Semitones vs perfect fifth

Question

This website:

• https://www.musictheoryacademy.com/understanding-music/intervals/perfect-fifth/)

is saying that:

"A perfect fifth is an interval of 7 semitones between 2 notes."

The internet is saying that:

"A semitone is a difference in pitch between one note and one immediately higher or lower after/before it."

(check photo attached)

I am going little bit crazy now; what's right and what's wrong?
I don't see 7 semitones anywhere. All I see is 2 semitones in a major scale/Diatonic scale.

Answer 1

This is why sometimes we want to use hyper-accurate words like "pitch" rather than colloquial words like "note." Let's give some more accurate definitions:

• When we're talking about music, an interval is a measurement of distance, like a mile or a centimeter. If we name two pitches, we can also name the interval that separates them.
• A semitone is a very small interval. It's the smallest distance apart that two pitches could be, at least on a modern piano.* On a piano keyboard, two keys that are right next to each other, including a black key and the white key that's next to it, are a semitone apart. Other words for the same idea are "half step" or "minor second." The two images are a bit confusing because the only semitones they label as such are the places where there are two white keys with no black key between them: E to F and B to C. But from C to C#, for instance, is also a semitone.
• A perfect fifth is also an interval, along with thirds, fourths, and so on. The easy but imprecise definition is "It's two notes that are five notes apart—like from C to G: C, D, E, F, G." The stricter definition is "It's two pitches that are separated by seven semitones," since if you had a C# and a G you could still name five notes, but they would be only six semitones apart. To count the seven semitones from C to G, you play all the white and black keys: C, C#, D, D#, E, F, F#, G.

* In general, and over-simplifying, you could think of a semitone for now as "the smallest musical interval." The way we name notes in western traditional music, "C", "C#" etc., gives names to pitches that are semitones apart. Other musical systems include smaller intervals, and even in western music, smaller intervals are often played or sung, but describing and naming these is a bit trickier.

Answer 2

If you start on C: the first semitone is C#, the second semitone is D, 3rd is D#, 4th is E, 5th is F, 6th is F#, and the seventh semitone is G.

So the interval of a perfect fifth up from C is G.

Answer 3

I am going little bit crazy now; what's right and what's wrong? I don't see 7 semitones anywhere. All I see is 2 semitones in a major scale/Diatonic scale.

Look more closely at the image. On the left, there are note names; on the right, there is a diagram of a piano keyboard. With the note names, you can see that some notes are connected by lines labelled Tone; these notes correspond to the white keys on the keyboard.

However, there are also black keys on the keyboard, that correspond to in-between notes. These are labelled in lighter text. On the left side, we see C# (pronounced "C sharp") in between the C and D; this note can also be called Db (pronounced "D flat"), with the standard system of notes we are using here. The piano keyboard diagram puts both labels on that black key.

You can also see that the E and F, and the B and (upper) C, are connected by lines labelled Semitone. This is because these notes/white keys don't have another note/black key in between them.

In English, "semi-" is a prefix meaning "half". The conceptual "distance" between the notes of a tone - i.e., "how much higher" D is than C - is twice that between the notes of a semitone (e.g. between F and E).

In the approximations that we use in teaching Western musical theory at an elementary level, we say that the note played by the G key on the piano is a perfect fifth above the C note - that is the name we use for that interval. The notes played by each key in sequence are equally spaced out: it takes 12 semitones, or 6 tones, to get from one C to the next C, an octave up. Similarly, from the C to the G we count 7 semitones: C-C#-D-D#-E-F-F#-G. (Or, using the "flat" names: C-Db-D-Eb-E-F-Gb-G.)

The reasons for all of these alternate names, for having black keys between some but not all white keys, etc. will become clear as you learn more of the theory - it would be out of scope to try to explain them here. Similarly, "perfect fifth" is actually intended to mean an interval that is very slightly larger than the distance from a piano's C to a piano's G (smaller than most people would be able to notice by listening).

There are specific reasons why we divide the octave up specifically into 12 notes, and then make these kinds of approximations, that will become clearer if you study more theory. (There's also history behind why we have the name "Tone" for the distance e.g. from C to D - i.e., two notes up in our sequence of adjacent notes.) It's also possible to do it any number of other ways. There aren't any specific "notes"; you can make sounds at any pitch want - they just won't be ones that Western musical instruments are designed to play, and won't necessarily sound good, or make sense to include in music written in Western styles, or have any natural representation in Western music notation.

Answer 4

A "semitone" uses a Greek prefix to say "halftone". On the assumption that all tones are identical, all semitones are identical, that two semitones constitute the same frequency interval as one tone, the three tones and one semitone making up a fifth in a scale are the same as seven semitones or 7/12 of an octave.

That is called 12-EDO. There is also 19-EDO where a semitone is 2/3 of a tone and a fifth is thus 3⅔ tones or 5½ semitones or 11/19 octaves in size. Then there is 31-EDO where a semitone is 3/5 of a tone, and a fifth is 3⅗ tones or 6 semitones or 18/31 of an octave in size.

An accidental will cover the difference between a semitone and a tone. As you can see, as soon as "semi" is no longer taken at its literal meaning "half" but more like "small", your question stops having a trivial answer.

But those more elaborate and much less used temperated scales built from an equal-tempered microscale are not typically available from most instruments. They do offer some purer intervals than the customary 12-EDO scales do.

Answer 5

The 12 keys (C, C#, D, D#, E, F, F#, G, G#, A, A#, B) on a piano represent 12 "equally spaced" pitches. This is called 12-tone equal temperament. Each piano key has a frequency that is exactly 2^1/12 times larger than its immediately preceding piano key. ^{[see footnote 1/2]}

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Recall:

2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 = (2^1/12)¹² = 2

Thus, when you take 12 steps up from any given piano key, you will end up at a note that has exactly double the frequency. This is called an octave. ^{[see footnote 8]} When two notes that are an octave apart are played simultaneously, their sound waves will overlap every 2:1 cycles, which sounds breathtakingly beautiful.

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Furthermore:

2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 = (2^1/12)⁷ ≈ 1.498 ≈ 1.5 = 3/2

Thus, when you take 7 steps up from any given piano key, you will end up at a note that has roughly 1.5 times the frequency. This is called a perfect fifth. ^{[see footnote 5]} When two notes that are a perfect fifth apart are played simultaneously, their sound waves will overlap every 3:2 cycles, which sounds simply sublime.

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Additionally:

2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 · 2^1/12 = (2^1/12)⁵ ≈ 1.334 ≈ 4/3

Thus, when you take 5 steps up from any given piano key, you will end up at a note that has roughly 1.33 times the frequency. This is called a perfect fourth. ^{[see footnote 4]} When two notes that are a perfect fourth apart are played simultaneously, their sound waves will overlap every 4:3 cycles, which sounds suitably striking.

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^{1/2 Every adjacent piano key is a semitone apart. From C to C#. From C# to D. From D to D#. And so on.
8 An octave up from any given white key is seven white keys to the right. Example: from C4 to C5.
5 A perfect fifth up from any given white key except B is four white keys to the right. Example: from C to G.
4 A perfect fourth up from any given white key except F is three white keys to the right. Example: from C to F.}

Answer 6

It's true, there is only one semitone between two of the notes on the way to making up a perfect fifth.(Between notes ^3 and ^4 in the quoted case), BUT there are also three tones, and they must be counted too.

Each tone is worth two semitones, making the P5 7 semitones altogether. The phrasing is unfortunate, and could have been improved.

To further muddy the water, 7 semitones also make the interval of a diminished 6th, so don't go away with the 'knowledge' that every two notes 7 semitones apart will constitute P5 - the letter names and appropriate accidentals (♯/♭) will have some bearing on the calculation.

So, without reading the 'important note' at the end of that site, the statement is not actually true - there are 7 semitones making up a P5 - it's only half true!

Answer 7

Think about it like this:

1. First the octave (which means doubling the frequency) is divided into 12 equal parts. These are the notes that we are talking about. These 12 equal parts are the semitones.

2. Some of these notes are labeled A through G using this pattern:
   A * B C * D * E F * G *

   It's the pattern that you get if you label the white keys on a keyboard with letters while labeling the black keys with *. This is the diatonic scale used in either A-minor or C-major.

3. The large steps that skip over a * are called tones while the small steps are called semitones as above. So, the octave is divided into five tones (A-B, C-D, D-E, F-G, G-A) and two semitones (B-C, E-F) by the scale of the labeled notes.

4. As you can see, a perfect fifth (like the one from A to E) contains a total of seven small steps in this sequence. That is equivalent to saying that a perfect fifth is seven semitones. And it's the same as saying that a perfect fifth contains three tones and a semitone. (Makes sense: 3.5 = 7/2)

5. However, intervals are named for the letters, only. As such, the fifth from A to E is called a fifth because it encompasses the five letters A, B, C, D, and E. The interval A-C is called a third (A, B, C), and the interval C-E is also called a third (C, D, E), irrespective of the fact that the later is one semitone larger than the former. That is why the interval B-F is called a diminished fifth while the interval F-B is called an augmented fourth. It's six semitones (or three tones = tritone) either way, but there is one more letter on the way in B, C, D, E, F than in F, G, A, B.

Answer 8

You're mixing up a couple things and skipping the important fact that 1 tone = 2 semitones (hence the name semitones).

The images you posted tell you how to obtain a (major) scale: you begin at a certain note, in this example C, then get the next note by adding a tone (= two semitones = move two keys -> D), then another tone (-> E), then a semitone (->F since there's no black key in between), then repeat...

With this in mind, adding 7 semitones (= going to the note that is 7 semitones above yours) means adding 3 tones and one semitones... or simply counting 7 keys including the black ones. So from C: C# (1 semitone away), D (2), D# (3), E (4), F (5), F# (6), G (7)