question: why is there no half step between B and C as well as E and F?
comment: An octave is divided into 12 equally spaced notes, and only after that do we pick out scales. Western music theory specifies scales with 7 notes with very specific spacings between them as sounding the best and most useful for making harmonically functional music. With these spacings, B can slide smoothly up to C and F down to E in what we call a resolution. Compare that with a bigger jump like A to G and you'll see it's not as smooth because there's a note in the middle. The keyboard was designed around those 7 notes. There actually is only a half-step between those notes and a whole step between the other neighboring notes in the C major scale, but I know that you meant "extra note" instead. Also note that this applies for any key, not just C major.
passages: In the diatonic scale, a step is either a minor second (sometimes also called "half step") or a major second (sometimes also called "whole step"), with all intervals of a minor third or larger being skips. For example, C to D (major second) is a step, whereas C to E (major third) is a skip.
  
  More generally, a step is a smaller or narrower interval in a musical line, and a skip is a wider or larger interval with the categorization of intervals into steps and skips is determined by the tuning system and the pitch space used.
   Doh     Ray      Me       Fah      Soh      Lah      Te        Doh
  
  This can be applied in ‘Absolute’ (or, ‘Fixed-Doh’) Form — also known as ‘Solfege’ — or in a ‘Relative’ (or, ‘Movable-Doh’) Form — also known as ‘Solfa’ — where ‘Doh’ starts on the first pitch of the scale (ie. for A Major, ‘Doh’ is ‘A’; for G Major, ‘Doh’ is ‘G’; E Major, ‘Doh’ is ‘E’; and so on).
   In this stack, G and F have a similar frequency, and G is typically discarded. Also, the note between C and D is called D rather than C, and the note between G and A is called A rather than G. The C-based symmetric stack is rarely used, possibly because it produces the wolf fifth in the unusual position of F–D instead of G–E, where musicians using Pythagorean tuning expected it).
  
  Section::::Justly intonated quarter-comma meantone.
   In the first and second scale, B and D are exactly the inversion of each other. This is not true for the third one. This is the reason why these two scales are regarded as symmetric (although the removal of G makes all 12 tone scales, including those produced with any other tuning system, slightly asymmetric).
   Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possible temperaments having a step size between 30 and 120 cents. These were called "alpha", "beta", and "gamma". They can be considered as equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes:
  
  BULLET::::- "alpha": (78.0 cents)
  
  BULLET::::- "beta": (63.8 cents)
  
  BULLET::::- "gamma": (35.1 cents)
  
  Alpha and Beta may be heard on the title track of her 1986 album "Beauty in the Beast".
  
  Section::::Other equal temperaments.:Proportions between semitone and whole tone.
   Melodic motion in which the interval between any two consecutive pitches is no more than a step, or, less strictly, where skips are rare, is called "stepwise" or "conjunct" melodic motion, as opposed to "skipwise" or "disjunct" melodic motion, characterized by frequent skips.
  
  Section::::Half steps.
  
  In the major scale or any of its modes, a step will always be a movement of 1 or 2 semitones, and a skip a movement of 3 or more semitones.
  
  In other scales an augmented second—an incomposite step equivalent to 3 semitones—and/or a diminished third—a skip of 2 semitones—may be possible.
  
  Section::::Melody.
   Finer distinctions among the scale degree are sometimes made, with the D below the final called the "mediant", the lowest note, C, the "participant" (a tone functioning as an auxiliary to the mediant), the G, B, and B the "conceded modulations" (subsidiary degrees), and the lowest C, the final, F, and (rarely) the D the "absolute initials" .
   Since this is below C, one needs to move up by an octave to end up within the desired range of ratios (from 1:1 to 2:1):
   Since this is below C, you need to move up by an octave to end up within the desired range of ratios (from 1/1 to 2/1):
   In a melodic half step, listeners of the time perceived no tendency of the lower tone toward the upper, or the upper toward the lower. The second tone was not the 'goal' of the first. Instead, musicians avoided the half step in clausulas because, to their ears, it lacked clarity as an interval. Beginning in the 13th century, cadences begin to require motion in one voice by half step and the other a whole step in contrary motion.
  
  Section::::In medieval and Renaissance polyphony.:Plagal cadence.
   If the notes G and E need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based Pythagorean tuning would produce a stack of fifths running from D to F, making F-D the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all keys in tune.
  
  Section::::Size of intervals.
   If we contrast that against the earlier scale, we see that for five pairs of successive notes the ratios of the steps remain the same, but one note, D, the steps C-D and D-E have switched their ratios.
  
  The three major thirds are still 5:4, and three of the minor thirds are still 6:5 with the fourth being 32:27, except that now it's BD instead of DF that is 32:27. FAC and CEG still form just major triads (4:5:6), but GBD is now (108:135:160), and BDF is now (135:160:192).
   On the contrary, the intervals from D shown in the table above, since D is at the center of the stack, do not include wolf intervals and include a pure m6 (from D to B), instead of an impure augmented fifth. Notice that in the above-mentioned set of 144 intervals pure minor sixths are more frequently observed than impure augmented fifths (see table below), and this is one of the reasons why it is not desirable to show an impure augmented fifth in the construction table. A "C-based symmetric" stack might be also used, to avoid the above-mentioned drawback:
   Beckel was born in Marion, Ohio, where he studied with the trombonist Irey Halt. A diligent pupil, Beckel would often practice late into the evening, only to get up early and finish his homework. He wanted to become an orchestral trombonist, so he wrote Edward Kleinhammer, then the bass trombonist of the Chicago Symphony Orchestra. Kleinhammer suggested, simply, that Beckel practice not two hours a day (as Beckel had done), but four hours a day.
   The existence of the "wolf fifth" is one of the reasons why, before the introduction of well temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G and E).
   Common-practice conventions dictate that melodic lines should be smooth and independent. To be smooth, they should be primarily conjunct (stepwise), avoid leaps that are difficult to sing, approach and follow leaps with movement in the opposite direction, and correctly handle tendency tones (primarily, the leading-tone, but also the , which often moves down to ). To be independent, they should avoid parallel fifths and octaves.
   After the initial presentation, the opposing characters of the two layers is gradually evened out, in a process of development over seven cyclical statements of the formula, until the two layers are merged into a single melody . This is accomplished by progressively transposing the upper level downward by one semitone per cycle, and the lower level upward by the same degree. In this way, the entirely separate ranges in the first cycle (F–F and F–F are brought into the single octave C–B in the seventh .
   On the other hand, consider the enigmatic scale, which follows the pattern 1-3-2-2-2-1-1. It is possible to find intervals in this scale that are narrower than other intervals in the scale spanning fewer scale steps: for example, the fourth built on the 6th scale step is three semitones wide, while the third built on the 2nd scale step is five semitones wide. Therefore, the enigmatic scale is not proper.
  
  Section::::Diatonic scale theory.
   The second is as a mode with a final on F and an ambitus extending to F an octave higher and in which the note C was regarded as having an important melodic function. Many theorists of the period observed that B is used more typically than B in compositions in Lydian mode .
  
  Section::::Modern Lydian mode.
   Only one accidental is used commonly in Gregorian chant—B may be lowered by a half-step to B. This usually (but not always) occurs in modes V and VI, as well as in the upper tetrachord of IV, and is optional in other modes except III, VII and VIII .
   If "c" and "c" are not integers they need to be rounded to the next larger integer, and distributed as evenly as possible among "k."
  
  Section::::Advantages.
   “As late as the 13th century the half step was experienced as a problematic interval not easily understood, as the irrational remainder between the perfect fourth and the ditone formula_1.” In a melodic half step, no “tendency was perceived of the lower tone toward the upper, or of the upper toward the lower. The second tone was not taken to be the ‘goal’ of the first. Instead, the half step was avoided in clausulae because it lacked clarity as an interval.” 
   Accordingly, a note pair of B–L "(cum proprietate et cum perfectione)" could be written with the most basic of ligature shapes, those inherited from plainchant, namely the descending "clivis" () and the ascending "podatus" (). Likewise, three-note groups of B–B–L could be written with some of the inherited ternary neumes, such as the "porrectus" (, direction down–up), the "torculus" (, direction up–down), or the "scandicus" (, direction up–up).
   Major triads are formed out of both major thirds and perfect fifths. If either of the two intervals is substituted by a wolf interval (d6 instead of P5, or d4 instead of M3), then the triad is not acceptable. Therefore, major triads with root notes of C, F, G and B are not used in meantone scales whose fundamental note is C.
  
  Now look at the first two columns on the left. Nine of the twelve 3-semitone intervals have a ratio of
   Steps and skips
  
  In music, a step, or conjunct motion, is the difference in pitch between two consecutive notes of a musical scale. In other words, it is the interval between two consecutive scale degrees. Any larger interval is called a skip (also called a leap), or disjunct motion.
  