Octave Types

8/13/2019 Octave Types

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Achieving 21stCentury Standardsof

Excellence in Tuning

Aural Tuning Tests

for

2:1, 4:2 and 6:3 Tye !ctaves

"ill "re##er $%T&adison, 'isconsin

&arch, 2(()


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There have recently been many questions posed about octave sizes, the differences between them and how totune them. The fact is, that you can construct an Equal Temperament within any size octave, a 2:1, 4:2 or :! oreven a wider than a :! octave or a sli"htly narrow octave if that is your "oal. #t is simply the consensus ofopinion today that a compromise between a 4:2 and :! octave is the optimum. The very sli"ht beat you will hearin such an octave will ma$e your %& a little purer and your %4 and '! a little faster than what is theoreticallycorrect and what all the old boo$s on tunin" say is correct.

(one of these boo$s that # $now of, say anythin" other than to ma$e a pure soundin" octave. #n reality, anoctave can only be truly pure, meanin" havin" no audible beat, between any two matchin" sets of coincident

partials)e*planation to follow+ but there is indeed a small ran"e within which an octave may sound beatless orpure to the ear. rom what we $now today however, tunin" an octave from -4 to -! that sounds pure to the ear,when startin" the tunin" process with no rapidly beatin" interval chec$s, would most probably create an octaveanywhere from a 2:1 to a 4:2 type. This article will e*plain in detail in how to tune any size octave you chooseand how to $now the difference.

efore "oin" any further, let/s define 0ust what these numbers, said to be octave types,are. They are ratios, yes,but they are not the equivalent of each other as they would seem to be theoretically. This has been a source ofconfusion to some because ratios are used in definin" other intervals. - &th is a !:2 and a '! is a &:4 fore*ample. These are fi"ures which also point topartialswhich coincideand indeed a &th can often have morethan one set of audible coincident partials )the definition to follow+ but a '! and a &th are not usually discussedas typesthe way octaves are, only whether they are tempered or not and to what de"ree.

Theoretically, an octave has a ratio of 2:1 and any multiple thereof, such as 4:2, :! and beyond are theequivalent, merely fi"ures unreduced to the lowest common denominator. The ratio of 2:1 indicates simply thatthe upper note of the octave theoretically has twice the frequency of vibrations per second as the lower. utwhen discussin" octave types, these fi"ures indicate what are $nown as coincident partials. The words,partial,harmonic and overtone all refer to the same phenomenon and are "enerally interchan"eable but pianotechnicians usually prefer to use the wordpartial)used as a noun+ when discussin" tunin".

- partial is one of the hi"her and usually fainter )but not always+ sounds that we hear alon" with the fundamentalwhich is the pitch we thin$ about and reco"nize when any "iven strin" of the piano is struc$. The fundamentaltone or pitch is also considered to be the first partial. Theoretically, an upper or hi"her partial is an e*act multipleof the fundamental pitch, such as twice the frequency, two and a half times the frequency, three times thefrequency, etc. 'ost people, even musicians do not perceive )hear+ partials. Even many piano technicians "otheir entire career sayin" that they do not hear them. thers consider hearin" partials as essential to tunin".(ot to worry however, because when we hear beats in any interval, we are hearin" partials which aremismatched or not e*actly in tune with each other. #ndeed, the very essence of aural tunin" depends upon theability of the tuner toperceiveand control beats.

-ny strin" of the piano has a series of partials which, if transcribed to loo$ li$e music, appear to be a lar"e,spread out, dominant seventh chord with e*tensions. There can be as many as 24 audible partials from thelower wound strin"s. sually however, we only concern ourselves with the first 8 partials which ma$e up whatappears to be when written out on a music manuscript "rand staff, a lar"e dominant seventh chord which coversthree octaves. 3hen you play any particular note of the piano, you of course do not hear what sounds li$e achord )one usually only perceives the fundamental+ but try this tric$ to actually hear them: press slowly enou"hone of the lower notes of the piano, ta$e 2 for e*ample, so that it does not sound and hold it open )usin" thesostenuto pedal if the piano has one+ and play in staccato fashionthe partial series up to the ei"hth partial, youwill clearly hear each note of the chordcome eerily out of the open 2 strin". 5old open 2 and then stri$e !,

6!, 4, E4, 64, -7sharp 4 and &. 2 is the first partial or fundamentaland & is the eighth partial in thise*ample.

3hen you play any octave, you will naturally have sets of partials from each strin" which sound side by side orcoincide, thus you havewhat are $nown ascoincident partials. ut to complicate matters "reatly and "ive thetuner the ultimate and nearly unsolvable problem and challen"e, all piano strin"s bear a phenomenon $nown asinharmonicity. asically, this means that the upper partials are not e*act mathematical multiples of thefundamental tone, as they are thou"ht of theoretically. They are always somewhat sharper than a theoreticalpartial would be and the hi"her the partial, the more e*a""erated the sharpness. #nharmonicity occurs because a

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piano/s strin" is made of steel which has a certain stiffness factor to it which distorts what otherwise would fit atheoretical mathematical model.

#f there were no inharmonicity as there is none in an instrument such as a pipe or"an, all coincident partialswould match each other when an octave is tuned so that it has no beat but with piano strin"s, it is only possiblefor one set of coincident partials to match at any "iven time when tunin". 3hen one set of coincident partialsmatch, all others must be and are off or out of tune with each other, to some de"ree or another, dependin" onthe inharmonicity and how hi"h the partials are. Thus, we have the types of octaves8 the most commonly used

are the 2:1, 4:2 and :! types, althou"h there are other possibilities.

#n the 2:1 type octave, the second partial of the lower note is e*actly in tune with the fundamental or first partialof the upper note. #n the 4:2, the fourth partial of the lower note is e*actly in tune with or matches the secondpartial of the upper note and li$ewise, with the :! type, the si*th partial of the lower note matches the thirdpartial of the upper. #t is also possible that none of the partials match e*actly, all may be a sli"ht mismatch butthe octave can still have a pleasin" or desirable sound, dependin" on the "oal of the tuner.

To be"in the fine tunin" process, you must first tune your -4 to the pitch source. 9ou start by simply tunin" abeatless unison to the for$ or other source such as a tone from a metronome that has an -744 pitch or anyother -744 tone that you consider reliable. Then, to verify that the match is e*act, you "o down to the note, 2.(ow, this is afinetunin" procedure, not what you would do for a pitch raise or lowerin", somethin" only to verifythe ultimate precision and perfection such as when attemptin" the Tunin" E*am or for a public performance orprofessional recordin" where the specification on the contract is -744. 9our piano needs to already be nearlyperfectly tuned for any rapidly beatin" interval chec$ or test to be of any useful or practical value.

%lay the interval, 27-4, a '1;. 9ou should hear a rapid beat but probably not very rapid and the 27-4seventeenth must be wide of bein" a pure or beatless interval, not narrow. latten 0ust temporarilythe 2 so thatit creates a very rapid beat, as fast as you can discern without it becomin" an indiscernible blur. -lternately playthe 27-4 '1; on the piano, then the 27-4 interval from 2 on the piano and the -4 from the pitch source.3hen both intervals beat exactly the same, it confirms that the piano/s -4 and the pitch source are e*actly thesame. #t is possible to have an aural accuracy which would measure to within one cent electronically usin" thistest. -ccuracy within one cent would score a perfect 1 on the Tunin" E*am and would satisfy any contractobli"ation.

#f you hear a sli"ht discrepancy, all you have to do is ad0ust your -4 up or down by the sli"htest amount to ma$ethe 27-4 test match e*actly. #f your piano/s 27-4 is sli"htly faster than the piano


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"et these Equal Beatingtests absolutely perfect so that you will $now and understand the difference, one fromthe other.

(ow, from a perfect 4:2 octave, flatten sli"htly a"ain the -! until there is a very slow beat between the -! and-4, 0ust a little more than you mi"ht feel comfortable hearin". (ow play the -!74 m!. #f your piano is alreadynearly perfectly tuned, the -!74 m! will beat very rapidly at nearly the limit of what may be discernable. - m! isnormally about 1 cents narrow in an optimally tuned Equal Temperament and at that spot on the $eyboard8 thebeat is close to bein" an indiscernible blur. Therefore, you may temporarily sharpen the 4 sli"htly, to ma$e the

-!74 interval easier to hear. (ow, alternately play the -!74 m! and the 47-4 '. 3hen both the -!74 m!and the 47-4 ' beat exactly the same, the -!7-4 octave is a perfect :! octave.

(ow, to achieve the optimum compromise between a 4:2 and :! octave, sharpen the -! very sli"htly so thatneitherchec$ for a :! or a 4:2 octave tests perfectly. #n other words, the test for a 4:2 octave should reveal asli"htly faster !7-4 '1 than !7-! '! and the test for a :! octave should reveal a sli"htly slower 47-4 'than -!74 m!. 3hen you have found the spot for -! which reveals this sli"ht discrepancy between the 4:2 and:! tests, you will hear that the -!7-4 octave has a very slow beat to it, about one beat in every two seconds.This is now considered the optimum width for the initial -!7-4 octave in Equal Temperament and what is used bymost TE=s to set up the 'aster Tunin" for the Tunin" E*am.

#t is also the equivalent to within a very small and ne"li"ible de"ree of a 4:2 octave plus one cent. This is thewidth of octave that >r. -l ?anderson used to obtain the amount of stretch needed for optimal piano tunin" whenhe created the calculation for his Electronic Tunin" >evice, the ?anderson -ccu7Tuner. #n this $ind ofcompromise, note that noneof the coincident partials are in tune or match e*actly with each other. This mayalso be an e*ample of the whole octave sound of which @ir"il ?mith often spea$s. This, almost but not quiteperfect tunin", is an e*ample of the $ind of compromise which is necessary to defeat the problem and challen"eof inharmonicity, the way to achieve the finest tunin" possible from the modern piano, throu"hout its entireran"e.

-lso a note of importance, :! type octaves are considered optimum for ass tunin". The m! with lower noteand ' with upper note test is useful down to about the note 1. 3hen the octave has a reasonably "ood soundwith 0ust a sli"ht and very slow beat when a sin"le octave is played and both m! and ' interval chec$s beatexactly the same, the octaves in the ass are optimally tuned.

To tune the seventh octave in the hi"h treble in perfect 2:1 octaves with the si*th octave below it, as is requiredfor the Tunin" E*am, use the test for the 2:1 octave as described, a '1 below the bottom note and a '1;

below the top note. #n most other situations however, the seventh octave tuned so that it matches other partialsfrom further down the $eyboard is considered ideal, dependin" upon the perception and taste of the pianistwhom we serve. - future article is planned which will discuss and e*plore all of these options.

#n summation, the tests for the 2:1, 4:2 and :! octaves are as follows:

2:1 use the test note a '1 below the bottom note and a '1; below the top note.

4:2 use the '! below the bottom note and the '1 below the top note.

:! use the m! above the bottom note and a ' below the top note.

The Equal eatin" techniques are attributed to wen Aor"ensen B%T.

The understandin" of coincident partials is from Aim oleman, ?r. B%T and his Coleman Beat Locater chartsfrom red Tremper B%T.

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Octave Types