## 1 Introduction  
Pianos cannot be tuned like other string instruments. The strings of a piano have a significant stiffness which causes the tones produced to be inharmonic [1; 2; 3; 4]. This means that the partials above the fundamental have a frequency which is slightly higher than the expected integral multiple of the fundamental. In a piano, due to the shifted frequency of the partials, the frequency ratio of known intervals is changed. Every piano is different, and the inharmonicity varies from string to string. Thus, tuning is not as simple as tuning every string to a known fundamental frequency, as is the case with most other string instruments. Professional tuners listen to the tones produced by each key of a piano, and make them in tune with an already tuned tone, at some interval from it, based on their experience. The ultimate goal of this study is to produce results as good as those of a professional tuner with just a computer. This has the potential to lead to an automatic piano tuner providing a cheaper and faster alternative to piano tuning by a professional. Previous work has been carried out to find a method to automatically tune the piano. Most approaches involve matching the frequency of specific partials for two tones in an interval [3; 5; 6]. One approach involved calculating the sensory dissonances between tones and minimizing them [4]. Another unique approach was to minimize the Shannon entropy of all tones in a piano [7]. All these seem to produce quite good results. However, the results have not been backed with formal listening tests; and since it is ultimately humans who decide the tuning process and what should be a \"good\" tuning, it should be them who decide how well these tuning methods do. In order to create an automatic tuner, it is important to first identify what a \"good\" tuning is. We can say that one tuning is better than another if it is preferred by listeners. Based on the fact that professional tuners primarily use beating rates to tune [8], we can hypothesize that a good tuning would be one with minimum beating for commonly used intervals. Previous papers seem to consider a good tuning to be one that is exactly the same as a professional tuner. However, every tuner has their own style of tuning, and there is no one perfect tuning curve. Rather, there are many possible curves which sound acceptable [2]. It is shown later in this paper that the comparison of two different tuning rules on two differently tuned pianos gives different results. Most previous studies judge the success of their proposed methods by comparing the deviation of the fundamental frequency of the automatically tuned tones from the fundamental frequency of the tones tuned by a professional tuner. As mentioned above, different tuners have different tuning curves. Combined with the fact that almost every approach has been tested on a single tuning of a single piano, the results do not provide as much insight on the method proposed as it may seem initially. While trying to obtain a tuning accuracy that will be able to tune even concert pianos in the future, it is important to come up with some other means of judging the success of a tuning method. This means of assessment should not be biased towards any specific piano tuning. In this paper, one such method of analysis is proposed, which is based on the beating rates of tones. Piano tuning is generally conducted in three segments starting with the reference octave (usually from F3 to F4), followed by the tones above the reference octave, and finally tones below the reference octave [8]. The best tuning for high piano tones involves adjusting octaves so that partials of upper and lower tones match as closely as possible. One way to do this is to make the first partial frequency of the upper tone exactly the same as the second partial frequency of the lower tone. These two partials are the first partials of each tone that would overlap (i.e., be of the same frequency) when tuning harmonic tones, and so we call them the first set of matching partials.