question: How are two notes that are an octave apart "the same note" if they have different frequencies.
comment: notes one octave higher have exactly twice the frequency and have the same "sound" to humans. One of the earliest diatonic scales (similar to the modern do re mi....) have notes generated by using frequency ratio of 3/2 for each fifth note. (a fifth note to musicians is the interval between C and G for example) by multiplying this ratio repeatedly you can generate a circle of fifths (all the tones and semitones) Modern tuning adjusts this a bit to be less discordant so that you can transpose music from one key to another and yet sound fairly harmonious. & #x200B;
passages: Compound Intervals are formed and named as follows:
  
  BULLET::::- 2nd + Octave = 9th
  
  BULLET::::- 3rd + Octave = 10th
  
  BULLET::::- 4th + Octave = 11th
  
  BULLET::::- 5th + Octave = 12th
  
  BULLET::::- 6th + Octave = 13th
  
  BULLET::::- 7th + Octave = 14th
   As the Railsback curve shows, octaves are normally stretched on a well-tuned piano. That is, the high notes are higher, and the low notes lower, than they are in an equal-tempered scale. Railsback discovered that pianos were typically tuned in this manner not because of a lack of precision, but because of inharmonicity in the strings. Ideally, the overtone series of a note consists of frequencies that are integer multiples of the note's fundamental frequency. Inharmonicity as present in piano strings makes successive overtones higher than they "should" be.
   When pianos are tuned by piano tuners, the technician sometimes listens for the sound of "beating" when two notes are played together, and tunes to the point that minimizes roughness between tones. Piano tuners must deal with the inharmonicity of piano strings, which is present in different amounts in all of the ranges of the instrument, but especially in the bass and high treble registers. The result is that octaves are tuned slightly wider than the harmonic 2:1 ratio. The exact amount octaves are stretched in a piano tuning varies from piano to piano and even from register to register within a single piano—depending on the exact inharmonicity of the strings involved.
   claimed, therefore, that their results were not consistent with those of Deutsch in relation to hand dominance and ear dominance versus location of pitch.
   Section::::Ancient Greek theory.:Octave species.
  
  The basis of octave species was the smaller category of species of the perfect fourth, or "diatessaron"; when filled in with two intermediary notes, the resulting four notes and three consecutive intervals constitute a "tetrachord" . The species defined by the different positioning of the intervals within the tetrachord in turn depend upon genus first being established . Incomposite in this context refers to intervals not composed of smaller intervals.
   Two notes with fundamental frequencies in a ratio equal to any integer power of two (e.g., half, twice, or four times) are perceived as very similar. Because of that, all notes with these kinds of relations can be grouped under the same pitch class.
   In tuning, the relationship between two notes (known musically as an interval) is determined by evaluating their common harmonics. For example, we say two notes are an octave apart when the fundamental frequency of the upper note exactly matches the second harmonic of the lower note. Theoretically, this means the fundamental frequency of the upper note is exactly twice that of the lower note, and we would assume that the second harmonic of the upper note will exactly match the fourth harmonic of the lower note.
   In the case of a musical note, the lowest frequency of its timbre is designated as the sound's fundamental frequency. For simplicity, we often say that the note is playing at that fundamental frequency (e.g. "middle C is 261.6 Hz"), even though the sound of that note consists of many other frequencies as well. The set of the remaining frequencies is called the overtones (or the harmonics) of the sound. In other words, the fundamental frequency alone is responsible for the pitch of the note, while the overtones define the timbre of the sound. The overtones of a piano playing middle C will be quite different from the overtones of a violin playing the same note; that's what allows us to differentiate the sounds of the two instruments. There are even subtle differences in timbre between different versions of the same instrument (for example, an upright piano vs. a grand piano).
   Two intervals are considered "enharmonic", or "enharmonically equivalent", if they both contain the same pitches spelled in different ways; that is, if the notes in the two intervals are themselves enharmonically equivalent. Enharmonic intervals span the same number of semitones.
  
  For example, the four intervals listed in the table below are all enharmonically equivalent, because the notes F and G indicate the same pitch, and the same is true for A and B. All these intervals span four semitones.
   Octave equivalence is a part of most "advanced musical cultures", but is far from universal in "primitive" and early music. The languages in which the oldest extant written documents on tuning are written, Sumerian and Akkadian, have no known word for "octave". However, it is believed that a set of cuneiform tablets that collectively describe the tuning of a nine-stringed instrument, believed to be a Babylonian lyre, describe tunings for seven of the strings, with indications to tune the remaining two strings an octave from two of the seven tuned strings. Leon Crickmore recently proposed that "The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week".
   However, in stringed instruments such as the piano, violin, and guitar, or in some Indian drums such as tabla, the overtones are close to—or in some cases, quite exactly—whole number multiples of the fundamental frequency. Any departure from this ideal harmonic series is known as inharmonicity. The less elastic the strings are (that is, the shorter, thicker, smaller tension or stiffer they are), the more inharmonicity they exhibit.
   Species of the perfect fifth ("diapente") are then created by the addition of a whole tone to the intervals of the tetrachord. The first, or original species in both cases has the "pyknon" or, in the diatonic genus, the semitone, at the bottom and, similarly, the lower interval of the "pyknon" must be smaller or equal to the higher one . The whole tone added to create the species of fifth (the "tone of disjunction") is at the top in the first species; the remaining two species of fourth and three species of fifth are regular rotations of the constituent intervals, in which the lowest interval of each species becomes the highest of the next (; ). Because of these constraints, tetrachords containing three different incomposite intervals (compared with those in which two of the intervals are of the same size, such as two whole tones) still have only three species, rather than the six possible permutations of the three elements . Similar considerations apply to the species of fifth.
   The Lesser and Greater Perfect Systems exercise constraints on the possible octave species. Some early theorists, such as Gaudentius in his "Harmonic Introduction", recognized that, if the various available intervals could be combined in any order, even restricting species to just the diatonic genus would result in twelve ways of dividing the octave (and his 17th-century editor, Marcus Meibom, pointed out that the actual number is 21), but "only seven species or forms are melodic and symphonic" . Those octave species that cannot be mapped onto the system are therefore rejected .
  
  Section::::Medieval theory.
   Finally, it can be seen from this formula that octaves automatically yield powers of two times the original frequency, since "n" is a multiple of 12 (12"k", where "k" is the number of octaves up or down), and so the formula reduces to:
  
  yielding a factor of 2. In fact, this is the means by which this formula is derived, combined with the notion of equally-spaced intervals.
   In chant theory beginning in the 9th century, the "New Exposition" of the composite treatise called "Alia musica" developed an eightfold modal system from the seven diatonic octave species of ancient Greek theory, transmitted to the West through the Latin writings of Martianus Capella, Cassiodorus, Isidore of Seville, and, most importantly, Boethius. Together with the species of fourth and fifth, the octave species remained in use as a basis of the theory of modes, in combination with other elements, particularly the system of octoechos borrowed from the Eastern Orthodox Church .
   Species theory in general (not just the "octave" species) remained an important theoretical concept throughout Middle Ages. The following appreciation of species as a structural basis of a mode, found in the "Lucidarium" (XI, 3) of Marchetto (ca. 1317), can be seen as typical:
  
  We declare that those who judge the mode of a melody exclusively with regard to ascent and descent cannot be called musicians, but rather blind men, singers of mistake... for, as Bernard said, "species are dishes at a musical banquet; they create modes" .
   Inharmonicity "stretches" harmonics beyond their theoretical frequencies, and higher harmonics are stretched proportionally more than lower. Thus, in our example of an octave, exactly matching the lowest common harmonic causes a slight amount of stretch, matching the next higher common harmonic causes a greater amount of stretch, and so on. If the interval is a double octave, exactly matching the upper note to the fourth harmonic of the lower complicates the tuning of that upper note with the one an octave below it.
   When a string is bowed or tone in a wind instrument initiated by vibrating reed or lips, a phenomenon called mode-locking counteracts the natural inharmonicity of the string or air column and causes the overtones to lock precisely onto integer multiples of the fundamental pitch, even though these are slightly different from the natural resonance points of the instrument. For this reason, a single tone played by a bowed string instrument, brass instrument, or reed instrument does not necessarily exhibit inharmonicity.
   In equal temperament tunings, the generating interval is often found by dividing some larger desired interval, often the octave (ratio 2:1), into a number of smaller equal steps (equal frequency ratios between successive notes).
   Chambers, Moss, and Mattingley believe that the illusion is caused by a combination of harmonic fusion and binaural diplacusis, a condition in which a pitch is perceived slightly differently between ears. In experiments employing a few subjects, none reported the percept most commonly obtained by Deutsch’s subjects. This does not necessarily conflict with other studies, as the number of subjects used in this experiment was small and there are individual differences in perception of the illusion. The authors also argued that experiments conducted by others did not clarify whether the high tone and low tones were perceived as being an octave apart. They 
   The two tones used were pitched at 400 Hz and 800 Hz, corresponding roughly to G4 and G5 in modern pitch notation. Each tone was played for 250 ms before switching ears. No gaps were allowed between tones. Both tones were therefore always present during the experiment; only the ears perceiving the high and the low tone at any one time changed repeatedly. After the initial test, the headphones were reversed, and the test was repeated.
  
  Section::::Results.
   Octave species
  
  In early Greek music theory, an octave species (εἶδος τοῦ διὰ πασῶν, or σχῆμα τοῦ διὰ πασῶν) is a sequence of incomposite intervals (ditones, minor thirds, whole tones, semitones of various sizes, or quarter tones) making up a complete octave . The concept was also important in Medieval and Renaissance music theory.
  
  Section::::Ancient Greek theory.
   The nature of the chromatic genus, too, is an attribute of the "kinēsis phonēs" (, "potentiality of the sounds"), so that certain melody types are "brought into being". In other words, "being composite" and "being incomposite" are attributes of the dynamic character of melodic motion. "None of these consists in the voice's coming to rest at points separated by distances of specific and determinate sizes".
   As can be seen, no note always corresponds to a certain degree of the scale. The "tonic", or 1st-degree note, can be any of the 12 notes (pitch classes) of the chromatic scale. All the other notes fall into place. For example, when C is the tonic, the fourth degree or subdominant is F. When D is the tonic, the fourth degree is G. While the note names remain constant, they may refer to different scale degrees, implying different intervals with respect to the tonic. The great power of this fact is that any musical work can be played or sung in any key. It is the same piece of music, as long as the intervals are the same—thus transposing the melody into the corresponding key. When the intervals surpass the perfect Octave (12 semitones), these intervals are called "compound intervals", which include particularly the 9th, 11th, and 13th Intervals—widely used in jazz and blues Music.
   A vibrating wire subdivides itself into many parts vibrating at the same time. Each part produces a pitch of its own, called a partial. A vibrating string has one fundamental and a series of partials. The most pure combination of two pitches is when one is double the frequency of the other.
  
  For a repeating wave, the velocity equals the wavelength times the frequency ,
  
presuppositions: If notes are an octave apart and have two different frequencies they can't have the same note. 
corrections: Notes that have one octave higher have twice the frequency of the other note, which tends to sound exactly the same to humans. 
raw_presuppositions: If notes are an octave apart and have two different frequencies they can't have the same note.  Notes octaves apart should have the same frequency.
raw_corrections: Notes that have one octave higher have twice the frequency of the other note, which tends to sound exactly the same to humans.  Octaves are defined as twice the frequency. It just sounds the same to humans. 