AP08_4.vp 1 Introduction In 1202, the Italian mathematician Fibonacci (also known as Leonardo da Pisa) asked a simple question in his book Liber Abaci: If a pair of rabbits beget a pair of new rabbits after one year of maturing, and these rabbits beget another pair after maturing, how many pairs of rabbits will there be after n years? To make things simple, Fibonacci assumed that rabbits never die and breed every year. Fig. 1 shows the first 5 years of the rabbit population. The number of rabbit pairs in the nth year Fn (F for Fibonacci) is the number of pairs one year earlier Fn �1 (because the rabbits do not die) plus the number of pairs two years earlier Fn �2, because they are all mature and can reproduce: F F Fn n n� �� �1 2 (1) Starting with one immature pair of rabbits (F1 1� , F2 1� ), it is easy to calculate the number of pairs for the next years: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 114, 223, 347… If an immature pair of rabbits is represented by a “0” and a mature pair is represented by a “1”, a binary Fibonacci Sequence can be built. In the first year a “0” would represent one immature pair, and in the second year a “1” would represent the matured pair. In the third year a “10” would represent the same matured pair and a newly born immature pair. Table 1 shows the binary Fibonacci Sequence for the first 8 years. The self-similarity of the Fibonacci Sequence can also be found in the binary sequence: After the first two years, the nth binary sequence can be built if the sequence of year n � 2 is attached to the end of the sequence of year n � 1. The sequence of the number of pairs Fn is the original Fibonacci Sequence. The number of mature pairs (“1”) M are the Fibonacci Numbers starting in year 2, and the number of immature pairs (“0”) N are the Fibonacci Numbers starting in year 3. The quotient of the ones and zeros M/N converges to the Golden Ratio: M N F F gn n n � � �� � � �� 1 . (2) The Golden Ratio is a geometric relation: A straight line is sectioned in such a way that the ratio of the total length to the longer segment equals the ratio of the longer segment to the shorter one. Calling the total length l and the longer segment a, the following equation describes the Golden Ratio: g l a a l a : � � � . (3) The only positive solution of this equation is © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 3 Acta Polytechnica Vol. 48 No. 4/2008 Psychoacoustic Properties of Fibonacci Sequences J. Sokoll , S. Fingerhuth 1202, Fibonacci set up one of the most interesting sequences in number theory. This sequence can be represented by so-called Fibonacci Numbers, and by a binary sequence of zeros and ones. If such a binary Fibonacci Sequence is played back as an audio file, a very dissonant sound results. This is caused by the “almost-periodic”, “self-similar” property of the binary sequence. The ratio of zeros and ones converges to the golden ratio, as do the primary and secondary spectral components intheir frequencies and amplitudes. These Fibonacci Sequences will be characterized using listening tests and psychoacoustic analyses. Keywords: Fibonacci Sequence, psychoacoustics, listening tests. Fig. 1: The Fibonacci rabbit population Year Fn M N M/N 1 0 1 0 1 0 2 1 1 1 0 � 3 10 2 1 1 1 4 101 3 2 1 2 5 10110 5 3 2 1.5 6 10110101 8 5 3 1 66. 7 1011010110110 13 8 5 1.6 8 101101011011010110101 21 13 8 1.625 Table 1: The binary Fibonacci Sequence g � � � 1 5 2 1618033989. � (4) The Golden Ratio is considered the most irrational of all ratios. The Fibonacci Number Fn can be easily calculated with this approximation (the wavy equal sign means: take the near- est integer) [1]: F g n n � 5 . (5) 2 Acoustical realization of the Fibonacci Sequence As a very dissonant sound is expected if the binary Fibonacci Sequence is played as an audio file, it would be interesting to analyze these sounds. For the acoustical realiza- tion of the Fibonacci Sequence, a binary sequence of any length Fn can be generated and played back with a digi- tal-analog-converter. Although there is no definite period, certain sequences recur and the signal shows a distinct dis- crete spectrum. The analysis of longer sequences shows only small differences, so that the spectrum of a sequence with a length of F20 6765� , as shown in Fig. 2 can be seen as the spectrum of a sequence of infinite length [2]. In the spectrum of the Fibonacci Sound, the Golden Ratio g can be found again. In the top plot of Fig. 2 the second high- est peak has the magnitude of the highest peak divided by g ( .1 0 618g � ) and the frequency of the highest peak multiplied by g (392 634 �g ). The binary Fibonacci Sequence was interpolated with a sinc-algorithm to minimize the spectral influences of the rectangle-characteristic of a binary sequence. Because of the low-pass characteristic of the sinc-interpolation, the high spectral components were cut off. The sinc-interpolated binary Fibonacci Sequences were integrated in an internally developed wavetable synthesizer (see Fig. 3). With this syn- thesizer it is possible to play intervals and melodies with Fibonacci Sounds. MIDI files are supported. To characterize the psychoacoustic properties of the Fibonacci Sequences, listening tests and psychoacoustic analyses have been per- formed. For the listening tests, sequences of length F16 987� have been used. 4 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 4/2008 Fig. 2: Binary sequence vs sinc interpolation 3 Listening tests For a subjective psychoacoustic analysis, listening tests have to be performed. In the first test the subjects heard pop- ular melodies played with the Fibonacci Synthesizer. All sub- jects were able to recognize the melodies, though the subjects could not identify the intervals. A “Fibonacci octave” cannot be perceived as a “real” octave with a frequency ratio of 1:2. The Tritone Paradox phenomenon [4] occurs as well. [start frequency: result of last pass] [start frequency: 0 Hz] [start frequency: 10 kHz] The next test examined whether one of the peaks in the spectrum could be spotted as the most remarkable peak. The subjects were told to change the pitch of a sine tone until it matched the Fibonacci Sound. The sounds were presented with an electrostatic Stax headphone with a linear frequency response: the pure tone on one ear, the Fibonacci Sound on the other ear. The effect of the binaural beat [3] was intention- ally used to help to find the right pitch. As there is more than one noticeable peak in the spectrum of the Fibonacci Sounds, different subjects identified different peaks. Fig. 4 shows the results of this listening test. Each dot represents the choice of one subject. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 5 Acta Polytechnica Vol. 48 No. 4/2008 Fig. 3: The Fibonacci Synthesizer Because the subjects started each pass with the sine fre- quency of the last pass, two further tests were performed to find out whether the start frequency of the sine tone has an in- fluence on the results. The subjects were asked to start each pass with the lowest sine tone of the frequency synthesizer and the highest respectively. Fig. 4 shows the results of the listen- ing test starting each pass with a sine frequency of 0 Hz. It can be seen that many more low frequencies were identified than in the first test. In the last listening test every pass started with a sine frequency of 10 kHz (see Fig. 4). Almost all subjects have marked frequencies above the actual spectrum of the presented sounds. The different results of the three different listening tests refer to the dissonant spectrum of the Fibonacci Sounds. Al- though there is a major peak in the frequency domain which is higher in amplitude by factor g than the second major peak, this peak was only rarely identified in the pitch perception test. 6 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 4/2008 Fig. 4: Listening test: pitch-perception of Fibonacci Sounds 4 Psychoacoustics For an objective analysis, psychoacoustic parameters like loudness and Pitch Strength were calculated. Fig. 5 shows the loudness according to ISO 226:2003 [5] of a Fibonacci Sound. The relative Pitch Strength [6] was calculated for the Fibonacci Sequence that was used in the listening tests. If a sine of 80 dB and 1500 Hz has a Pitch Strength of 1, the Fibonacci Sequence has a relative Pitch Strength of 0,1878. 5 Summary / Outlook The Fibonacci Numbers are a very interesting mathemati- cal phenomenon. As the Golden Ratio g occurs permanently in the spectrum of the binary Fibonacci Sequence, a very dis- sonant sound results. Although melodies can be recognized, it is not possible to identify specific intervals. Listening tests were performed in order to find the subjective pitch of the sig- nals. It is not possible to define one most outstanding peak, as all subjects marked different frequencies. In the future, more psychoacoustic analyses will be per- formed. Another project of the Institute of Technical Acous- tics will investigate consonance and tonality. Acknowledgements The authors would like to thank Prof. Michael Vorländer for supporting this work and discussion, and also all the par- ticipants in the listening tests. References [1] Schroeder, M. R.: Number Theory in Science and Communi- cation. 3rd Edition. Springer Verlag 1999. [2] Schmidt, H.: Eigenschaften und Akustische Realisierung von Fibonacci-Folgen. Proceedings of DAGA ’88, Braun- schweig, p. 621. [3] Terhardt, E.: Akustische Kommunikation. Springer Verlag 1998. [4] Deutsch, D.: The Tritone Paradox: Effects of Spectral Variables. Perception & Psychophysics, Vol. 41 (1987), p. 563–575 . [5] ISO 226:2003: Acoustics – Normal Equal-Loudness-Level Contours. [6] Fruhmann, M.: Introduction and Practical use of an Al- gorithm for the Calculation of Pitch Strength. Journal of the Acoustical Society of America (JASA), Vol. 118 (2005), No. 3, Pt. 2 , p. 1894. Jan Sokoll e-mail: jso@akustik.rwth-aachen.de Sebastian Fingerhuth sfi@akustik.rwth-aachen.de Institute of Technical Acoustics RWTH Aachen University D-52056 Aachen, Germany © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 7 Acta Polytechnica Vol. 48 No. 4/2008 Fig. 5: Loudness of a Fibonacci Sound << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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