Understanding Octave Behavior

In tuning Octaves it is important to gain an understanding of how coinicedent partials in octaves actually interrelate to one another as we listen to octaves. Many beginning tuners will notice that no matter what they do, no matter what octave test is applied, the octave sounds bad. Understanding fully the nature of octave coincidentals and the nature of inharmonicity can help explain why this is the case. It may also help the beginning and intermediate tuner to avoid the temptation of spending too much time on an octave in an attempt to force it to come clean. The basic point is this: When we are "tune" indivdual octaves, we are only comparing directly one set of coincident partial pairs. If  we regard the other partial pairs we do so only in relationship to neighboring octaves in order to achieve as smooth as possible curve in their behavior. In the end it will be seen that this curve is indeed the main goal of the piano tuner and that as a tuner becomes more advanced he/she will tune with increasing regard to this curve. However, as the saying goes, you have to learn to walk before you can learn to fly.

To start with, lets digress a bit into basic single string behavior. When we play a single string a series of partials (harmonics) are sounded in addition to the fundemental tone we are playing. In theory, this harmonic series is all nice and packed into whole number ratios and looks just great. The octave is 2:1, the fifth is 3:2, the fourth is 4:3 and so on. But in practice these nice neat ratios do not exist in vibrating strings. This is due to string inharmonicity. There are some basic properties of  string inharmonicity we need to have very clear in our minds.. Lets start with a notion about string inharmonicity that until recently was held to be true, and assume for the moment that the following statements are true.

    1: The higher the partial, the more it deviates from its' theoretical value.
    2: This deviation occurs in a exponential fashion as we increase the frequency of the fundemental (ie moving upwards one
        or more semitone).
    3: Lastly, this deviation is always is such that the frequency of the partial is greater then its theoretical value.

Now this sounds really nice and predictable. Figure 1 is a graph resulting from taking frequency measurments of fundementals and partials up to number 8 (excluding number 7) from a well tuned grand, and displays the nice curves upwards we'd expect from the assumptions we've taken so far. So lets proceed. Notice that the higher the partial, the greater its growth coeficient. (ie higher partials become increasingly sharp at a faster rate then lower partials as we increase the frequency of th fundamental)

Diagram Actual Harmonic Series from a well tune piano C3-C5
figure 1

When we compare coincidental partials of accending octaves then, it follows that the the partials of the lower note should become increasingly sharp in relationship to corresponding partials of the upper note as we progress upwards.   Think about this for a second. Remember that in compairing octave partials, the partial number of the lower note is always twice the partial number of the higher note.  The partial of the lower note  "grows" faster then its higher note coincident as we progress in an ascending manner up the keyboard. In other words, coincidental partials found in octaves should become increasingly narrow in relationship to each other as we progress upwards. In the reality of what tuners actually do, this is only eh.. partially true.

When we play an octave we are in a sense superimposing the partial sets of the two notes onto one another. Figure 2 shows the results of this superimposition from our well tuned grand. Lets examine what the tuner did. Starting at C3-C4, octaves are graphed chromatically in an accending fashion until we reach C4-C5.  Looking at  the 8:4 and 6:3  coincidents we see that they become increasingly narrow. This is what we expected. The graph shows this clearly in the 8:4 relationship The line representing this pair of coincidents drops off dramatically, showing that the coincidents beat faster and faster. The 4th partial of the upper note simply cant "keep up" with the 8th partial of the lower note as we accend the scale. The same can be easily seen in the 6:3 partial pairs.  However, the 4:2 partials hold constant at about +0,5 bps (+ is wide, - is narrow)  and the 2:1 partials become increasingly wide.   This needs an explanation.

figure 2

When we tune octaves, we decide to hold certain partial pairs relatively constant over several succesive octaves depending on the area of the piano we are tuning. One reason we do this has to do with the process known as Octave Stretching.   Octave stretching allows us to match double octaves more closely, and also keeps the lower coincidentals beating relatively slow over a greater area of the scale. In this section of our well tuned grand,  the tuner decided to hold the 4:2 pair relatively constant.  In other words tuner has "forced" the 2nd partial of the upper note to "keep pace" with the 4th partial of the lower note.  This explains why the 2:1 coincidentals become increasingly wide. As a consequence of holding 4:2 constant, the 2nd partial of the upper is forced to "outrun" the 1st partial of the lower.
If the section of the piano in our example above had been tuned with the 2:1 being held constant, then the 8:4, and 6:3 would have fallen off narrow much quicker, and the 4:2 would also show this tendency. This pattern would be even more emphasized if the 2:1 was held beatless, or beating slowly narrow.

Octave stretching is in a sense, stretching out the area of the piano where the beating of the lower coincidentals in octaves is very slow. In another sense it is a process which enables us to match or even widen double octaves which otherwise would be significantly narrow. The amount of stretch is in the end dependant on the tuners judgement of what sounds best on any particular piano, and on the tuners own taste for stretching. Some like them wide, and some like them narrow. But all like them smooth. The smoother the better.

Holding to the above view of inharmonicity, we observe then the following.

Any partial pair with a ratio less then that of the partial pair being kept constant (here the 4:2) will become increasingly wide. Likewise any partial pair with a ratio greater then that of the partial pair being kept constant will become increasingly narrow.

Octave Types

In tuning (stretching) octaves we use several different coincidental octave partials for different areas of the piano and employ test intervals which share these coincidentals in each instance. These are in practice limited to the 12:6, 10:5, 8:4, 6:3, 4:2, and 2:1. All of these share the basic 2 to 1 relationship of the octave but as we have seen, the higher the actual partial number the more it diverges from its theoretical value and the faster this divergence increases with relation to increases in the fundamental frequency. What factors, then, governs our decisions as to where to use each of these ? What happens when we move from one octave type to another ?

Every piano has its own degree of inharmonicity. Factors that determine this include the scaleing of the piano, the impedance matching between the scale tension and the bridge and soundboard constructions, and climatic conditions. The degree of inharmonicity at any given time then is one factor that governs how much stretch we apply to our tunings. The disired degree of double and triple octave matching also plays a part in how much stretch we apply, as well as how much the tuner is willing to sacrifice with regard to lower partial coincidentals in order to accomplish this matching.

The amplitude (volume) of partials in strings also play a large part in where we apply any given octave type. In low bass strings, the higher partials sound much louder then in higher strings. At the extremes of the keyboard the fundamental almost disapears in the mesh (lowest bass) and it is reasonable that we tend to pay more attention to the behavior of partials higher up the harmonic ladder then in the highest treble where all but the lowest partials disapear. Since the appearence (soundwise) of partials follows roughly a curve relative to the frequency of the fundemental, this provides us with natural partitions for which to apply different octave types. As each successivly higher partial emerges (decending fundamentals) we need to place more weight on them, insuring that they become evenly and steadily slower within the octaves as we decend. Likewise in accending, as each partial fades from being easily heard we need to pay less attention to them, and can pay more attention to the lower coincidentals.

Where it gets confusing is in the areas of transition from one octave type to the other. And there are at least two very good reasons why this is confusing. A consequence of changing from one octave type to another is that the beating rate tendency of the previous type will reverse. Think about this for a moment. As we tune octaves decending from the temperament area we start typically with the 4:2 as our anchour. As we decend it at some point becomes beatless, and as we make the transition to the 6:3 it will start becoming wider as demonstrated above and stated in our last observation. This means the beat rate of the 4:2 will increase slightly as we decend. We normally think in terms of beating slowing down as we decend so this can get confusing. In the case of the 4:2 this may not be so readily apparent to the ear. This is because it has a relatively low set of partial numbers and therefore change less in relation to changes in the fundmental then higher partial numbers do, and because in the lower ranges of the bass area where this occurs, they become harder to hear in relation to higher numbered partials that have greater amplitudes. But in cases where there are great degrees of inharmonicity (small pianos) you can definately hear this "reversal" in the lower coincidents, as one makes the transition to types with higher partial numbers. In fact, our example above shows this happening in the 6:3 at about middle C. . It moves from beating narrow to beating wide (decending). A good stretch allows for this reversal, but only allows it to develop to a slow beat rate which changes very very slowly or not at all once the reversal is made.

(I need to measure an entire piano to substantiate some of this, and to make further observations as to partial behavior in the bass area. The development upwards seems logical enough to extrapolate from the existing data.)

        --Define Inharmonicity more correctly

                Para or Unexpected Inharmonicity
                    define, explain significance
                    what can we do about it ?
                    causes.. not known at this time.

                Can we visualize a piano tuning with Octave partial behavior as a viewpoint.
                How would this be of benifit (awareness of general tendencies, what can we expect to hear) / pitfalls (unreliable as a direct method for tuning).

written by Richard S.L. Brekne
all rights reserved. May 14th 1999

Richard Brekne
Sydneskleiven 1
5010 Bergen, Norway

E-mail Richard Brekne
Richard Brekne Website


The frequencies for the graphics above were taken from the June 1978 edition of the Piano Technicians Journal.