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878 L. E. WAD1) INGTON

materials will often be determined by suchfactors as the rate of aging or deterioration underthe action of oil, water, solvents, ozone, andrange of operating temperatures.

Although elt is not effective vitration isola-tioii for exciting frequencies )elow 40 1o 50 c.t).s.,:I highel' fre(tten'ies l i]/ the :tdible ralgC

it produces ffective solation and the thickness,pressure, nd type of felt are not at all critical.

The authors wish to express heir appreciationlo the Western Felt Works for permission tpublish the results of the tests mid to Dr. H. A.l,eed for initiating lhe investigation and eCOl.'agi the wrk lhnt14hnl he troje-I,

THE JOURNAl. OF TIlE ACOUSTICAL SOCIETY OF AMERICA VOLUME 19, NUMBER 5 SI,P I'IMBk'R, 1947

A Slide Rule for the Study of Music and Musical AcousticsL. E. WADDINGTON*

Miles Laboratories, Inc., Elkhart, Indiana

(Received May 25, 1947)

Musicians are seldom concerned with the mathematical background of their art, but ailunderstanding f the underlying physical principles f music can be helpful n the study ofmusic and in the considerations of problems related to musical instrument design. Musical dataand numerical standards of the physics of music are readily adaptable to slide-rule presentation,since hey involve relationships hich are the same or any key. This rule adjusts elativevibration rates, degrees f scale, ntervals, chord structures, cale ndications, nd transpositiondata, against base of the piano keyboard. t employs nd relates everal tandard vslems ffre!ency evel spe('itication.

HEheoryfmusics ommonlyhoughtfn terms of scales, ntervals, and harmonicrelationships. eople who have studied heoryhave learned all of those fundamentals and sup-posedly ave them in mind for instant use. Onthe other hand, the physics f music s consideredin terms of numbers applied to those samemusical elationships, nd the physicist an per-form all kinds of musical calculations, oftenwithout any understanding f harmony as amusical concept. The need for a referenceshowing he physical nd the harmonic elation-ships ncountered n music rompted he prepa-ration of this slide rule.

The Acoustical Society of America, through anappointive ommittee n music, s encouragingimprovements n musical nstruments hroughresearch n musical matters, where the science of

This s an adaptation f a paper n this subject hichwas ecorded n disks with appropriate musical backgroundand demonstration sound effects, and illustrated with 26Kodachrome lides. t was presented t the AcousticalSociety meeting n New York, May 9, 1947.

* Formerly Design ngineer, . G. Corm, id., Elkhart,Indiana.

acoustics and the art of music possess ommolinterests.

Although many members of this Society arehighly trained in music as an art as well as inacoustics as a science, t may be well, through thediscussion of this slide rule, to reaffirm some of theacoustical standards which have been previouslypresented o the Society, and to tie them in withsome of the fundamentals of music and musical

instruments. These data are readily adaptable toslide-rule presentation since they involve manyrelationships which hold true regardless of key inwhich they are considered. A simple guide such asthis can serve as a reference for the acoustician as

well as the musician when studies are pursued ineither direction. (Figure 1.) (Sections of the ruleare shown in this and succeeding iews for clarityof presentation.)

The face of this rule has a lower rail on which is

imprinted a piano keyboard consisting of thestandard 7-} octaves, identified by the black andwhite keys and designated by letter notation(A, B, C, etc.), standard piano key numbers(there are 88 keys on the piano), and by a

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A MUSICAL SLIDE RULE 879.

subscript otation which will be further described.The top rail is equally divided to correspond othe lower rail, and the divisions are identified bychromatic notation as well as by an additionalfrequency level designation known as semitone

count. Subscript umbers nd semitone ount 'eprin ed in red.Our standard of musical pitch, having moved

up and down the scale since the 17th century, isnow fixed at A-440, this having been adopted byA.F. of M. in 1917, accepted by Music IndustriesChamber of Commerce n 1925, and approved byAmerican Standards Association in its standards

on acoustical erminology rom 1936 o date.Moreover, steps have been taken to make this theinternational standard. In the equally temperedscale based upon the American standard ofA=440 cycles per second, here is a "C" whichis just about at the threshold of hearing. It isconvenient o use this as a reference requency asproposed y Fletcher and, following oung, tocall it Co. The tones of the first complete octaveof the piano keyboard starting with the lowest C

are thus identified by the subscript one (1), andthe three tones below that octave drop into thefrequency range of the octave identified by thesubscript zero (0) (Fig. 2). Using Co as a referencefor counting semitones and remembering that

there are 12 semitones n any octave, then C canbe designated as number 12. Similarly, MiddleC4, four octaves above Co bears the number 48(equal to 4X12) as a semitone count for fre-quency-level determination.

The frequencies of all of the equally temperedscale tones within the frequency range of thepiano keyboard are printed in the groove, andindicated by either end of the slide (A. is indi-cated as 110.0 cycles per second. Three octavesabove is A5 equal to 880.00) (Fig. 3).

For purposes of reference and comparison, theJust major scale fractional ratios from a givenstarting tone, and the ratios between tones, areindicated on one space of the slide. (Fig. 4.) Inanother position, the ratios of the' equally tem-pered scale are indicated for one octave in termsof relative vibration numbers: Remember 'one:

Fro. 1. Face of the rule with slide in normal position.

Musique et Instruments 29,' 237, 263, 283, 287 (1938); also British National Acoustics Committee Resolution of 1938.Fletcher, J. Acous. Soc. Am. 6, 59-69 (1934).R. W. Young, J. Acous. Soc. Am. 11,134-139 (1939).


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A MUSICAL SLIDE RULE 881

Fro. 4. Just major scale fractional ratios based on C.

"While this may not be true today, yet the con-struction of the equally tempered scale s clearlyscientific, and it is no doubt true that the rela-tions of the major and minor scales, and thenature of chords and their various forms and

progressions, as well as various other funda-mental principles can be explained better byscience han precept."

Another section of the slide shows that divi-

sions of the octave can be indicated as degrees of

FIG. 5. Equally tempered scale ractional ratios, or relative vibration numbers, based on C.Also the harmonic series, showing irst five modes of vibrations.


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882 L. E. WADDINGTON

Fro. 6. Intervals of scale indicated for the C scale.

scale in which the scale is spelled by names,Tonic, Super Tonic, Mediant, Sub Dominant,Dominant, Super Dominant, and Sub Tonic. Stillanother approach outlines the divisions of the

octave when indicated as Intervals of Scale from

Prime, minor 2, M aj. 2, min. 3, M aj. 3, Perf. 4,Aug. 4, Perf. 5, min. 6, M aj. 6, min. 7, M aj. 7 andthrough he Octave (Fig. 6). And finally the slide

Fro. 7. The major, minor, ugmented, nd. iminished hords ndicated n the key of Fo


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Fro. 8. Major and relative minor scale spelling s indicated. or the key of F.

Fro. 9. With key note set on F, transposition f all tones f the scale s ndicated or nstrumehts uilt incommon keys, by reference o the degrees f scale.

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termined for the 7th, 9th, 11th, and 13th chords,and all chords may be determined in any key byadjustments of the slide so that the indicator sset to the key note (Fig. 7).

If information about scales s desired, a sectionof the slide ists tones composing ll major scales,relative harmonic and melodic minor (both upand down), in addition to whole lone andpentatonic cales. he key note of each of theseis the first degree of the scale and is printed inred. It can be set to any desired key and thedegrees f scale will indicate he proper composi-tion of that scale. n Fig. 8 the key note is set at Gand the major scale s spelled G A B C D E F G.The relative minor starts a third lower on E and

is spelled E F G A B C D E. The melodic minor

is similarly spelled up and down. Also includedare the whole tone spelling and the pentatonic asused by the Chinese.

Musicians are often required 'to play in otherkeys han that in which he music s written. Forsimple note ransposition, he concert ey degreesof scale are also named by letter notation, and thefundamental has only to be set to the desired keyand all intervals of both scales will be properlyaligned or transposition. For example, f it werewished to transpose n intervals equal to thatfrom F to C, those two tones are lined up and

automatically all other tones in the scale indi-cated by letter notation would fall in line (Fig. 9).

By use of the transposition scales, correct keysmay be determined for all B[, A, F, E[, and D[instruments (the commonly used band andorchestra instruments) working from any givenconcert key. To demonstrate this, the slide ismoved so that the red fundamental of the C

instruments s set to F, supposing hat a compo-sition under consideration s written in the keyof F. Now the red fundamental of the B[ lineindicates hat the B[ instruments would play inthe key of C, and the E instruments in the keyof D. The same setting would indicate the trans-position of any tone or series of tones, or with thefundamental of the C scale set to F, the Cinstruments would play a major scale in F asindicated by the degrees of scale, the B[ instru-ments would play the scale of G, and so on. Withthis information on the movable slide, transposi-tions may be determined, from any given majoror minor key to any desired major or minor key.

On the back of the movable slide, a way isprovided for locating the tempered approxima-tions to a harmonic series of sounds. These maybe the modes of vibration from 1 through 20nominally expected from brass instruments. Forexample, if a trumpet is of such length that itsfirst mode is B[., then the other modes wouldproduce Bla, F4, Bt4, Ds, Fs, etc. If tone qualityis being considered, hen the slide can be similarlyused to locate the names of the tones nearest to

the harmonic partials. (See Figs. 5 and 6 for theharmonic series showing modes of vibration basedon C.)

Other miscellaneous bits of musical informa-tion have been added to the back of the rule. A

range chart is included (Fig. 10) which shows he

"as sounded" range of string, reed, and brassinstruments, related to subscript notation. Forexample, the violin range is Ga to C7. This can bechecked against the staff location of subscriptnotation, or against the piano keyboard on theface of the rule.

There will also be found a "Circle of Fifths"

from which the names of major and relativeminor scales may be seen along with the signa-tures (number of sharps or flats) of both, and theorder of key progression n a clockwise irection.The names of sharps and flats and their order of

addition may be determined from a table(Fig. 11).

Common tempo markings in terms of metro-nome markings are listed from Larghissimothrough Prestissimo. Adagio, for example, withmetronome-mark of 66, means a rate of 66 beatsper minute and is useful as the composer's ndica-tion of the standard time of a composition.) Timesignatures re shown elated to their basic unit ofmeasurement, the eighth note, quarter note orhalf-note. And finally it is again pointed out thatthe standard pitch is based on the A =440 cyclesper second.

Future acoustical research will explore thevarious fields of musical and scientific heory andwill establish additional basic standards of ex-

pression. Musical instruments will be improved.If each worker in this field is aware of pastprogress n both the science and the art, theadvances of the science an be applied to the artwith resulting appreciation and enjoyment ofmusic by increasing numbers of people.

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Jas 000878