Why 12 notes to the Octave? Why 12 notes to the Octave? Any moderately curious person will ask themselves at some point why, in western music, is the octave divided into 12 'semi-tones'. From a mathematical point of view, we can easily explain why 12 works nicely. The Greeks realized that sounds which have frequencies in rational proportion are perceived as harmonius. For example, a doubling of frequency gives an octave. A tripling of frequency gives a perfect fifth one octave higher. They didn't know this in terms of frequencies, but in terms of lengths of vibrating strings. Pythagoras, who experimented with a monochord, noticed that subdividing a vibrating string into rational proportions produces consonant sounds. This translates into frequencies when you know that the fundamental frequency of the string is inversely proportional to its length, and that its other frequencies are just whole number multiples of the fundamental. (actually, the notion of consonance is more complicated than rationality- see, for example, this fascinating article ). First, we should examine what ratios are "meant" to exist in the western scale. The prominence of the major triad in western music reflects the Greek discoveries mentioned above.

Starting with the note C as a fundamental, we get the major triad from the 3rd and 5th overtones, dropping down one and two octaves respectively, obtaining ratios of 3/2 (G:C) and 5/4 (E:C) respectively. Two other prominent features in western music include the V I cadence, and the I,IV,V triads. Both reflect the importance of the 3/2 ratio, with the IV further taking into account the reciprocal of 3/2, namely 2/3 aka 4/3. Musically, the reciprocal ratio corresponds to going down rather than up. While 3/2 corresponds to going up a fifth, 2/3 corresponds to going down a fifth, and 4/3 corresponds to going down a fifth and up an octave. Together, 3/2 and 4/3 divide the octave, so that going up by 3/2 followed by 4/3 gives an octave. The IV and V triads give us the four new notes, B and D of G,B,D, and F and A of F,A,C.

Their ratios, relative to C are 15/8 for B, 9/8 for D, 4/3 for F, and 5/3 for A. The notes formed from the I,IV, and V major triads produce the C major scale: C D E F G A B C. Throwing in reciprocals for each of these intervals yields all the intervals that made up western music until the rise of chromaticism.

1/1 unison C 2/1 octave C 3/2 perfect fifth G 4/3 fourth F 5/4 major third E 8/5 minor 6th Ab 6/5 minor 3rd Eb 5/3 major 6th A 9/8 major 2nd D 16/9 minor 7th Bb 15/8 major 7th B 16/15 minor 2nd C# While this list of intervals does include a few of the most basic intervals and their reciprocals: unison, perfect 5th, major 3rd, major 6th = 3rd above a 4th (or also a 4th above a 3rd), major 2nd = a 5th above a 5th, and major 7th = a 3rd above a 5th (or also a 5th above a 3rd), some obvious ones are missing (such as 7/4, 25/16 = a 3rd above a 3rd, or 9/5 = a fifth above a minor 3rd).

The tritone (such as C to F#) is also omitted from this list, an interval that did not affect the evolution of the western scale as it was not used in western music until twelve note chromaticism had become firmly established. Actually, a tritone refers to two different possible intervals: 7/5 tritone 10/7 also called a tritone. The idea behind twelve is to build up a collection of notes using just one ratio. The advantage to doing so is that it allows a uniformity that makes modulating between keys possible. Without a compromise most keys would be unusable as most of the basic intervals would not be captured in the different keys (see the table at the end of this essay). Unfortunately, no one ratio will do the trick exactly. However, the ratio of 3/2 happens to work reasonably well using 12 steps. With 3/2 as the basis for the scale, none of the above ratios besides a unison, fifth, and major 2nd are captured exactly. However the most important constraint- namely that we get a repeating pattern going up in octaves, is almost satisfied by this scheme.

Namely, after 12 applications of the ratios 3/2, we come back very close to where we started from (always dropping down by an octave, i.e.

dividing by 2, each time the ratio exceeds 2): (3/2)^0 = 1 (3/2)^1 = 1.5 (3/2)^2 = 1.125 (after dividing by 2) (3/2)^3 = 1.6875 (after dividing by 2) (3/2)^4 = 1.2656 (after dividing by 4) (3/2)^5 = 1.8984 (after dividing by 4) (3/2)^6 = 1.4238 (after dividing by 8) (3/2)^7 = 1.0678 (after dividing by 16) (3/2)^8 = 1.6018 (after dividing by 16) (3/2)^9 = 1.2013 (after dividing by 32) (3/2)^10 = 1.8020 (after dividing by 32) (3/2)^11 = 1.3515 (after dividing by 64) (3/2)^1

2 = 1.0136 (after dividing by 128) we have returned close to where we started from.

(these 12 frequencies correspond to the circle of 5ths. Starting from C, we then get G D A E B F# C# Ab Eb Bb F and back to C). The chromatic scale reflects this fact. In the 18th and 19th centuries, the chromatic scale was tuned using the idea of 3/2. In the most elegant of these, Thomas Young's tuning , several of the fifths were set exactly to 3/2, and the others were tempered slightly (to make octaves exact). In the modern equal temperament (which came into practical use during the early part of the 20th century), all fifths are tuned to 2^(7/12)=1.49651..., slightly less than 3/2, and 12 repetitions of this ratio gets us back to where we started (after dropping down 7 octaves). Between the two methods of incorporating 3/2, the former gives the various keys character, and I prefer it highly. See the essay I wrote on this.

Of the various intervals, the only ones that are really well captured by tempered versions of the 3/2 scheme are: unison, 5th, major 2nd, and their reciprocals (octave, 4th, minor 7th). Two questions: why 3/2? The choice of 3/2 says that, next to the octave, it should be regarded as the most important interval. One can also use a major 3rd (i.e. ratio of 5/4) to build up a scale. This is discussed towards the end of this essay. Why do 12 steps work nicely? Interestingly, this can be explained in terms of simple number theory, namely continued fractions. We want to understand when a power of 3/2 will be close to a power of 2: a b (3/2) = 2 where a and b are natural numbers Of course, this equation can't be solved exactly using natural numbers a and b, since this would imply, once we clear denominators, that a power of 3 was a power of 2 (an impossibility since powers of three are odd, whereas powers of 2 are even.

Such an expression would also contradict the fact that each integer is *uniquely* expressible, up to order, as a product of primes). But, taking the ath root of both sides: b/a (3/2) = 2 theory - Why are there twelve notes in an octave? - Music: Practice & Theory Stack Exchange Why are there twelve notes in an octave? up vote 78 down vote favorite 30 I know that one scale consists of 12 half-tones. But my question is still: Why? Why not 13 or 11? 4 Do you mean "given the interval we call the 'half step,' why do 12 of them make an octave" or "given the interval we call the 'octave,' why do we split it into 12 half steps"? – rshallit Apr 26 '11 at 19:41 1 Presumably the latter, but I could be wrong.

– Ben Alpert Apr 26 '11 at 19:44 In addition to some good answers here - this book provides a fairly good explanation amazon.com/dp/0962949671/?tag=stackoverfl08-20 – DRL Jun 24 '11 at 19:51 1 Another in-depth answer can be found here . A nice demonstration of other tunings is here . – Michael Litvin Dec 7 '13 at 21:30 up vote 71 down vote accepted This requires an excursion into musical history. Originally, instruments were made to simply play notes that sounded "right" together. Why some notes sounded right and others wrong wasn't of great concern for most of humanity's history, until Pythagoras , (yes, the guy with the theorem) noticed that it had to do with intervals, and made a music theory based on perfect fifths.

This theory had its problems, however, and was improved upon by later people, eventually ending up on what is called a " just intonation " Basically, notes sound harmonious if the frequency of the notes is close to a simple interval, like 3/2 or 5/4. These theories were important because it meant it was possible for different instrument makers to make instruments that could play scales together, thereby making orchestras. But just tuning has a problem: you can basically only play the scale that the instrument is built for, because the intervals between the notes are different. If you play a tune on the wrong scale, it will sounds out of tune. This means that if you want to sing along with the instrument, you have to find a singer whose range fits the song in the scale the instrument is built for. You can't transpose the song to fit the singer. Also, musicians were exploring the limits of what you could do with just intoned instruments. So out of this came then the equal temperament . It splits the scale into equal intervals, meaning you can transpose a tune into other keys, and also means you can do dramatic chord changes and other interesting things.

You can indeed split the octave into 11 or 13 notes if you should wish to do so, but to most people it will sound out of tune . But when you split it into 12 notes, you get close enough to the seven notes of just intonation for it to be bearable, except to some unlucky few supposedly burdened with overactive perfect pitch. The five tones that are in between the basic seven are, as expected, called "half-tones". There are equal temperaments other than the 12 tones per octave that will sound fine, but they don't generally have a integral number of notes per octave. Wendy Carlos experimented a lot with this, and made such scales as the Gamma scale with a slightly mindboggling 34.29 notes per octave.

2 there was a lot of practical and theoretical exploration going on for centuries but equal temperament came specifically out of the standardization of keyboard instruments (especially church organs), the question of fretted instruments and the renewal of a mathematical approach of tonality (see Mersenne treatise for instance) – ogerard Apr 30 '11 at 8:47 Among odd (no pun intended) scales there is also the Bohlen-Pierce scale that is built upon odd number ratios. en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale – Ulf Åkerstedt Jun 4 '12 at 23:58 Actually this was known before Pythagoras. He was just the first whose followers wrote it down. Also, modern theory shows that small integer ratios are only applicable to harmonic sounds. Inharmonic sounds or sounds with only odd harmonics produce different scales. – endolith Jul 23 '12 at 18:10 1 That's the whole point. Small integer rations = harmonic sound. I don't see what is modern with that. :-) And how do you know people knew it before Pythagoras if they didn't write it down?

– Lennart Regebro Jul 23 '12 at 20:29 Here's an image of just vs ET side by side flic.kr/p/7rNope – endolith Dec 22 '16 at 0:45 I think more fundamentally, (3/2)^12 (129.75) is close to a power of two (128). Thus, the fifths on a 12-note equal-tempered scale have a ratio of 1.498:1 (ideal would be 1.5:1), which is closer to perfect than for any other reasonable number of notes. – supercat May 19 '13 at 17:47 I've read discussions of 19-TET (19-tone equal temperament) in which a diatonic scale would have five "large" intervals of 3/19 octave and two "small" intervals of 2/19 octave. Such a scale would be amenable to normal music notation if one regards e.g. C# and Db as being 1/3 step apart.

The biggest oddity would be that key signatures with up to nine sharps or flats would be distinct (rather than having C#/Db, F#/Gb, and B/Cb as pairs of sound-alike key signatures). – supercat May 19 '13 at 17:51 Two points that may have not been completely answered. Why is C major the reference scale for natural tones ? The anglo-saxon notation obscures the history a little. Tradition from church music led in Italy (then shortly after France and Spain) to naming notes of the reference major scale by conventional syllables: Ut Re Mi Fa Sol La Si (this corresponds to C D E F G A B) coming from the latin lyrics of a very well known piece of that time. The latter single-letter notation takes another starting point, but the reference character of the C major scale has persisted across Occidental countries even if you can find evidence of notations and keyboards using other notes as reference. One of the main influences has been the construction of keyboard instruments (notably the church organ).

The current keyboard layout is a compromise between the typical width of the hands, playing the Ut (now mostly called Do or C) major scale easily and having access to all semitones and a few other things. Other designs have not been as successful. You also have to know that the theorization and standardization of music at least up to the 19th century was made under the patronage of the churches (orthodox, catholic, reformed, ...) pushing for uniformity. The nineteenth century has seen an even larger standardization and internationalization of tuning, music teaching and piano domination as reference and composition instrument. The last three centuries have progressively suppressed or put into oblivion most of the divergent traditions (as to scales, modes, tuning) in Europe. Nowadays, people learning about music are taught as an evidence the C major scale as a foundation of music theory and the minor scale and his variants is not always treated fairly. Why is there a semitone between E & F and B & C and not elsewhere ?

There are several scales/modes outside of the major scale, with a varying number of notes, where the semitones are not placed between the 3rd and 4th note and between the 7th and 8th. The three minor scales (harmonic, ascending, descending) for instance, but also dorian , phrygian , you can read an encyclopedia article about them. up vote 9 down vote It has to do with harmony. Notes clash the least when their frequencies match up . For example, a note and its octave match up every two cycles, or a 2/1 ratio. Other ratios that sound good are 3/2, 4/3, 5/3, 5/4, 6/5, and 8/5; these are called the basic consonant intervals. Intervals that clash are the dissonant intervals. So why twelve notes? The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent — and contains more consonant intervals than dissonant intervals.

This page (from which I quoted) provides greater detail: up vote 5 down vote A fifth is the smallest non-octave consonant interval, with a frequency ratio of 3:2. If you start stacking pure fifths, the first result reasonably close to stacked octaves (2:1) is 12 fifths, which turns out to be 531441:4096 as opposed to 128:1 for 7 octaves. That's as close as you can get for a reasonable number of notes per octave. So if you are looking for a tonality built from stacked octaves and almost perfect fifths, a twelve-tone division will be pretty much what you'll arrive at. This also happens to serve a few other intervals (major and minor thirds, for example), but worse so than fifths. "mean tone temperament" tries getting a number of major thirds pure at the cost of making several other intervals as well as some thirds sound worse, and "well-tempered tuning" gets several pure fifths and some nice thirds in exchange for some more distasteful fifths.

So over the millennia, tuning has changed its focus from pure thirds to pure fifths and finally settled on making only the octaves pure and building the rest of the scale around an equally-tempered fifth, resulting in 12 equal-tempered semitones. that was a very good explanation. thank you. i am still interested in splitting the octaves up into various numbers of semitones and playing with the results. It makes me wonder if the 12-semitone octave sounded good before the advent of "music as we know it" or if it is something of an acquired taste, in which case alternative breakdowns of the octave could be adapted to, like in the case of western vs indian vs east asian music. – sova Aug 24 '15 at 0:23 up vote 2 down vote When two notes are played together, they sound pleasing only if their wave curves come together every few cycles. We call them harmonic sounding. If the wave curves never come together, or don't do so within a few cycles, they sound discordant. Wave curves will only come together if the two frequencies are multiples of each other.

For example, if one frequency is 200 cycles per second and the other is 600 cycles per second, their sound curves will coincide exactly 3 times every second, and they will sound harmonic. By dividing each octave into 12 intervals, you maximize the number of pleasingly sounding pairs of notes. That is because the number 12 is divisible by more small numbers than any other number less than 60. It is divisible by 1,2,3,4,and 6. The number 60 would allow more pleasing combinations (1,2,3,4,and 5), but it would be ridiculous to divide an octave into 60 intervals. So in modern western music they use 12 intervals. That provides the maximum number of pleasingly sounding combinations to create harmony. 2 I don't see why the divisors are important here. Because for example the equal tempered tritone has a frequency ratio 2^(6/12) which is one of the worst approximations (compared to just intonation) in the scale whereas the perfect fourth (2^(5/12)) is one of the best (see the link in Matthew's answer).

Another little comment: If one frequency is 200Hz and another is 600Hz then, assuming they're synchronized, they will be in the same phase 200 times every second, i.e. every 3rd cycle of the faster one. – nonpop Jun 20 '13 at 8:23 The frequencies don't need to be multiples of each other; they need to share a small common mutiple. See my answer here . – Matthew Read♦ Dec 9 '13 at 5:28 60 semitones per octave! that is an excellent experiment to try :D – sova Aug 24 '15 at 0:24 1 @nonpop is right. If we divide the octave into n equal intervals, it is not important for n to have many factors. 16et has no usable approximation to a perfect fifth. 30et has no intervals better than those of 15et, whose best fifth is 18 cents wide (12et's is 2 cents narrow).

On the other hand, some equal temperaments with excellent intervals have prime n, for example 19et, 31et and 53et. – Rosie F Nov 13 '16 at 12:53 up vote -1 down vote Great answer by @john Baldwin above. Jut wanted to add that these minimum divisions are also the most practical to use. Taking the case of singing for instance between one note say C and its higher Octave C, 7 intervals produce the most distinct sound, plus 5 sharps and flats = 12. And then if we start dividing it further it slowly starts getting very fine sub harmonies for the human hearing to discern. And these 12 divisions then also repeat in the higher and lower octaves and so on. The easiest to identify is 4 divisions which is a divisor of 12, which makes up a pentatonic scale with the higher note, and is why is easily enjoyable. This doesn't make a lot of sense to me. What do you mean by "distinct"? I would think that consonant intervals are less distinct that dissonant ones, for example, and the twelve tone scale is designed around consonant intervals.

Sharps and flats aren't something you can disclude when counting intervals either, unless you're working within a particular key or harmonic theory or seomthing (and you haven't specified one). Finally, how can 7 intervals produce "the most distinct sound" if 4 (or rather 5) intervals are "the easiest to identify"? – Matthew Read♦ Dec 8 '13 at 4:53 The reason is THE BRAIN. THe brain likes frequencies that are simple proportions. It thinks they go together. You should really ask, first, why are there octaves? Well the octave represents a doubling / halving of hertz (cycles per second). So, midi middle C is 256 hz, and if you know your computer numbers, you'll realise that the next octave C's are at 512, 1024, 2048, etc and the lower octaves are at 128, 64, and (pimp your ride) 32. Earthquakes, by the way, show up at around 11 hertz. Every society starts with the octave. 'Cos 1/2. Got it?

(I propose that the 2nd Viennese school abandon the octave by the way, and also tuning the instruments. Niether make any sense for them. The current state of things with octaves and tuning and suchlike is pure hypocrisy. Let it go, boys! Also scores. And playing in public. Nobody comes anyway.) Hh HHm... How to divide the octave? If we start it on C and divide it into 3 (which is a nice brain-friendly proportion) we will get a lovely 3 note scale: C, E, G#, C How about dividing it in to four: C, Eb, F#, A, C "THat's nice", says the brain, "but it's too SYMMETRICAL. Both of these scales just seem to go on forever and ever, I can't tell what's what. I know! Why don't you mix and match the proportions so they are slightly more uneven? Then I can figure out the bass note.".

And thus was born the "Proto Major Thingy": C, E, G, C and the "Proto Minor Thingy": C, Eb, G, C "Hang on a bit", says the brain, "you missed a note, didn't you?". "Where?" "Between G and C, I'm pretty sure you had something between G and C". C, E, G, A, C ? "Thas NICE! Rock and Rollish. Go on then, what about the other one?" C, Eb, G, Bb, C ? "Hey, what's with the Bb? We never heard that before. What kind of proportion is that?" "It's 10/12ths". "You mean 5/6ths. Alright. Play it again". C, Eb, G, Bb, C "Kay, that's bluesy. All right! But its 70,000 years ago and there's loads of poor bastards buggerin' round the the scenery getting crunched and munched by sabre tooth tigers and suchlike. Lotta funerals. Mucho sadness. Like Trump nowadays, you should know! Need variety.

" "Permutations?" C, D, E, G, A, C C, D, E, G, Bb, C C, Eb, F, G, Bb, C C, Eb, F, G, A, C "Whats the F proportion?" "4/3" "Great! I like it. 5 notes. Let's give it fancy Greek name. Tart it up a bit. Penta...?" "Tonic?". "That's wonderful". "I was kidding. You know, too literal..." "Nevermind. It's awesome. we'll go with Pentatonic. More! We NEED MORE! Now there's chieftains, mud huts, jewelry" "I need some rules". "kay. Er.. keep the Minor third or the Major third and the Fifth where it is, and just move the others about...

I know, like this: move the seventh up, the sixth down, the fourth up, and the second down!" C, D, E, G, A, C C, D, E, G, Ab, C C, D, E, G, Bb, C C, D, E, G, B, C C, Eb, F, G, Bb, C C, Eb, F#, G, Bb, C C, Eb, F, G, A, C C, Eb, F#, G, A, C C, Db, E, G, A, C C, Db, E, G, Ab, C C, Db, E, G, Bb, C C, Db, E, G, B, C "Hey, then if we superimpose them all we''ll get 12 subdivisions of the octave! Brilliant!" C, Db, D, Eb, E, F, F#, G, Ab, A, Bb, B, C "That's why I'm called the BRAIN, son.

Oh, and you're welcome." Jazclass - Jazz Theory 1 : TONES and SEMITONES An interval is the distance (difference in pitch) between two notes. When the notes occur one after another the interval is melodic. When the notes occur simultaneously the interval is harmonic. The smallest interval in use in the Western musical tradition is the semitone (also called half step). It is the interval between any note and its nearest neighbour on the keyboard. Two semitones together form one tone (or whole tone or whole step). From C to C# (or Db) is a semitone. From C# to D is also a semitone. Therefore from C to D is 2 semitones or 1 whole tone. Going up, the black notes on the keyboard are usually written as sharps. C - C# - D - D# - E - F - F# - G - G# - A - A# - B - C Going down, the black notes are written as flats. C - B - Bb - A - Ab - G - Gb - F - E - Eb - D - Db - C This keeps the accidentals required (flats, sharps, naturals) to a minimum.

Ten of the semitone intervals within the octave are between a white note and a black note on the keyboard . But two of the semitones are from one white note to another white note. They are the intervals from E to F, and from B to C. G# A# C D E F# G# A# whole-tone scale A# C D E F# G# A# The whole-tone scales gained prominence in the beginning of this Century through modern composers such as Claude Debussy and Olivier Messiaen. Jazz improvisers were quick to take up this new sound for improvisation over augmented triads and augmented 7th chords (+7 or 7aug). JT 1.4 - 8-note Symmetric scales Other symmetric scales can be formed by alternating note spacings of tones and semitones. There are two possibilities. 8-note Dominant scale (semitone - tone - semitone - tone -) This scale starts with a semitone between the tonic and 2nd note, then a tone between the 2nd and 3rd notes, again a semitone between the 3rd and 4th note, and so on.

JT 1.6 - Ear tests 1 - 3 Ear test 1 - tones and semitones. First listen to the Audio Demo . It plays : first a semitone interval three times, then a tone interval, also three times, finally the beginning of the Harry Lime motif, which consists of semitones only. (The Harry Lime motif is from the legendary 1950s movie thriller The Third Man, starring Orson Welles.) Ear test 1 - 10 intervals. Tone and semitone intervals only. Each interval is played three times. First twice as two separate notes (the lower note first), then as two tones together. Ear test 2 - symmetric scales (3-notes segments). Each ear test plays a 3-note segment, forming interval combinations of tones and semitones. These combinations are : Ear test 2 - 12 three-note segments. Each 3-note segment is played twice, in ascending order only. Ear test 3 - symmetric scales (5-notes segments). The ear test plays segments of the four types of symmetric scales.

- s - s - s - s - = chromatic scale segment - t - t - t - t - = whole-tone scale segment - s - t - s - t - = 8-note dominant segment - t - s - t - s - = 8-note diminished segment Each scale segment is 5 notes long (otherwise you could recognise the scale just by counting the number of notes !). Ear test 3 - 8 five-note scale segments. Each 5-note scale segment is played twice, in ascending order only. Complete scales First Music Lesson - Frequency, Octave, Semitone, Sound Names, Major Scale First Music Lesson Next Lesson >> Frequency We can describe sounds in terms of frequency. You can click on the buttons below to hear the sounds of piano. When you hear the first sound the membrane in your speaker makes 440 vibrations per second, and when you hear the second sound it makes 220 vibrations per second. The frequency is measured in Hz. (1Hz = one vibration per second) Octave and Semitone We can measure the distance between the frequencies of two sounds by deviding the frequency of one by the frequency of the other. Such a distance is called interval.

An octave is the interval between two sounds one of which has twice the frequency of the other. We devide octave into 12 semitones. In our terms, one semitone = 1.05946... (basically the 12th root of 2). Names of sounds Apronus convention The sound with frequency 440Hz is called here A1. The sound which is an octave lower is called A and has the frequency of 220Hz. We define the other names below. 130.8 Hz 246.9 Hz 493.9 Hz Note that 130.8Hz x 1.05946 = 138.6Hz which means that the interval between C and C# is one semitone. It is also one semitone between C# and D, D and Eb, Eb and E, and so on. The frequency of C1 is 261.6Hz. Note that 130.8Hz x (1.05946)^12 = 130.8Hz x 2 = 261.6Hz.

The interval between C and C1 is one octave and the interval between B and C1 is one semitone. You can continue the above table. For example you can have lower C=65.4Hz and C2=523.2Hz and so on and so forth. Note that in our Apronus didactic notation A(-1) = 110Hz, A = 220Hz, A1 = 440Hz, A2 = 880Hz, etc. You must have in mind that there are some other conventions. For example in so called scientific notation A0 = 55Hz, A1 = 110Hz, A3 = 220Hz, A4 = 440Hz, etc. There is also Helmholtz system where A = 110Hz, a = 220Hz, a' = 440Hz, a'' = 880Hz. Scale When you click on the button below you will hear the diatonic scale C major. You can play it manually by pressing the buttons below from left to right.

Note that in the C major scale there are only "pure" capitals without # and b. Now look at the C major scale in terms of semitones. sounds Tuning and Ratio : nrich.maths.org A new problem posed by Lyndon Baker who has devised many NRICH problems over the years. Tuning and Ratio Stage: 5 Challenge Level: You don't have to know any music theory to do this question as all you need to know is explained here. You only need to know about logarithms. The actual problems to solve are given in bold italic text. Diagram showing part of a piano keyboard. Wherever you are on the keyboard, the interval between one note and the next with the same letter name is always an octave. In the same way, the interval between C and the G above it, or D and the A above it, or anything similar (seven steps on the keyboard, including both black and white notes), is always a fifth. Each of these intervals, the octave, the fifth, and others, corresponds to a particular ratio of string lengths which produce the notes. The octave corresponds to 2/1, halving the length of the string.

The fifth corresponds to 3/2, taking two thirds of the length of the string. The interval from C to F, called a fourth, has the ratio 4/3.'Fourth' and 'fifth' etc. are musical terms and do not refer to the fractions 1/4 and 1/5. The interval from F to G, between the fourth and the fifth, has the ratio ${3/2 \over 4/3} = 9/8$. This interval is called a tone and it is also the the interval from C to D. Notes tuned in these ratios produce pleasant harmonies but compromises have to be made in tuning the other notes because the ratios do not relate to each other exactly. This problem is about discovering what the compromises might be. The table below shows some musical intervals and the corresponding ratios of the notes with respect to the note C. C octave The Pythagorean Scale You might expect there to be three tones in the interval from C to F, changing the ratio by 9/8 then 9/8 again and then 9/8 a third time, that is ${(9/8)}^3$. How does this compare with the ratio 4/3?

Pythagorean tuning kept the ideal ratios for the octaves, fifths and fourths and tuned the note E with a ratio ${(9/8)}^2$ and the notes A and B with the ratios $(3/2)\times (9/8)$ and $(3/2)\times {(9/8)}^2$. This system preserves the ratios of the fourth and fifth but produces a major third, from C to E and a major sixth, from C to A, which sound unpleasant. The Greeks were not interested in these intervals, but when composers in the middle ages and Renaissance wanted to use them, a new system of tuning was needed. Just intonation (shown in the table below) has much better major thirds and major sixths, of 5/4 and 5/3. It tunes the B so that the interval from E to B is a perfect fifth and the interval from G to B is a perfect third. But it does this by having two different sizes of tone: the intervals C-D and D-E are different here, but they were the same in the Pythagorean scale. Find the ratio corresponding to B. Also compare the D-A ratio with the ideal fifth (3/2).

C octave Just Intonation You might expect there to be six tones and twelve semitones in an octave (because there are 12 notes) but again that is not exactly so. Find exactly how many tones there are in an octave by finding what power of $9\over 8$ gives 2. Find also exactly how many major thirds there are in an octave. In the equal tempered scale, the standard tuning nowadays, these mismatches between different ratios are removed by defining the intervals differently. The tone is made a little smaller, so that there are exactly six of them in an octave. The semitone (half a tone) is a twelfth of an octave: so its ratio is ${(2)}^{1/12}$. If, for example, the note C is tuned with a string length of 32 units then the C# is tuned with a string of length ${32\over {(2)}^{1/12}}=30.2$ (to 3 significant figures). This applet by Benjamin Wardhaugh demonstrates the relationship between the string lengths and the notes.

The ratio ${(2)}^{1/12}$ is used to build up the other intervals, so that each interval is a whole number of semitones, and the ratio between its frequency and the frequency of the lowest note in the scale is given by a power of ${(2)}^{1/12}$. For example the fifth is ${(2)}^{7/12}$. Instrument tuners customarily use a logarithmic unit of measure, the cent, where 1200 cents are equal to one octave, a frequency ratio of 2/1, so that a cent is a 1200th root of 2. The table below shows the Equal tempered, Pythagorean and Just systems of tuning given in cents showing how many cents the note lies above the starting C. Fill in the table. Semitone and Hz - Whats the correlation? | Cakewalk Forums Semitone and Hz - Whats the correlation? Author Max Output Level: -64 dBFS Total Posts : 1332 Semitone and Hz - Whats the correlation? Hi, Has anyone got the correlation between Hz and semitones?

I need some info on what a semitone's range is over frequencies Max Output Level: 0 dBFS Total Posts : 11998 RE: Semitone and Hz - Whats the correlation? 2008/01/11 02:28:59 ( permalink ) Hi Steven- post edited by Susan G - 2008/01/11 02:47:28 2.30 gigahertz Intel Core i7-3610QM; 16 GB RAM Windows 10 x64; NI Komplete Audio 6. SONAR Platinum (Lexington) x64 Max Output Level: 0 dBFS Total Posts : 10031 Max Output Level: -39 dBFS Total Posts : 3631 Status: offline RE: Semitone and Hz - Whats the correlation? 2008/01/11 02:41:15 ( permalink ) There is no linear correlation. It depends on what octave you mean. (i.e.

Frequency [in hertz] is a linear measurement system, tones/semitones/cents are a non-linear measurement system.) The above sites help you to convert between the two. Think of it this way: Every time you double the frequency, you raise the pitch by an octave (12 semitones). So from A-440Hz to A-880Hz is 12 semitones. But then from A-880Hz to A-1760Hz is also 12 semitones. In one case you have a change of 440Hz spread over 12 semitones. In the other you have a change of 880Hz spread over 12 semitones. So as you go up the pitch scale, each semitone consists of a slightly larger change in frequency than the semitone below it. That is why the pitch system is non-linear. There is no set number of (change in) Hz per semitone. Mike Max Output Level: -64 dBFS Total Posts : 1332 RE: Semitone and Hz - Whats the correlation?

2008/01/11 02:42:11 ( permalink ) WOW Cool n the gang! Thanks guys! Max Output Level: -17 dBFS Total Posts : 5817 Status: offline RE: Semitone and Hz - Whats the correlation? 2008/01/11 02:49:09 ( permalink ) Celebrate Good Times, Come ON!! (Its a Celebration....(this low part is dedicated to Jose...no idea why, i just picture him with such a bassy/baritone/bellowing voice) No offense, Jose....y, por favor, regalanos con la identidad del "hombre blanco" que nos mira con los ojos tan mediocerrados y llenos de sangre.... Max Output Level: 0 dBFS Total Posts : 10031 RE: Semitone and Hz - Whats the correlation? 2008/01/11 10:57:30 ( permalink ) ORIGINAL: Infinite5ths There is no linear correlation. It depends on what octave you mean. (i.e.

Frequency [in hertz] is a linear measurement system, tones/semitones/cents are a non-linear measurement system.) The above sites help you to convert between the two. Think of it this way: Every time you double the frequency, you raise the pitch by an octave (12 semitones). So from A-440Hz to A-880Hz is 12 semitones. But then from A-880Hz to A-1760Hz is also 12 semitones. In one case you have a change of 440Hz spread over 12 semitones. In the other you have a change of 880Hz spread over 12 semitones. So as you go up the pitch scale, each semitone consists of a slightly larger change in frequency than the semitone below it. That is why the pitch system is non-linear. There is no set number of (change in) Hz per semitone. To add to this, the non-linearity that Mike speaks of actually comes from the fact that we use the 12-Tone Equal Temperament Tuning System.

This system is full of compromises (with the exception of the octave interval) that were purposely created in order to enable some instruments to play in all 12 keys without retuning. These compromises include having a 5th that's a little smaller than a pure 5th and a 3rd that's a little bigger than a pure 3rd, etc. The reason for the emergence of this system was because not all instruments have the capacity of adjusting their pitch on the spot. Instruments like the voice or the violin can play pure intervals without the need to retune. On the other hand, instruments like the piano obviously can't adjust tunning on the spot so they end up sounding good on some keys and out of tune in others. But the 12-Tone Equal Temperament System solved this problems, and was actually very well recieved by most (although some musicians didn't like the sound of impure perfect intervals). Take care! Max Output Level: 0 dBFS Total Posts : 10031 RE: Semitone and Hz - Whats the correlation?

2008/01/11 11:05:16 ( permalink ) ORIGINAL: jinga8 Celebrate Good Times, Come ON!! (Its a Celebration....(this low part is dedicated to Jose...no idea why, i just picture him with such a bassy/baritone/bellowing voice) No offense, Jose....y, por favor, regalanos con la identidad del "hombre blanco" que nos mira con los ojos tan mediocerrados y llenos de sangre.... ( Hahaha Thanks Jinga8, for dedicating a post to me. Although I have to correct you in that I'm actually considered a Tenor (at least when I used to sing in the college choir). Hopefully, sometime this year, I will build a website which I'll link here. I just wanna get some songs done before that though. Take care! RE: Semitone and Hz - Whats the correlation? 2008/01/11 13:21:56 ( permalink ) ORIGINAL: s_barber I believe the correlation is the 12th root of 2.

So...if you take the square root of 2 and the root of the result, repeating 12 times that end result is the interval between semi-tones. Supposedly you can multiply that number times the HZ of say A 440 the result will be B flat. I think the number you get has a gazillion places though. I guess science does have it's place in music. I've never tried it but that's the theory as I understand it. You recall correctly. The equal temperament scale is based on the twelfth root of two which is roughly 1.059463 â€¦ (2 raised to the 1/12 power). Assuming that A natural (in the U.S.) is 440 Hz, then B-flat (a single chromatic step) is 440 * (1.059463 raised to the first power) which comes out to 631.8883ish Hz.

Continuing on, the pitch for B natural (two chromatic steps above our reference A 440) is calculated as 440 * (1.059463 raised to the second power) , or 493.883ish Hz. Each pitch is calculated by changing the exponent used with the 12th root of 2. Eventually, you come to the octave A 880, which is 12 chromatic steps from A 440. The calculation is 440 * (1.059463 raised to the 12th power) which comes out to 879.999ish Hz (not quite 880, but the error here is due to rounding the 12th root of two to 7 decimal places. post edited by feedback50 - 2008/01/11 13:36:52 RE: Semitone and Hz - Whats the correlation?

2008/01/11 14:02:17 ( permalink ) ORIGINAL: feedback50 ORIGINAL: s_barber I believe the correlation is the 12th root of 2. So...if you take the square root of 2 and the root of the result, repeating 12 times that end result is the interval between semi-tones. Supposedly you can multiply that number times the HZ of say A 440 the result will be B flat. I think the number you get has a gazillion places though. I guess science does have it's place in music. I've never tried it but that's the theory as I understand it. You recall correctly. The equal temperament scale is based on the twelfth root of two which is roughly 1.059463 â€¦ (2 raised to the 1/12 power). Assuming that A natural (in the U.S.) is 440 Hz, then B-flat (a single chromatic step) is 440 * (1.059463 raised to the first power) which comes out to 631.8883ish Hz.

Continuing on, the pitch for B natural (two chromatic steps above our reference A 440) is calculated as 440 * (1.059463 raised to the second power) , or 493.883ish Hz. Each pitch is calculated by changing the exponent used with the 12th root of 2. Eventually, you come to the octave A 880, which is 12 chromatic steps from A 440. The calculation is 440 * (1.059463 raised to the 12th power) which comes out to 879.999ish Hz (not quite 880, but the error here is due to rounding the 12th root of two to 7 decimal places. According to your calculations Bb is much higher in pitch then B natural.

Question: How many semitones are there in an octave?
Answer: 12