Octaves are named from one C to the next higher C. For example, all the notes between "great C" and "small C" are "great". One-line c is also often called "middle C". No other notes are called "middle", only the C. Exercise 4. Go to Solution. The word "octave" comes from a Latin root meaning "eight". It seems an odd name for a frequency that is two times, not eight times, higher. The octave was named by musicians who were more interested in how octaves are divided into scales, than in how their frequencies are related.
Octaves aren't the only notes that sound good together. The people in different musical traditions have different ideas about what notes they think sound best together. In the Western musical tradition - which includes most familiar music from Europe and the Americas - the octave is divided up into twelve equally spaced notes.
If you play all twelve of these notes within one octave you are playing a chromatic scale. Other musical traditions - traditional Chinese music for example - have divided the octave differently and so they use different scales. You may be thinking "OK, that's twelve notes; that still has nothing to do with the number eight", but out of those twelve notes, only seven are used in any particular major or minor scale.
Add the first note of the next octave, so that you have that a "complete"-sounding scale "do-re-mi-fa-so-la-ti" and then "do" again , and you have the eight notes of the octave. These are the diatonic scales, and they are the basis of most Western music.
Now take a look at the piano keyboard. The eighth note would, of course, be the next A, beginning the next octave. To name the other notes, the notes on the black piano keys, you have to use a sharp or flat sign. Whether it is a popular song, a classical symphony, or an old folk tune, most of the music that feels comfortable and familiar to Western listeners is based on either a major or minor scale.
It is tonal music that mostly uses only seven of the notes within an octave: only one of the possible A's A sharp, A natural, or A flat , one of the possible B's B sharp, B natural, or B flat , and so on. The other notes in the chromatic scale are usually used sparingly to add interest or to temporarily change the key in the middle of the music. For more on the keys and scales that are the basis of tonal music, see Major Keys and Scales and Minor Keys and Scales. Solution to Exercise 4.
Return to Exercise. Free Tools. Other products. Back Free versions Previous versions. Back Forum Join our street team. Back Ear training. Whole-cell patch-clamp recording reveals subthreshold sound -evoked. J Neurosci, 16 9 : , Cynx J. Auditory frequency generalization and a failure to find octave generalization in a songbird, the. Demany L, Armand F. The perceptual reality of tone chroma in early infancy. J Acoust Soc Am ,.
Demany L, Semal C. Dichotic fusion of two tones one octave apart: Evidence for internal octave templates. Harmonic and melodic octave templates. J Acoust Soc Am 88 5 : , Deutsch D. Octave generalization of specific interference effects in memory for tonal pitch.
Deutsch D, Academic. Eargle JM. Ehret G, Merzenich MM. Auditory midbrain responses parallel spectral integration phenomena. Friauf E, Kandler K. Galbraith GC. Two-channel brain-stem frequency-following responses to pure tone and missing fundamental. Heffner H, Whitfield IC. Perception of the missing fundamental by cats. J Acoust Soc Am 59 4 : ,. Houtgast T. J Acoust Soc Am 60 2 : , The central origin of the pitch of complex tones: evidence from musical.
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Langner G, Schreiner CE. Periodicity coding in the inferior colliculus of the cat. Neuronal mechanisms. Triadic comparisons of musical intervals. Liberman MC. Auditory-nerve response from cats raised in a low-noise chamber. J Acoust Soc Am. Licklider, JCR. J Acoust Soc Am, , Lipps T. Consonance and Dissonance in Music, , trans. Thomsen W. Everett Books, San Marino,. Spontaneous otoacoustic emission frequency is modulated by heartbeat.
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Acustica ,. Pitch of the residue. Schreiner CE, Langner G. Laminar fine structure of frequency organization in auditory midbrain. Frequency selectivity of central auditory neurons without inner. The role of resolved and unresolved harmonics in pitch perception and. Shepard RN. Circularity in judgments of relative pitch. J Acoust Soc Am, 36 12 : , Human auditory frequency-following responses to a missing.
Organization of do rsal cochlear nucleus type IV unit response maps and the ir. Click train encoding in primary. J Neurophys , Disproportioate tonotopic representation for processing CF-FM sonar signals in the. Harmonic-sensitive neurons in the auditory cortex of the mustache bat. Musical octaves and pitch. J Acoust Soc Am, 54 4 : , Terhardt E, Zick M.
Evaluation of the tempered tone scale in normal, stratched [sic], and contracted. Perception of the missing fundamental in nonhuman primates. Uncrossed and crossed inhibition in the inferior colliculus. J Neurosci, 5 7 : , White L, Plack C. Temporal processing of the pitch of complex tones. J Acoust Soc Am, 4 : ,. Abstract The octave, a relation between two tones whose fundamental frequencies stand in the ratio , is a foundation of tonal musics worldwide: notes separated by an octave are considered harmonically equivalent, and melodies are often sung or played in parallel octaves yet are considered identical melodies.
I hypo the size that our perceived similarity of octave-related tones derives from o the r properties of the octave interval — namely spectral fusion, sensitivity to interval tuning, and generalization of response to the common fundamental — which are qualitatively similar to properties of larger intervals in the harmonic series.
Consequently, do uble- octaves ratio should have less similarity in the above sense than twelfths ratio have; the octave equivalence present in music is the result of a learned transitivity applied to the natural perceptual similarity.
I fur the r hypo the size a mechanism for the se properties in which neural circuits in the brainstem inferior colliculus detect coincident firing of neurons tuned to harmonics of a fundamental to compute the periodicity pitch; occasional skipped firings lead to excitation of subharmonically tuned neurons, causing a note to sound like its subharmonic octave.
Animals trained to respond to one note generalize this response to o the r notes Blackwell and Schlosberg ; raising the se animals in artificial acoustic environments may alter the patterns of generalization and thus provide a test for learning of octaves. Direct extracellular recording from the brainstem of animals could provide evidence for the proposed neural mechanism of harmonic equivalence.
Contents Introduction Harmonic structure of sound the Octave: definition and psychophysics Pitch perception a possible neural mechanism of pitch perception Hypo the sis Experiments Specific Aims Psychophysics Animal Behavior Auditory System Development Modification Electrophysiology Implications, Interpretations, and Renovations Conclusion Acknowledgments References Introduction Harmonic structure of sound 3 Biologically relevant sound s are often produced by the periodic vibration of a mem- brane and air column such as our vocal cord and tract or that of our predator, pet, or prey.
This method of creating noise produces vibrations not only with the fundamental frequency of the membrane, but also with higher frequencies or harmonics. The relative strength and phase of the se harmonics will tend to covary as the sound source moves, and thus the auditory system can use this information to aid in localization or attention. A single representation of the se correlated sensory responses would thus be a useful representation to have, and we have one: when presented simultaneously, harmonically related frequencies fuse into a single percept, pitch.
Pitch is the subjective analog of physical frequency: the percept ranges from low to high and has a nearly linear, one-to- one correspondence to frequency of pure tones. The harmonic se- ries is a set of frequencies with arithmetically constant frequency separation between the m, yet we hear the se- ries as notes that get closer and closer toge the r.
The natu- ral comparison between two pitches is the ratio between the ir respective frequencies. These ratios define the in- terval between two pitches. A list of important interval names is found in Table 1.
All non-pitch audible spectral attributes of a sound , including the harmonic spectrum, relative phase relationships between component frequencies, and slow modulations be- low audible frequencies, are typically subsumed under the term timbre. Voices and in- struments may produce sound s which share the same pitch, but the y are distinguishable based on timbre. These spectral qualities give each instrument its unique sound type. Temporal structures built of pitch intervals are melodic and harmonic musics.
One consonance, however, is absolutely universal across all music cultures. This is the octave. In a more culture-independent definition, an octave is the relationship between two pitches whose frequencies are related by a ratio of Physically, it is the interval between a funda- mental and its first harmonic. Vocally, when men and women sing toge the r the y typi- cally mirror the melodic pitches one octave apart, and somehow this sound s like the same note.
In western music the ory, the principle of octave equivalence asserts that notes related by one or more octaves function the same in a harmonic or melodic struc- ture. When listing pitches in increasing order, written systems use a circular scale in which notes separated by octaves are given the same name: DO-re-mi-fa-so-la-ti-DO in the West, SA-re-ga-ma-pa-dha-ni-SA in south Asia.
Octaves just seem to sound the same. Does the cause lie in ma the matics? Is the octave just an arbitrary, a culturally impressed pitch relation? Circa BCE, Pythagorus explained that simple ratios of the lengths of vibrating strings gave consonance. Much later, von Helmholtz correctly hypo the sized from physical principles that the ear takes a Fourier transform of the pressure wave When two waves are close in fre- quency, an audible beat pattern of amplitude modulation results.
Harmonics of two tones can overlap and produce beat patterns as well. Lipps modified this argument, saying that it applied not to physical vibrations but to mental excitations An octave interval is merely the very first interval which allows greatest alignment of har- monics, 2 being the smallest integer greater than 1.
Yet a frequency tripling and a qua- drupling also allow alignment of harmonics. If the harmonic-alignment hypo the sis is in fact true, the n the tripling would actually sound more similar than a quadrupling. A 1 When I refer to an interval sound ing similar, I mean that the constituent notes of the interval sound similar.
If octaves are important for reasons that extend beyond just overlap of harmonics, the n the do uble- octave should sound more similar than the twelfth. Can one test such a hypo the sis by simply asking subjects to compare the similarity of each harmonic to its fundamental? Kallman attempted just such a study, and found that similarity decreased mono- Figure 1: Pitch has been represented as a curve in two dimensions called pitch height and pitch chroma.
Height ranges from low, like a thundering rumble, to high, like a shrill birdcall. Chroma represents the musical function of a pitch; in essence, chroma is pitch that has been normalized to a single octave, by assunming all-octave equivalence from Shepard One of his mu- sical subject did rate octaves as more similar than neigh- boring intervals, but this is uninterpretable: for this subject, do octaves sound the same because musical training taught the y did, or was the re a genuine perceptual similarity?
Allen found precisely the same dichotomy between musi- cally trained and untrained subjects. To investigate the oc- tave, one must use measures of similarity that do not rely on subjective judgments.
Using better psychophysical methods, several studies have demonstrated some perceptual similarity between two oc- tave-related tones that are not present in tones whose inter- val differs from an octave by a semitone or more. People with absolute pitch sometimes place notes in the wrong oc- tave with the right note name Bachem Pure-tone pairs are sometimes confused with the ir inversions, coun- terparts in which one note is transposed in the direction of the o the r by an octave Plomp Deutsch showed that when subjects are asked to compare two tones sepa- rated by several intervening pitches, the intervening pitches interfere strongly with memory of the target pitches if the y match the target, even if it is an octave above or, more weakly, an octave below the target, but not o the rwise.
Shepard demonstrated that a harmonic complex containing all frequencies standing in a ratio of 2 n to a given reference frequency — all octaves of a given pitch — fuse into a single organ-like pitch with ambiguous tone height; and that pairs of the se tones are perceived non-transitively.
Demany presented evidence that 3-month-old infants are less surprised by tonal sequences in which one note is replaced by its octave than those in which that note is replaced by a seventh or a ninth. Blackwell and Schlosberg showed that rats trained to respond to one frequency have stronger generalization to an octave subharmonic than to o the r lower inharmonic frequencies.
Skin galvanometry measurements by Humphreys revealed that after mild shock conditioning against one frequency humans have subconscious generalization to octaves , shown by greater skin conductance response to an octave than to a slightly smaller interval. It might seem fair to conclude that, based on perceptual studies of animals, conscious and uncon- scious human faculties, absolute and relative pitch, and sequential and simultaneous presentation of pitches, the re really is something special about the octave.
Risset generated Shepard-tone-like sound s, with stretched octaves in lieu of pure oc- taves, that when transposed up by an octave actually sound like the y decrease in pitch Both Deutsch and Risset thus conclude that for certain tasks, proximity of tones or, alternately, melodic contour determine perception of pitches more strongly than do octave transpositions. Octaves are not even subjectively fixed intervals. When any given human subject is asked to tune an oscillator to an octave above a given tone, the frequency ratio upon which the y decide is usually greater than The precise amount of stretch is consis- tent within one session of trials, but varies significantly from day to day Ward Even for trained musicians, stretched octaves and scales are preferred in melodic lines, at least when such intonation is possible as on fretless instruments such as violins and trombones Terhardt The amount of stretch is dependent on both the reference 7 frequency and intensity Sundberg The subjective octave was actually shown to be additive: tuning to an octave above a reference, and the n tuning an octave above that, yields the same frequency as tuning two octaves above the first reference Ward Most psychoacoustic literature explores relationships among pitches that vary within one octave, or perhaps a tone higher or lower; scientists seem to not have been inter- ested in the se larger multi-octave intervals.
Yet our perception of the se tones is ex- tremely relevant to discovering the nature of the octave. Most natural sound s have a strong first harmonic; in the bat this harmonic is far stronger than the fundamen- tal Suga Is the harmonic structure of sound somehow responsible for octave similarity?
This general question leads to a much more specific one: do higher harmonics evoke the same kind of similarity percept, or are octaves really qualitatively different? If octaves are special, the do uble- octave should be more similar, or be more mistakable, or have a lower discrimination threshold, than the twelfth. If not, any aforemention psychophysical measure should show a monotonic progression in harmonic number.
Every conclusion reached supposedly about the octave is not actually a conclusion about the octave but about the first harmonic. Some of the studies investigate the importance of smaller intervals such as the fifth, which is related to the second harmonic an octave below a twelfth but the two may give different results; and the do uble-octave has not been thoroughly investigated.
Thus, no one knows whe the r infants are less surprised by twelfths than by elevenths or thirteenths, or whe the r rats will generalize to twelfths and do uble- octaves. Shepard-like tones composed of stacked non-octave intervals do still give rise to circular illusions; no one knows whe the r stacked twelfths would fuse and give rise to ambiguous pitch ordering. Would skin galvanometry reveal a subconscious conditioning generalization to any harmonic, or just octaves?
It is thus useful to notate a distinction between two types of octave equivalence. All- octave equivalence is the perception that all octave-related notes, e. Nearest-octave equivalence is the perception that single octaves sound more similar than multiple- octaves do , and that non-octave harmonics sound more similar than even slightly higher octave harmonics. My first hypo the sis is, 8 with this terminology, that only nearest-octave equivalence is present in the musically untrained; training or musical exposure creates all-octave equivalence by transitively associating octaves.
Fur the rmore, the se low-order harmonics have been found to be most influential in determining the fundamental of a stimulus Ritsma Indeed, the spatial resolution of the tonotopic map of the cochlea is inadequate to explain our abil- ity to discriminate pure frequencies below 1 kHz Langner If pitch is perceived through analysis of harmonic or temporal structure, and octaves might be caused by harmonic structure, the n it makes sense to see what studies have been do ne on the perception of harmonics.
Most work on the subject applies to the perception of harmonic complexes — simultaneous tones composed of frequencies all of which are harmonics of a single fundamental. Pitch perception Psychophysics shows us just how good we are at resolving harmonic spectra into a single pitch: if we artificially stimulate with a few upper harmonics of some base fre- quency, or mask the fundamental with noise Licklider , our brains recreate a missing fundamental which we perceive as a pitch.
Importantly, note that the re is no resolved spectral energy at the frequency corresponding to this pitch. Over time, various experiments discovered that more and more limited spectra could give rise to a perception of the missing fundamental.
First, a complete harmonic com- plex without the fundamental Schouten the n a set of five de Boer , the n three Ritsma , the n two harmonics Houtsma were all found to create a low pitch corresponding to the missing fundamental.
Amazingly, even a single harmonic was shown to elicit a pitch at a subharmonic, though only if noise were present and if attention were directed to the expected pitch region Houtgast Low pitch perception emerges only when information from different auditory nerve fibers is brought toge the r centrally.
We are not only able to detect a missing fundamental whose spectral components are resolved as separate peaks in the peripheral auditory system, but we can also detect a missing fundamental in higher harmonics which are unresolved Houtsma Here, only temporal Figure 2: a wave composed only of harmonics 5—7 blue has the same periodicity as the ir fundamental red but no spectral energy at the fundamental frequency.
Auditory nerve fibers are known to phase-lock with acoustic waveforms, firing stochastically at peaks of vibration. Cariani and Delgutte demonstrated that the perceived pitch for both resolved and unre- solved harmonic stimuli was predicted by pooling the temporal information of all auditory nerve firings and identifying the most common interspike interval one could predict; one could even explain the origin of the stretched perceptual octave as due to the refractory period and of neural firings McKinney.
However, certain pitch-evoking click series caused serious prob- lems for this model. Thus, the simple temporal or autocorrelation model is insufficient to explain certain pitch per- cepts.
None the less, stimuli with resolved and unresolved harmonics seem to be perceived by different mechanisms Steinschneider Shackleton and Carlyon found com- parison amongst tones of unresolved harmonics, or comparison amongst tones of re- solved harmonics, was easier than comparison between resolved and unresolved tones.
One implication is that resolved harmonics generate useful information in spatially separated neural chan- nels, whereas unresolved harmonics yield information only through temporal structure of pooled responses which must be interpreted for a longer time.
The conclusion is that both place- and time-coding are both at work in pitch perception, to differing degrees based on the resolvability of harmonic structure. For a wide range of spectral types, our brain tries to compute the single subharmonic pitch whose har- monics best match the perceived acoustic spectrum.
A possible neural mechanism of pitch perception This is interesting in its own right, but if we can understand neurophysiologically how we perceive a missing fundamental, we may be stumbling onto the very same mecha- nism that makes us perceive octaves as the same. After all, the spectrum of a note is just like the spectrum of a note an octave lower which is missing its fundamental and even harmonics.
If a circuit exists which extracts a missing fundamental from spectral infor- mation, the n that same mechanism might be partly excited by its octave. Recent psy- chophysical, encephalographic data, brainstem single-unit recordings, and anatomical evidence suggests a specific neural circuit in the inferior colliculus may generate the missing fundamental Braun In psychoacoustic studies, the percept of two pure tones depends on the ir frequency separation.
For a difference of 11 neuron recordings from the inferior colliculus Ehret Neurons in the inferior colliculus have are excited maximally by a particular frequency, and detailed three- dimensional mappings of the se characteristic frequencies reveal distinct layers of closely tuned neurons separated by discrete jumps in tuning Schreiner Critical bands are hypo the sized to arise from this structural arrangement Schreiner ; anatomical evidence supports this notion.
Inhibitory effects of stimuli within a critical bandwidth were found to exist at a cellular level and were localized to individual lamina, measured by activity-induced labeling of tissue in the inferior colliculus Webster and extracellular recording upon suppression of GABA inhibition Palombi This jives with morphological studies which show extensive, oriented dendritic con- nections within a lamina and between both neighboring and second-nearest laminae Oliver Braun emphasizes the finding of some specific next-neighbor axonal connec- tions in Malmierca , and claims this as a possible anatomical substrate of a previ- ously unnoticed do uble-critical bandwidth 2CB which he observes in pooled psycho- physical data.
The do uble-critical bandwidth is, simply, twice the bandwidth of the critical band. Whereas tone pairs within the CB sound dissonant, tones whose frequencies stand in a relation between the CB and 2CB appear consonant; and tones pairs whose fre- quencies are separated by more than 2CB elicit neutral reports Figure 3. A similar weak trend may exist in the number of spontaneous otoacoustic emission SOAE pairs as a function of frequency ratio Braun Otoacoustic emissions are physical vi- 12 brations of the basilar membrane, induced by contractions of the Outer Hair Cells which are thought to actively regular the nonlinear properties of the basilar membrane based on descending input from the brainstem Kim , Long The connection between the single- and do uble-critical bandwidth and the computation of a missing fundamental arises when one notes the frequency separation of the psycho- physically most important harmonics, partials , in perceiving the fundamental.
The difference between each subsequent frequency is of course constant when measured in frequency; but when measured in terms of frequency ratio, the spacing between har- monics grows smaller with increasing harmonic number.
The critical bandwidth is ap- proximately proportional to center frequency, so as harmonic number grows, neighbor- Figure 4: Harmonics 3—5 fall between one and two critical bands from Braun The fundamental and second harmonic fall outside of the 2CB, and thus may be considered less necessary to pitch extraction by the brain.
For harmonics , whose ratios fall between CB and 2CB, the consonance observed in pairwise psycho- physical experiments may reflect a physiological coalescence of the information from the ir respective laminae. Thus, the circuitry of the inferior colliculus may be designed to extract the fundamental from the do minant harmonics.
The temporal model of pitch perception may also be relevant in this computation. Neurons are synchro- nized to the acoustic waveform: the y tend to fire with highest probability at a particular phase in the stimu- lus period.
If such phaselocked neurons correspond- ing to two consecutive partials converge, the greatest coincidence of spikes will occur with the period of the fundamental. A coincidence detector may have already been found in the whole-cell patch-clamp recordings of Covey et. They measured post- stimulus currents in several neurons of the bat inferior colliculus, and found subthresh- old oscillations over a wide range of frequencies.
Inputs whose frequencies are com- mensurate with the intrinsic oscillations could excite spiking output. Pure tones could not have the ir frequencies calculated by this mechanism, but frequency-sensitive neurons that respond directly to the fundamental could be used in tandem with the peri- odicity information Langner EEGs from horizontal current dipole correspond to auditory nerve fibers; those from vertical dipoles correspond to brainstem activity.
Vertical bars represent 0. Note that the vertical current dipole response to the missing fundamental is larger than the horizontal response from Galbraith Noninvasive studies of human brains have discerned a signal strongly correlated to acoustic stimuli.
Electrodes that measure current dipoles in the head have detected cur- rent oscillations with the same waveform as pure tone stimuli, the so-called Frequency Following Response FFR , associated with the synchronized firing of action poten- tials. Galbraith discovered that FFRs would arise even when the stimulus lacked the fundamental frequency; but importantly, this evoked response was measured only for the vertical current dipole associated with brainstem activity, and not for the hori- zontal dipole associated with the auditory nerve fibers Figure 5.
This is evidence that the missing fundamental is indeed computed in the brainstem, not the auditory cortex, and is not present in the peripheral auditory sense. Figure 6: Neural networks for calculating the missing fundamental or periodicity of a stimulus. Frequencytuned neurons cyan circles in the inferior colliculus are arranged in bands of similar tunings light magenta.
Connections are inhibitory red within a layer, and excitatory blue between neighboring layers. A coincidence detector yellow receives input green from nearby neurons and the n fires with periocity of the stimulus waveform. If different frequency channels converge in the inferior colliculus and temporal coincidence de- tection the re calculates the periodicity, the n the stochastic nature of neural firings could affect the calculation.
If such a mistake were to be given as input to a neuron which is most sensitive to that lower frequency, the n it would respond with a low level of excitation. A given pitch would the reby also excite a subharmonic an octave below; it would truly re- semble ano the r pitch one octave lower.
If extraction of the fundamental is realized as hypo the sized by Braun, the re would be both an innate and a learned component.
The rough laminar structure of the inferior colliculus is developmentally programmed but fine-tuning of the auditory system in general requires appropriate auditory input Friauf This suggests that appropri- ate teaching soon after birth might allow subtle but measurable reconfiguration of the spectral pattern recognition developed.
Is it necessary to suppose this fundamental extraction circuit to make this leap? After all, the first harmonic of a natural fundamental is an octave, and this is much more strongly correlated with the fundamental than any ran do mly occurring subharmonic octave mistakes. He hypo the sized that through pattern recognition the brain learns to associate harmonically related frequencies as a single percept, the virtual pitch.
O the r harmonic complexes could excite subpatterns, and this would give percepts of consonance. For example, the spectral composition of a harmonic series missing its fundamental would closely match the excitation pattern of the complete spectrum, and thus elicit the pitch of that missing fundamental.
However, when the first upper harmonic is present, so is the second — 15 which is not an octave — and a whole slew of o the r non-octave intervals. There is extra information in the sound ed tone which unambiguously distinguishes it from an octave above.
The more super-personal you get, the more we refer to timeless themes of inspiration and teleology. We are dazzled by the perfection of the ratios, but we no longer care what they are actually ratios of. Many people will be benefited from your writing. Thank you! You are commenting using your WordPress.
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Instead of consciousness c arising as an unexplained addition to an unconscious, non-experienced universe u of matter and information mi , material and informative appearances arise as from the spatiotemporal nesting dt of conscious experiences that make up the universe.
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