Logarithms of Primes in Bases

Properties of logarithms enable calculations to be conducted by simpler operations, provided the logarithms and antilogarithms are available for consultation from tables or by a mechanical slide rule or similar device if not by computation. A logarithm of a multiple of two quantities is the sum of their logarithms. This enables multiplication calculations to be reduced to addition. A logarithm of a fraction is the difference between the logarithms of its numerator and denominator, enabling division to be replaced by mere subtraction followed by conversion of the result to the power term by the antilogarithm or exponentiation. It follows from these rules that the logarithm of a composite whole number can be reduced to a sum of multiples of the logarithms of its prime factors, and indeed the logarithm of any rational number can be decomposed into addition and subtraction of multiples of the logarithms of the prime factors of the numerator and denominator of the fraction. Furthermore, logarithms can be used for calculating irrational numbers that result from algebraic operations on rational numbers. A rational number can be raised to any fractional power by applying a logarithm, which can then be manipulated by the rules of logarithms to provide the fraction multiplied by the logarithm of the rational number. For example, the rational number three to the power the fraction a half is the same as the irrational square root of three and can be calculated with the aid of logarithms by taking the logarithm of it to any base, resulting in half the logarithm of the prime number three to that base. That result is then converted from a logarithm to a power term by exponentiation to furnish the square root of three to an accuracy of the number of significant figures allowed by the logarithms and antilogarithms available.

While the rules of logarithms apply no matter what consistent positive real base they have, it is clear that logarithms would be especially useful if the base chosen were one such that the logarithms of the simplest prime numbers to that base were approximated peculiarly well by a limited number of significant figures of their logarithms. Unfortunately, logarithms are usually transcendental irrational numbers that cannot be exactly represented by terminating strings of numerals in positional notation. Nevertheless, sometimes the logarithm of a number such as a prime number can be approximated well enough to a few significant figures, which would make arithmetic with them involve fewer steps and be faster. For example, if the base of the logarithms is chosen to be close to a root of the prime number two whereby two is raised to the power of a unit fraction or reciprocal of a counting number, then raising that irrational base to the power of that counting number will result nearly in the prime number two with the consequence that the logarithm of two to that irrational root base will be nearly a whole number capable of being represented by a finite number of significant figures. The creation of such a root of two as a base is equivalent to a temperament of the musical octave into equal geometric steps, with the ratio between the frequencies of adjacent notes a step apart being the base of the logarithms. The task of finding a base of logarithms such that the logarithms of the smallest prime numbers are approximated very well by terminating numbers is hence equivalent to finding temperaments of the octave that align with the frequencies of the first prime numbers as harmonics.

A number of temperaments of the octave are already known to provide good coincidences with the first few harmonics. The most widespread is the equal temperament of the octave into twelve semitones, which corresponds well with the harmonics that are powers of the prime numbers two or three, and to lesser accuracy the prime number five. Other temperaments can approximate more of the small prime numbers. For example, temperament of the octave into six dozen equal step notes can provide good enough approximation to the first few smallest prime numbers two, three, five, seven, and eleven. Since the twelve semitones are a subset of these six dozen steps and six dozen is half the second power of twelve, temperament of the octave by a power of base twelve is particularly useful for music and would be good as a base of logarithms for mathematical calculations. While there is little wrong with temperament of the octave into six dozen steps in music, a slight annoyance could arise with this formulation of the base of logarithms mathematically because there are two bases involved: binary, the logarithm to the base of which gives the number of octaves, and dozenal for writing the logarithms in numerals and expressing the number of steps within the octaves, but just one base for the number of different numerals and for the base of the logarithms and all computation would rather be desired. Is there a way to have the same base for the base of the logarithms and for the number of different digits by which the logarithms are notated? The answer is almost.

If the problem is approached initially by finding base B such that B^(1/B) is nearly the frequency ratio of the step in the octave desired, then the same base can be used as the base of the logarithms and as the number of different numerals for notating numbers such that the logarithms will tell the number of those steps, but the logarithms will not necessarily round well at the octaves to powers or whole number multiples of that base B. It so happens that the temperament to six dozens steps per octave is approximated when the base B is the square or second power of twenty-six. A base for the logarithms can be constructed by sectioning the square of twenty-six into twenty-six steps geometrically. The logarithms of the smallest prime numbers two, three, and five to that step as base, which is equivalent to half the logarithms to the base twenty-six, will be particularly accurate to two significant figures when written in base twenty-six. The logarithm of the prime number seven to this base is not as accurate at just two significant figures. Thus, computations using logarithms of numbers containing the first three prime numbers would be unusually convenient in base twenty-six, which could be called base two dozen plus two or twenzy-two in another form of dozenal parlance. There are as many upper case letters in the modern English variety of the Roman alphabet to use as symbols for the numerals for this base.

More accurate results for the logarithms of also the next prime numbers seven and eleven can be had to two significant figures by base thrice eleven, using a third of the logarithm to that base. There are said to be as many runes in the English futhorc as the number of different numerals required for this base thrice eleven. However, apart from such a base being an odd one to use for general purposes in society, like base twenty-six it would not round to whole multiples for the number of steps at the octaves.

Is there a way to allow the numbers of temperament steps at the octaves to be round multiples of the base? This is possible if the step size for the base of the logarithms can be defined in terms of powers of the number of equal temperament steps per octave. To make that number of steps a convenient base to use, it is made to be a power of twelve, either twelve itself or its square. For a step near two to the power of a square twelfth, the square of thrice thirteen is close to the base in B^(1/B) required for the logarithms, but the logarithm of two is not as accurate to two significant figures and the notation would have to be in base twelve for the numbers of steps at the octaves to be rounded.

More conveniently, the base of the logarithms for the semitone is given nearly by the cubic dozenth root of twelve to the power of forty. When such logarithms are written in base twelve, there are few significant figures to a good degree of accuracy for the first few prime numbers. The prime number seven requires more significant figures for a not worse level of accuracy, but they are not too many to be useful in approximate calculations. These logarithms give approximations for the numbers of semitones for the harmonics, which are easy to remember when musical theory is understood. These logarithms are defined in terms of powers only of the base twelve and can be reduced to simple arithmetic using the base twelve logarithms.

It should be mentioned that the fortieth root of ten also nearly gives the semitone step, but expressed that way the base of the logarithms and the base of numeration to allow rounded numbers of semitones at the octaves for this to be useful in conjunction with musical theory are not the same. The common decimal logarithms of the smallest prime numbers two, three, and five are accurate enough to two significant figures if multiplied by forty or four. A thousand to the power of a hundredth partitions the octave into ten equal steps. Not many useful bases B have that property of having whole number values of n and m to make B^(n/B^m) partition the octave into nearly B steps. Binary bases, decimal, and dozenal do. The dozenal option that I have described has the extra benefits of giving accurate values for the logarithms of the first few prime numbers by a small number of significant figures and agreeing well with the most normal musical temperament.

Invention of Musical Wind Instruments

Prior Art: Musical wind instruments are of various kinds functioning on slightly different principles. They all involve vibration of air in a cavity and differ in the shape and material of the cavity as well as the method by which the air is set in motion.

The earliest type of wind musical instrument preserved is probably the flute, which is a hollow mainly cylindrical tube with holes along the length that can be blocked by the fingers. With some exceptions, opening the holes shortens the column of air vibrating and increases the pitch or frequency of the note. The air is set vibrating by blowing against an edge at one end of the flute, either longitudinally or transversely. In whistles and recorders blown longitudinally, the mouth blows air across a fipple. The oldest flutes that survived were made of bone, but they could also be made from wood or cane. Folk or traditional flutes are made of wood, while the modern concert flute is made of metal. In pan pipes, there are no finger holes but rather a different length of tube for each note.

In woodwind and certain pipe instruments, the air is set vibrating at the mouthpiece by a thin flat reed, either double as in shawms, oboes and bassoons or single as in clarinets. The modern saxophone has a single reed mouthpiece like a clarinet but is made of metal and has a conical interior. These instruments change pitch by holes along the length. Organ pipes, typically made of metal, also use a reed, but tend to be of fixed length for each note.

Another ancient type of wind musical instrument is a horn or trumpet. The horn has a conical rather than cylindrical cavity. At first, these would have developed from animal horns, before being fashioned of metal, leading to modern brass instruments. The air is set vibrating in them by the buzzing of the lips on a cup mouthpiece. The pitch is lowered by increasing the length of the tube. In horns, the extra length is attached as crooks. In modern horns and trumpets, the crooks are fixed to the instrument and the air flow through them is regulated by valves and pressing lever or piston keys with the fingers. In the trombone the lengthening is achieved by sliding an extra length of tubing. Trumpets and trombones have a more cylindrical or less conical shape than horns and tubas. Trumpets are amongst the loudest of orchestral instruments. Mutes of various kinds can be inserted onto the bell end of brass instruments to muffle the tone to some degree. The pitch can also be altered to some extent by such mutes or stopping the bell end with the hand in the case of the horn.

The ocarina is a very ancient instrument that is a cavity of air of any shape enclosed within a vessel of any material, but often clay, and functions as a Helmholtz resonator. The air is set vibrating in a way similar to that in a flute or by a fipple. Pitch is altered by the opening or closing of holes by the fingers. It is the size of the holes that determines how much they alter the pitch, while their positioning is irrelevant to the pitch. Opening smaller holes increases the pitch less than opening larger holes.

My Proposal: For design of a wind instrument to be as simple as possible in its construction yet capable of producing all semitones in its range of discrete pitches, I propose first creating a chamber like an ocarina in the shape of a horn stoppered at the bell end by a detachable stopping mute. The stopping mute shortens the effective vibrating length of the waveform and raises the pitch by about a semitone. To overcome the decreased projection of the sound as a result of the stopping mute, holes drilled into the material along the length of the resonator body are to be opened by the action of the fingers to allow the air to escape. The diameters of the holes are such as to increase the pitch in intervals when opened. The smallest hole is to be such as to increase the pitch by one semitone; the second smallest hole is to alter the pitch by a whole tone or two semitones; the third smallest hole is to increase the pitch by a minor third or three semitones when opened; the fourth hole is to increase the pitch by a perfect fourth or five semitones; and the fifth hole is to increase the pitch by an octave when opened as an octave or descant key. The choice of intervals for the holes is to emulate the interval changes resulting from the customary keys on a brass instrument. Thus, the smallest hole of this design is to mimic the middle or second key of a horn which when depressed by the middle finger lowers the pitch by one semitone; the second smallest hole mimics the first key that when pressed by the forefinger lowers the pitch by one whole tone; the third smallest hole when pressed over by the finger resembles the third key of a modern brass wind instrument in lowering the pitch by a minor third. The fourth smallest hole resembles the mechanism of changing between the F and B-flat modes of a modern double horn in the interval by which it modifies the pitch, especially in those models in which pressing a thumb key opens the air flow to the extra length of tubing for the F-horn compared to the B-flat horn. The octave or descant hole mimics the operation of a descant horn in a modern triple horn. Thus, the design of the five holes serves an educational role in how the notes are to be keyed in brass instruments. By the combinations of fingering over these five holes, all of the semitones in a range of two octaves less a semitone may be played, which is a useful range for a melodic musical instrument. The air is set in motion at the mouthpiece end, which is to be detachable to allow adaptable insertion of the various kinds of mouthpiece. This much describes the stoppered mode of the instrument.

When the bell stopper is removed, the instrument behaves like a conventional unmuted wind instrument in regard to the quality of tone when all holes are still closed. However, the presence of opened holes along the length of the tube can affect the quality of the tone produced, similarly to wind instruments such as flutes that have finger holes.

Since the unstoppered instrument no longer acts as a Helmholtz resonator, the positions of the holes along the length of the instrument must be chosen so as to match the intervals by which they alter the pitch of notes. The effective length of vibrating air in the column is inversely proportional to the frequency. Thus, to increase the pitch by a semitone, the length of vibrating air ought to be shorter from the original length by a factor of the reciprocal of the twelfth root of two. Hence, the semitone hole should be positioned a twelfth root of the length of the column away from its original end. Similarly, the whole tone hole should be positioned from the bell end by a factor of the sixth root of two of the original effective vibrational length. The finger hole for modifying the pitch of notes by a minor third should be positioned at a distance of a quartic or fourth root of two of the original effective vibrational length from its bell end. The finger hole for the interval of the perfect fourth should be at three quarters of the original length of the tube. The octave or descant hole should be at half the length of the tube.

The coiling of the instrument may allow the finger holes to be reached ergonomically by the fingers of one hand despite their positions along the length of the tube. Unfortunately, in a conventional keyed brass instrument, the third key that alters the pitch by a larger interval than the first and second keys is positioned closer to the bell than the first two keys, whereas in the proposed design it ought to be upstream of the first two keys. The reason for this discrepancy historically may be that the first two keys are more frequently used and consequently have the more dexterous forefinger and middle fingers assigned to them.

In order for combinations of open holes to be effective in producing predictable interval changes to pitches as simple combinations of their individual frequencies, the diameters of the finger holes must be small enough to enable the waveform to bypass an upstream open hole and be affected in pitch to the right amount by a closed downstream hole. A similar effect can be seen for example in the flute. I devise notation whereby an open hole can be denoted by a circle or the symbol 0, a closed hole by a vertical line or the symbol 1 representing the finger, and a half open hole by a vertical line in a circle, or the symbol Φ. The left hand fingering is to be preceded by the abbreviation of the initials L.H., while the right hand fingers are to be denoted by the preceding abbreviation R.H. Thus, on a flute constructed in the pitch of C, the note G having the fingering L.H. 1111 R.H. 0001 may be altered to a semitone lower by closing holes downstream of the uppermost open one by the fingering L.H. 1111 R.H. 0111 or may be lowered by a whole tone to the note F with the fingering L.H. 1111 R.H. 1001 by the closing of the highest open hole. In the instrument the design of which I propose, the closing of the two holes merged and positioned as though they were a single hole in the flute to lower by a semitone is analogous to the closing of a single semitone hole in my design. Similarly, the lowering of the pitch by a single tone by depressing a finger hole in the flute is analogous to lowering by a whole tone in my proposal also by closing one hole by a finger.

In order for this mechanism to be as effective as possible without reducing the sonorous quality of the tone of notes produced, the holes should neither be too far apart not too large. For example, in the case of the octave or descant hole, the diameter of the hole must be small in the unstoppered mode of operation in order to allow the simple combination of its interval with those intervals of the other holes, whereas in the Helmholtz resonator mode, the octave hole must be the largest of the holes mentioned. To overcome this conflict, at the position of the octave hole, there may be two holes side by side, one smaller and the other larger, which when combined would have the correct large area for the Helmholtz resonator, while only the smaller of these two holes would be opened to access the descant in the unstoppered instrument, by a sideways motion of the finger or thumb as the case may be.

In a brass wind instrument without finger holes open to the atmosphere, harmonics above the fundamental for the length of the tube may be accessed by increasing the tension of the buzzing lips, and a great dynamic range is accessible through changes in the force of breath. In instruments such as flutes with finger holes open to the atmosphere on the other hand, changes in force of breath may negate their combined effect and there is an increased risk of the pitch jumping between registers with changes in amplitude intensity or volume because the register is controlled more by the speed of the breath than tension of the lips compared to brass instruments. A result expected for the proposed design therefore would be decreased accessibility of the loudest dynamics at lower registers compared to conventional brass instruments. However, this risk may be overcome to some extent in performance by plugging the finger holes of the larger descant octave and perfect fourth intervals when a greater number of harmonics are being accessed through tension of the lips such that fewer keys would be required when the bell is not stoppered. In conventional brass instruments, only three keys are necessary for accessing every semitone note in the entire range of the instrument. It would only be for playing the Helmholtz resonator bell stoppered mode that availability of opening of larger holes for the bigger intervals of the octave and perfect fourth would be required due to the fewer harmonics achievable.

The need for the holes to be closed by the fingers limits the size of the instrument and therefore prevents its fundamental pitch being too low. In conventional brass instruments with cup mouthpieces, the higher harmonics around the fourth power of two are easier to play by requiring less tension from the lips when the fundamental pitch of the instrument is lower. In smaller instruments, there is more reliance on the lower harmonics and keys to achieve the different notes. A smaller instrument of the proposed design could nevertheless reach higher notes than conventional brass instruments of the same size because of the opening of the hole to raise pitch rather than the mechanism of keys to crooks that lower the pitch.

In summary, my proposed invention offers advantages of a simpler construction without complicating moveable key mechanisms and a smaller size to achieve a high range in comparison to modern brass wind instruments.

I call my invention an ocorn or occorn. I also considered okhorn, but according to a trademark database search, apparently there is already that word in a trademark.