changing the text for the link ==================================================== 7.2.2 How to switch off the automatic links ==================================================== Perhaps you need the $ sign in an ascii diagram: $ $.$ $.1.$ $.1.1.£ $.1.2.1.$ $.1.3.3.1.$ $.1.4.6.4.1.$ This link will remain in its original form: see $$.1.2 Another link to show that it is working again: See 7.2.1.2 Adding a link to any page on the WWW. ....................................................................... 7.3. Test links from html original drafts - Robert Walker../robertwalker/test_html.html
Test of links to one of the faq entries from an html entry
You can use any of these methods:
See John deLaubenfels adaptive system (relative url)
html source has: href="../jdl/adaptive_tuning.html"
See John deLaubenfels adaptive system (abbreviated relative url)
html source has: href="../jdl/adaptive"
See John deLaubenfels adaptive system (link to original draft)
html source has: href="http://groups.yahoo.com/group/tuning/files/faq/adaptive/adaptive_tuning.txt"
See John deLaubenfels adaptive system (link to example on-site version)
html source has: href="http://homepage.ntlworld.com/robertwalker/site_ex/jdl/adaptive_tuning.html"
When the entry is placed into the on-site FAQ tree, all get converted to the relative url to the on-site document.
href="../jdl/adaptive_tuning.html"
....................................................................... 8. TL Posts ....................................................................... 8.1. Organs with split keys fewer than 17 tones per octave - Ibo Ortgies../iboortgies/20120.html
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....................................................................... 8.2. What is meantone? - Daniel Wolf../danielwolf/19368.html
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....................................................................... 8.3. What is the Scottish bagpipes scale? - David C Keenan../davidckeenan/20343.html
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