Music Theory/Scales and Intervals - Wikibooks, open books for an open world Music Theory/Scales and Intervals Jump to: navigation , search A musical scale is a set of notes, usually not arbitrary, of which most notes in a piece of music might be chosen. There exist many scales with highly distinctive sounds, though some are much more common than others. the term "scale" comes from the latin word 'scala' meaning 'ladder'. Thus a scale is a ladder of notes. Contents Simple intervals[ edit ] Before we progress, we must discuss intervals. An interval is usually defined as the distance between two pitches, that is, how many semitones lie between them. When the two pitches are the same, they are said to be in unison, and two notes played in unison can be impossible to distinguish from a single note when they are played by the same instrument and the instrument is properly tuned. When they are twelve semitones apart, they are an octave apart (we will learn why it is called an octave shortly). Simple intervals are defined as those intervals that are one octave or less apart.

Intervals are usually named according to the relationship of the higher note to the lower note in the major scale, though they also have alternative names depending upon the spelling of the particular notes on the page of music. Semitones perfect octave augmented seventh This table gives the most common nomenclature for each interval according to its relation to the major scale. For example, the interval of four semitones occurs as the third note of the major scale, and thus it is called a major third. The interval of seven semitones occurs as the fifth note of the major scale, and so it is called a perfect fifth. Whether an interval is "perfect" or "major" depends on mathematical ratios of frequencies as determined by the Greeks. Other possible names are given under "alternate names," and the most common of these are emboldened.

One may draw several inferences from this table: If any perfect interval is raised by one semitone, the interval becomes augmented If any perfect interval is lowered by one semitone, the interval becomes diminished If any major interval is raised by one semitone, the interval becomes augmented If any major interval is lowered by one semitone, the interval becomes minor If any major interval is lowered by two semitones, the interval becomes diminished Interval double octave octave The compound intervals work by following the same five rules as the simple intervals above (so the augmented eleventh might also be called a diminished twelfth!). Why even bother giving them separate names? The answer lies in their normal function within music. Complex jazz chords are built around stacks of thirds, and so the terms "ninth," "eleventh," and "thirteenth" are needed to designate intervals larger than a seventh. Mnemonic memorization examples[ edit ] The following chart intends to give some mnemonic support in recognising musical interval. For each interval, ascending or descending, a popular song is given that contains it prominently. Capitalized syllables or a ">" mark the stated interval: Trainear is an online ear trainer that's specifically for associating intervals to songs.

Here are some examples for each interval: Interval Traditional Scales[ edit ] The major scale[ edit ] The major scale is a diatonic scale. Originally a Church mode named by Heinrich Glarean in 1547 as the Ionian scale/mode. There is some confusion with beginners as to why the Church modes are named after Greek tribes. A common belief is that the Ionian scale was invented by the Greeks. The Ancient Greeks did not have an Ionian scale but they did lay the foundation for its existence. The Ancient Greeks did have a system of music and we know quite a lot about their music theories from the Ancient Greeks themselves. The real origin of the Ionian mode is the period known as the Renaissance which started in Italy with the 13th century writer Petrach. Renaissance is the French word for "rebirth". The "rebirth" in this case was the rediscovery of lost Ancient Greek texts that had been brought back from the East (including Jerusalem) by Crusaders and other parties. A lot of these books had been unavailable in Western Europe for centuries. These books sparked a desire to rediscover the roots of European civilization.

The Renaissance scholars took the music practices of their time which already had been in use for hundreds of years and codified them in reference to the Greeks. You can read about the Ancient Greek system of tetrachords and scales if you're interested. It must be pointed out that we have no idea about the sounds of the instruments of the Ancient Greeks and only a few fragments of music exists. It is now believed that the Church modes are derived from early Christian mass chants which themselves were adapted from Jewish chants. It may be that modern Western Music has for its origins Eastern Jewish liturgical chants and Ancient Greek theories codified from the view of a Renaissance mindset a thousand years later. The major scale (Ionian mode) is most simply described as the eight note progression consisting of the perfect and major semitiones, i.e., perfect unison, major 2nd, major 3rd, perfect 4th, perfect 5th, major 6th, major 7th, and perfect octave in that order. You have already seen the major scale: C D E F G A B; do re mi fa sol la ti; 1 2 3 4 5 6 7. Scales may be constructed according to their intervals.

You can see that the C major scale consists of two whole tones, then a semitone (moving from E to F), then three more whole tones, then again a semitone (moving from B back to C). If we add the implied C at the end of the scale, we would have eight notes: C D E F G A B C. The minor scale[ edit ] The minor scale, the Aeolian mode , is also a diatonic scale. The C minor scale is C D E F G A 3 4 5 6 7. You can see that it consists of one whole tone, then a semitone (moving from D to E ), then two more whole tones, then again a semitone (moving from G to A ), and a final whole tone. If we add the implied C at the end of the scale, we would have eight notes: C D E F G A B C. The intervals of the natural minor scale follow the following pattern: tone, semitone, tone, tone, semitone, tone, tone. The following chart demonstrates this natural minor scale construction.

The minor scale is the sixth mode of the major scale; that is, the minor scale starts on the 6th note of the relative major scale. In the case of the C minor scale, the relative major scale is E major. We can illustrate this with two octaves of the Eb major scale, highlighting the C minor scale. E F G A . You will learn more about modes later. Pentatonic and Blues Scales[ edit ] The pentatonic scales[ edit ] A pentatonic scale has five notes. Each note in the major pentatonic scale is a fifth (seven semitones) relative to another note. For example, the C major pentatonic scale starts with C, then from there we can get G, then D, then A, then E. Rearranging the scale to ascending order from C, we get: C D E G A. This is the C major scale with F and B removed! So, why use it? Sometimes less is more, and pentatonic scales are certainly easier to use when improvising.

The major pentatonic is the same as the major scale with the 4 and 7 notes removed, while the minor pentatonic has the 2 and 6 notes removed, that is, the minor pentatonic is relative to the major pentatonic. So, to use our earlier example contrasting the E major pentatonic with the C minor pentatonic: E F G B C E . Pentatonic scales are abundant in rock and blues music, though these are certainly not their only uses. Traditional Chinese and Japanese music has defined and named many more pentatonic scales, some of which do not use the western twelve-note basis. The blues scale[ edit ] The most common blues scale has six notes, and may be considered a minor Pentatonic scale with the diminished fifth added as a blue note . In a major blues tune, the minor third is also considered a blue note. Therefore, the C blues scale is: C E F G G B . Sometimes the raised seventh degree (B) is added to this scale but most often used as a passing note, much like the diminished fifth. The blues scale is most commonly used in jazz improvisation to create a "bluesy" flavor.

The Symmetric Scales[ edit ] Symmetric scales include scales such as the whole-tone scale, octatonic scale (also called the diminished scale), and chromatic scale, and their defining characteristic is that they are composed of repeating subunits within an octave. This property allows these scales to be transposed to another pitch (or "key"), yet retain exactly the same notes as the original scale. The chromatic scale[ edit ] The simplest of the symmetric scales, the chromatic scale, is composed of repeating semitones (half-steps). Thus, the chromatic scale built on C contains the notes C,D ,D,E ,G,A ,A,B ,B, and C. Notice that these are exactly the same notes as the chromatic scale built on C. In fact, a chromatic scale built on any note of the twelve-tone western music scale will share these notes, a property which warrants the inclusion of this scale among the symmetrics. Usually chromatic scales are spelled with sharps when ascending and flats when descending. As noted above, composers will often choose certain notes from this scale to use more frequently than others, thereby providing the listener with a sense of a "home" note, referred to as the tonic.

However, many composers in the twentieth century have demonstrated that using all twelve chromatic notes equally can defeat any sense of tonal center. This technique is called atonality or, less commonly, pantonality, and can have a very unsettling effect upon those unaccustomed to this music. An everyday occurrence of atonal music would be in the soundtracks to many horror films, documentaries, or other movies where there is a need for extreme dissonance and tension to match the onscreen action. The whole-tone scale[ edit ] The whole-tone scale is made of repeating whole tones (whole-steps). Therefore, a whole-tone scale built upon D would contain D ,E ,F,G,A, and B. Like the chromatic scale, these pitches are the same pitches that one would find in a whole-tone scale built upon E , or any of the pitches in this particular scale. For instance, a whole-tone scale built upon F would be F,G,A,B,D ,E , and a whole-tone scale built on B would be B,C ,D ,F,G,A. These two are really the same scale, since C =D .

For this reason, there exist only two possible whole-tone scales: the scale including the pitches C,D,E,F ,G the scale including the pitches D ,E ,F,G,A, and B. Any whole-tone scale within the western musical system will fall enharmonically into one of these two categories. The whole-tone scale was used widely by impressionists to create a floating, ethereal sound. The scale also finds a place in jazz improvisation, as it is among the most colorful scales to use where a raised-fifth scale degree is indicated. Incidentally, the scale contains all of the notes of two augmented chords placed side-by-side, a whole step apart. The octatonic (diminished) scale[ edit ] The octatonic, or diminished, scale is among the simplest scales possible, yet has been used to tremendous effect in nearly every genre. This eight-note scale may be conceived in two manners, but both of the approaches use a repeating subunit of alternating whole-steps and half-steps. The first manner, most often used by classical composers and termed diminished, encourages beginning with a whole-step, while the second, used frequently by jazz players and composers who call it octatonic, encourages starting with the half-step.

Beginning from C (using the second method), the octatonic scale would include the notes C,D ,E ,E,F ,G,A, and B . As with the other symmetric scales, this scale may be moved to a different starting note yet retain the same pitches as the original. Thus, E ,E,F ,G,A,B ,C,D is an octatonic scale (second method) that shares all eight pitches with the octatonic scale starting on C. There are, then, three different octatonic scales possible: C,D ,A ,A,B, and C Any other octatonic scales within the western system will fall enharmonically into one of these three groups. The use of the octatonic scale in western music can be seen as early as Bach, who used pieces of the scale within his counterpoint to imply diminished harmony. Modern composers of the classical canon use this scale as a colorful alternative to redundant diatonicism or austere chromaticism. Jazz improvisers often turn to the diminished scale to improvise over a dominant seventh harmony to imply the flat-ninth degree of a chord.

The octatonic/diminished scale is extremely versatile: a single octatonic scale (C,D ,E Understanding and Building Musical Scales: Definitions & Types of Scales - Video & Lesson Transcript | Study.com Understanding and Building Musical Scales: Definitions & Types of Scales Watch short & fun videos Start Your Free Trial Today An error occurred trying to load this video. Try refreshing the page, or contact customer support. You must create an account to continue watching Register for a free trial Are you a student or a teacher? I am a student Start Your Free Trial To Continue Watching As a member, you'll also get unlimited access to over lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed. Coming up next: Musical Timbre of Instruments and Singers: Definition You're on a roll. Keep up the good work! Your next lesson will play in 10 seconds 1:52 Half Steps and Whole Steps 3:25 Building a Scale Add to Add to Add to Want to watch this again later? Log in or sign up to add this lesson to a Custom Course. Custom Courses are courses that you create from Study.com lessons.

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Create an account to start this course today Try it free for 5 days! Lesson Transcript Instructor: Liz Diamond-Manlusoc Liz has taught music for K-12 and beyond. She holds a master's degree in Education Media and Design Technology. What is a 'scale' in music? How do tonality, pitch, and accidentals play a role in making a scale? What kinds of notes make up a scale? What is the difference between a Major scale, minor scale, blues scale, and pentatonic scale? Learn all about scales in this lesson! Scales: Definition You now know some of the building blocks of music, like pitch is how high or low a note sounds and is referred to with a letter note. And, you know that accidentals are sharps and flats, sharp being a raised pitch and flat being a lowered pitch. You also know that tonality is the character of the tones and harmonies heard in a piece of music, where Major is happy and cheerful and minor is serious and dramatic. But, tonality, pitch, and accidentals are not generally independent of one another, and they are most often found working together in the context of a scale.

For centuries, scales have been used to arrange and give context to pitches. A scale is simply a set of pitches. They are usually arranged according to pitch in ascending or descending order. Pretty much all of the music you listen to is based on scales, and the song 'Do-Re-Mi' from The Sound of Music is literally based on singing a scale. The different scales provide different tonalities but rely on the same basic pitches. It's the arrangement of these pitches that determines the scale's tonality and function. You can think of it like a racing game. Each character gets from point A to point B, but their cars and driving style change the feel of the game. So, how are scales made? How does the musician know what pitches are in a scale? Half Steps and Whole Steps Most scales are based on the intervals between pitches. You can think of this like the distance you travel on the Monopoly board depending on the number that you roll. In Western music, the smallest interval is called a half step. Half steps are pitches that are adjacent to one another, like C and Db or E and F, where there is no pitch in between.

On a guitar, a half step can be found easily because the neck is broken up into half steps. So, when you're shredding the green and red buttons in Guitar Hero, you're technically shredding half steps! The other common interval is a whole step. A whole step is really just two half steps put together. An example would be C and D or A and B. Pitches that are a whole step apart tend to have the same accidentals, like C# and D# or F natural and G natural. Because whole steps are just two half steps put together, whole steps on the guitar are one fret apart. On the Guitar Hero guitar, this would be the green button to the yellow button. Building a Scale When we start linking whole steps and half steps together in sequence, we get different types of scales. This can become somewhat confusing or even annoying, but it's really just like learning how to play a new game. Like our racing game characters, each scale has its own set of characteristics. In this case, the characteristics are a set pattern of whole steps and half steps. Each scale follows its own pattern of whole steps and half steps.

So, no matter what note you start on, you can always figure out a scale based on its pattern. The Major Scale The Major scale pattern is whole-whole-half-whole-whole-whole-half (W-W-H-W-W-W-H). It includes one of each letter of the musical alphabet. In Western music theory, scales tend to be measured or compared to the Major scale, so this is an important one to know. Let's examine how this pattern works. If you start on C and go a whole step, you land on D. From D to the next whole step is E. The next distance of a half step takes us from E to F, another whole step brings us to G, another to A, another to B, and finally a half step brings us to the original pitch C. This particular scale is called the C Major scale since the pattern starts on and revolves around the pitch C. If you wanted to build the F Major scale, you just start on F and apply the same whole and half step pattern.

This time, you have F moving a whole step to G, G moving a whole step to A, A going a half step to Bb, Bb to C, C to D, D to E, and E going a half step to F. Minor Scale Another scale you hear frequently is the minor scale. Since the minor scale has a different tonality than the Major scale, it has a different pattern. There are actually three types of minor scales, but we'll start with the most common, the natural minor scale. The natural minor scale pattern is W-H-W-W-H-W-W. So, if you found the A natural minor scale, you would have A-B-C-D-E-F-G-A. If you found the C natural minor scale, you would have C-D-Eb-F-G-Ab-Bb-C. Here, you can see that compared to the Major scale, the natural minor has a lowered 3rd, 6th, and 7th scale degree. The two other minor scales are similar to the natural minor, but have slight variations on the pattern. The harmonic minor scale pattern has a 1.5-step jump between the 6th and the 7th pitches.

So, the harmonic minor pattern is W-H-W-W-H-1.5-H. If we start this pattern with A again, we get A-B-C-D-E-F-G#-A. When compared with the natural minor scale, you can see that only the 7th note is different. In the harmonic minor scale, the 7th scale degree is raised. The third type of minor scale is the melodic minor scale. The melodic minor scale is an anomaly when it comes to scales because it has different ascending and descending pitches. The ascending pattern is W-H-W-W-W-W-H, where the 6th and 7th pitch are raised. The descending pattern follows the same as the natural minor scale, so overall, the scale sounds like this (please see the video at 07:52 to hear this scale). Diatonic and Non-Diatonic Scales Both the Major and minor scales are diatonic, in that they use each letter name once, A B C D E F G, with the repetition of the first pitch. The patterns for Major and minor scales - which are both diatonic - are virtually the same - it's just that they start in different places.

However, not all scales are diatonic. Some scales are non-diatonic in that they don't use each letter name or they use pitches which are not normally within the scale, like we saw in the harmonic minor scale with 1.5 steps. Non-diatonic scales have like a secret weapon to spice up the sound. Besides the harmonic minor scale, some common examples of non-diatonic scales are the chromatic scale and the blues scale. × Theory Basics: 12 Chromatic Tones Theory Basics: 12 Chromatic Tones How to Write Music Contents by Conrad Albrecht I wanted to explain a common set of chords called the diatonic chords. But to understand the diatonic chords, you first need to understand scales; and to understand scales, you first need to understand the 12 chromatic tones our music is based on. So this article will explain the basic theory of pitches in our music, including the 12 chromatic tones. The Chromatic Scale Each note in a piece of music has a pitch. The pitch is how "high" or "low" the note sounds. Between the lowest audible bass pitch and the highest audible treble pitch, there are thousands of possible pitches.

However, our music doesn't use all of these possible pitches. Pitches that are too close together sound "out of tune" to us, so our music doesn't use them and we don't have names for them. The pitches that we actually use and give names to are a series of pitches called the chromatic scale: The pitches in the chromatic scale are all the same pitch-distance apart. That particular pitch-distance is called a half-step (or, in Britain, semitone). So, the smallest pitch distance between any two notes in our music is the half-step. Between the highest and lowest pitches we can hear, there's room for about 100 half-steps. So, all of the pitches in our music come from this chromatic scale of about 100 different pitches. Pitch ladder doesn't play? The pitch ladder works in Internet Explorer only. The pitch ladder plays notes on your computer's built-in synthesizer. So, the synthesizer volume must be turned up in your Windows volume controls. The "chromatic ladder" on the left shows a section of the chromatic scale. Click some neighboring boxes in the ladder to hear what the half-step distance sounds like.

Letter Pitches G A We use letters to name some of the pitches. On the right is that chromatic ladder section again, showing the letter-named pitches. Notice: Only the letters A through G are used; then we start over with A again. This pattern repeats all the way up and down the chromatic scale. So there are actually about 8 different A's (ranging from a very low bass A all the way up to a very high treble A); about 8 different B's; and so on. Some of the letter pairs (B-C and E-F) are just one half-step apart; the other letter pairs are two half-steps (a whole-step) apart. Why do we use the same letter over and over for different pitches? Because to our ears, a higher C (for example) sounds like "the same note, but higher" as a lower C. Try playing two different C's (or D's, etc.) in the chromatic ladder to hear how they sound "the same, but different". The distance from one pitch to the next pitch with the same name (e.g. from an F to the next higher F) is called an octave.

If you count the half-steps in the chromatic ladder from one F to the next F, you can see that one octave equals 12 half-steps. Often, when we discuss the notes in a scale or chord, we don't care which octave the notes are in, because they have a similar effect regardless of their octave. So we can say, for example, "the C major chord contains the notes C E G" without worrying about which C, E, or G we mean. Sharps and Flats G# Ab A We still need names for the pitches in between the letter pitches, for example the pitch between G and A. These pitches don't get their own letter; to name them, we use a neighboring letter plus the sharp (#) or flat (b) symbols. The sharp (#) makes a pitch a half-step higher; the flat (b) makes it a half-step lower. For example, the pitch between G and A can be called either G# ("G sharp") or Ab ("A flat"). By either name, it sounds the same. Sometimes one of the names is preferred over the other for music theory reasons, but we won't worry about that yet.

On the left is our chromatic ladder section again, finally with all the pitch names filled in. Conclusion: 12 Chromatic Tones When we talk about the notes in a scale or chord, we usually don't care which octave they're in. And if we ignore the different octaves, there are only 12 chromatic tones: A 4. Pitches, Intervals, and Scales 4. Pitches, Intervals, and Scales This chapter is about how Western musical tradition treats pitch, and why. Since pitch is primarily heard (by most people) in terms of ratios of frequencies, it is natural to use a logarithmic scale to assign pitches (which are subjective) to (objective) frequencies. But one has to pick a scale, that is, a ratio that corresponds to one unit of interval. This ratio in the West is the twelfth root of two, approximately equal to 1.059. That particular number turns out to be such a good choice of interval to measure pitches by, that it came to rule over a millennium of Western art music.

Although we won't be concerned with all the historical details, this chapter will try to explain what's so special about the twelfth root of two as a unit of pitch. The punch line is that this particular logarithmic scale turns out to have a surprisingly high number of sweet-sounding intervals in it. To develop this idea we first have to figure out what makes some intervals sound sweeter than others; this is pretty well explained by what is known as the Helmholz theory of consonance and dissonance (section ). Then we will investigate the actual intervals that arise in the Western scale (section ). Finally we'll consider some of the consequences of the way pitch is organized in Western music and consider some alternative ways to organize pitches. It's a commonplace that some intervals sound sweet and some sound sour, like this: SOUND EXAMPLE 1 : a musical fifth (usually considered sweet sounding) SOUND EXAMPLE 2 : a tritone (sour by comparison). To call these sweet and sour is a rather clumsy metaphor. In musical language, we refer to a sweet-sounding interval as consonant and a sour-sounding on as dissonant--terms that can be taken to mean ``going together" and ``not going together".

(Even this more neutral-sounding terminology carries an implicit value judgment that should not be accepted unquestioningly.) It turns out that the two intervals above have a physical difference that correlates with people's judgment of consonance and dissonance (as they are measured by psychoacousticians in experiments), that fits into what we know today as the theory of consonance and dissonance. Although the theory of consonance and dissonance is usually associated with Hermann von Helmholz (1821 - 1894), many of its ideas and concepts date back further, even to ancient Greece; and the theory was much elaborated upon (and argued with) over the century since Helmholz published his contributions. The theory seems to have finally been brought to a definitive form in Plomp and Levelt's very readable and persuasive 1965 paper on the subject. In the theory, we consider two complex periodic tones, that is, tones that may be written as a sum of sinusoids with frequencies in the ratios 1:2:3:..; in other words, tones all of whose partials are tuned to multiples of a fundamental frequency.

Here is what happens when the two fundamental frequencies are chosen, for instance, as 100 and 150 Hz: (To make the picture easy to see, all the harmonics of the 100 Hz. tone are given the same power, and so are all the multiples of the 150-Hz. tone; but the theory doesn't rely on that fact. Also, the double peaks at 300 and 600 Hz. are in fact single sinusoids; they're drawn this way for clarity). Here, on the other hand, is the situation when the fundamental frequencies are 100 and 140: These pictures roughly correspond to the two sound examples above. The first one is consonant and the second one, dissonant.

The Helmholz theory explains the consonance of the first example and the dissonance of the second one, by the absence or presence of awkward pairs of sinusoids (in this example there are two: 280 and 300 Hz, and 400 and 420 Hz.) These pairs are far enough apart to be perceived separately but close enough to interfere with each other by vibrating in heavily overlapping regions of the cochlea (Section 3.4 ). Plomp and Levelt go so far as to posit the consonance and dissonance of two sinusoids as a function of their separation in critical bands, thus: Under this rule, the two pairs of sinusoids in the dissonant example above are almost as dissonant as possible (20 Hz. being close to 1/4 of a 100-Hz. critical band). The wider separations in the first example are about 1/2 of a critical band and contribute much less dissonance. It's unavoidable that multiples of two fundamental frequencies would give rise to close neighbors here and there.

The special reason the closely placed harmonics that occur in the first example didn't contribute to dissonance is that they landed right on top of each other. For this to happen, the ratio between the two pitches must be an integer ratio. For instance, for the third harmonic of one tone to coincide with the second harmonic of another, the fundamentals must be in a 2:3 ratio. Here are the definitions of some intervals given by integer ratios between one and two (that is, within an octave), arranged from the most consonant to the most dissonant. The names are what they are for music-theoretical reasons too abstruse to explain here: RATIO NAME 1:1 unison 2:1 octave 3:2 fifth 4:3 fourth 5:4 major third 5:3 major sixth 8:5 minor sixth 6:5 minor third 4.2 The Western Musical Scale In many situations it's a good, practical move to choose, out of the set of all possible musical pitches, a reasonably small set of pitches, called a scale, to which you would restrict yourself when writing music.

One reason for this might be that instruments, such as pianos or fretted guitars, are often designed to play a discrete set of pitches out of the whole continuum. (But if we consider that vocal music predates the development of keyboard and fretted instruments, this may cease to seem a compelling reason). Another consideration might be that you would want to be able to write music down. It would be impractical to write all the pitches as numerical frequencies, so in practice (in the West as well as elsewhere) musical traditions have settled on sets of pitches, typically between 5 and 21 in an octave, out of which a working musical context might use 5 to 7 at a time. For example, the Western scale has 12 pitches per octave, and one often chooses a musical key which implies a choice of 7 out of the 12. Now suppose we wanted to divide the octave into equal intervals to make up a musical scale. (Using equal sized intervals sounds like a good choice; it's like using a ruler whose marks are spaced regularly along it.

You could reasonably request, for instance, that the interval you hear when you play the first and third notes on the scale should be the same interval you get from the second to the forth, or the third to the fifth, and so on.) If we call the interval between two successive pitches in the scale , then the interval between the first and third is , and so on; the whole octave is a ratio of . Since we know an octave is a ratio of 2:1, we get or, solving for h, Would our scale (with equal steps per octave) have an interval that approximates a fifth (ratio of 3:2)? To answer this it's best to go back to computing things in octaves. One step is octave. How many octaves is the interval 3:2? From section 1.3 we get the interval in octaves is: Now for instance if we decided to put five steps in an octave the available intervals would be 0.2, 0.4, 0.6, or 0.8 octaves; we see that three steps gives us an approximate fifth with an error of octave.

If, instead, we divide the octave in six parts, the closest interval to the fifth we can find is octave for a much larger error, 0.081 octaves. How bad are those errors? If a critical band is taken as an 18% increment in frequency, it is then and the worst possible mis-tuning is then 1/4 of that (according to the Helmholz theory) or 0.06 octave. So our 6-tone scale, with its error of 0.081 octaves, has about the sourest fifth you could ask for. To gauge how well a scale is doing at hosting consonant intervals, we can study how its fifths and major thirds come out (the other intervals listed above can be formed from octaves, fifths, and major thirds.) Here is a table of the results.

The first row gives the number of steps we divide the octave into; the two other rows give the error, in thousandths of an octave, of the fifth and major third: division 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 fifth error 82 85 15 82 14 40 29 15 40 2 30 14 15 22 3 third error 11 72 78 11 36 53 11 22 42 11 14 35 11 9 28 division 18 19 20 21 22 23 24 fifth error 26 6 15 14 6 20 2 third error 11 6 22 11 4 18 11 The first column that gives even half-decent approximations for the fifth and the third has 12 steps per octave.

(Column 19 looks comparable and 22 even slightly better, but it would be hard to argue that the improvement is so great as to merit adding all those extra notes. Think what a piano would have looked like.) How good or bad is the 12-steps-per-octave scale at reproducing a major third and a fifth? Well, the approximations turn out to be four and seven half-steps: Starting with the better one (the fifth), we can now consider what happens when we play two pitches separated by seven steps on the 12-note-per-octave scale. Choosing 220 Hz. for the bottom pitch (A below middle C), we now see how closely the third harmonic of that tone meets up with the second harmonic of the tone 7 steps higher: So we are a mere 3/4 Hz off - the two partials will beat slightly less than once per second. Now for the major third, for which we compare the fifth harmonic of the base tone with the fourth harmonic of the tone four steps up: Whether this is a reasonable result (i.e., whether this interval should be regarded as consonant) is more a matter of taste than of measurable scientific fact.

Some people complain about it, and some instruments (brass and voice, for instance) have a tradition of slightly altering a pitch here and there to make thirds sound more consonant when they appear in the music. The other consonant intervals (minor third, fourth, major and minor sixths) may be built out of combinations of the octave, fifth, and major third. For example, to go up a fourth (4:3) ratio), we can go up an octave (2:1) and then down a fifth (2:3). This is approximated in the Western scale by going up 12 steps (the octave) and down 7 (the fifth), or in other words, by going up 5 steps. Similarly, the minor third (6:5) is up a fifth (3:2) and down a major third (4:5); this gives 7-4=3 steps.

Here is a summary of how the intervals fare in the Western scale (with the error now reported in steps--twelfths of an octave--to make them more readable): interval 4.4 The Just-Intoned Scale The Western chromatic scale is not without its discontents, who often complain about the poor accuracy of approximating major thirds as four twelfths of an octave. We can fix that if we are willing to relax the requirement that our scale have equal steps. (In fact, we could then do anything we wanted). Returning to the keyboard diagram above, we could simply assign frequencies to the diatonic scale so that all the marked intervals are exactly correct. An interval that is an exact ratio of integers, such as 3:2 or 5:4, is called a just interval, and the scale we then get is called a just-intoned scale. (The word intonation is music jargon for ``tuning".) To construct the just-intoned scale we figure out the frequency for each pitch as an interval from C. So for instance, the pitch A is a minor third below C, for a relative frequency of 5/6.

We then raise or lower it by octaves until it resides withing an octave above the original C; in this case we have to go up an octave, giving 5/3. Continuing in this way we get the just-intoned scale in C as shown: C D E F G A B 1 9/8 5/4 4/3 3/2 5/3 15/8 This is all excellent except for two things: first, there has to be a different scale starting at each key. (Even if you put in extra pitches for the accidentals, you can't get all the intervals the same and so transposing such a scale gives you a whole new set of pitches.) Second, certain of the intervals aren't what you'd wish for. In particular, the interval from D to A, ideally 3:2, is a thorny 40/27; and the interval from E to A, ideally 4:3, is instead 25/12.

If we were to move A and E over to fix those intervals, E would land at which is off by a factor of: This interval is called the syntonic comma, and it plays an amusing role in early, faltering attempts to find a musical scale in which all the intervals work. For instance, one approach was to nudge each of G, D, A, and E, successively, by 1/4 of the syntonic comma in order to get the E to be just - this was called mean-tone temperament. (In general, the process of slightly adjusting pitches up and down to compromise between incompatible interval constraints is called tempering. For this historical reason, the modern scale, whose intervals are all equal, is often said to have equal temperament.) Although it's easiest to relax and let the modern Western scale rule over your music, the investigation of alternative pitch scales has been, and continues to be, an exceedingly fruitful avenue for composers including Harry Partch, Alvin Lucier, Charles Dodge, John Chowning, and Rand Steiger (and certainly many others), who have found highly individual ways of organizing sets of pitches.

Sound and Music : Chromatic Scale Notes : How Music Works Chromatic Scale Notes Sound and Music Create Your Own Music with Songtrix - Free! Bring these music concepts to life with the free Songtrix Bronze Edition as you create songs from chords and scales. Then publish and share your ideas with the other musicians you meet on the ChordWizard Network. Have questions? Join the ChordWizard Network and post them in the Music Theory forum for answers and discussions on your topics of interest. Next Topic > (1.10) Chromatic Scale Notes The Chromatic scale could start on any note, but would always end up containing the same notes (because it contains all notes). So in practice, there is only one Chromatic scale. The table below shows the frequencies of the twelve notes between note A at 440 Hz, and note A one octave up from it. Higher pitched notes have larger frequency steps (in Hertz) between them, but each step makes an equal change to difference in pitch (one semitone) that we perceive. Even though there are twelve notes in an octave, only the first seven letters of the alphabet are used to name them (from A to G). We'll look at the reason for this later.

The other five notes of the Chromatic scale are named by placing a sharp (#) symbol after a note letter to make it one semitone higher, or by placing a flat (b) symbol after a note letter to make it one semitone lower. Often there are two name choices for these other five notes. For example, the note Ab in the table above could also be called G#. Sometimes, as we'll see later, there is a definite rule for which name is more appropriate. Chromatic Scales for Guitar | Cyberfret.com Chromatic Scales for Guitar By Shawn Bradshaw Learning how to play chromatic scales on guitar is important for a few reasons. First, they are just good guitar technique building exercises . But more important than that, guitar chromatic scales help you understand how the fretboard is organized a little better. In this video guitar lesson you will learn 3 different ways to play chromatic scales for guitar. (Video Guitar Lesson 1 of 2) Chromatic Scales on Guitar The word chromatic in music means 2 or more consecutive notes that are a half step (1 fret) apart from one another.

A guitar chromatic scale contains all 12 possible pitches before arriving back at the starting note name an octave higher. Here are the notes in a C chromatic scale. You will see that there are only 12 different pitches before arriving back at C. There are 12 different pitches, but you also see the 2nd note in the scale is a C# or a Db. That is because C# and Db are just different ways to spell the same pitch. What you will call that pitch really depends on the situation. C Chromatic Guitar Scale Often sharps are used when ascending in the chromatic scale, and flats are used when descending. When it comes to playing chromatic scales on the guitar, you are not necessarily starting and stopping on the “root” of the scale. In fact, you should not really think in terms of there being a root for chromatic scales. Any note could be considered the root, and you are not really going to hear all of the notes in relationship to any one note. Playing chromatic scales is really just an exercise in understanding how the notes are arranged on the neck of the guitar. Understanding where the next note higher or lower is, even when switching strings.

Like I mentioned they are also excellent guitar technique builders. Guitar Chromatic Scale #1 – 4 Notes Per String The first guitar chromatic scale form contains 4 notes on every string. As you move to the next higher string you will shift back one fret, except between the 3rd and 2nd strings which you will stay on the same fret. This is because of the way the guitar is tuned. For more on reading guitar scale charts , be sure and check out my video guitar lesson “ Intro to Scales on the Guitar ” The next 2 guitar chromatic scale forms stay in one position on the neck. This video guitar lesson will show you how to play these chromatic scales for guitar without shifting position like you did for the first form. (Video Guitar Lesson 2 of 2) If you want to stay in one position when playing a chromatic scale you will use the following forms. Use form #2 when ascending, and form #3 when descending. Guitar Chromatic Scale #2 – In Position Ascending Guitar Chromatic Scale #3 – In Position Descending Here is a graphic that shows how the notes in form #2 and form #3 compare to each other.

The red line is drawn between the notes that are the same. You certainly could play form #3 when ascending, and form #2 descending. But it’s easier to have your first finger start on a lower fret and move up than it is to do the opposite. You are starting out of position, and moving back into position. A position is a 4 fret area where a finger is assigned to each fret. Notes that are one fret lower than the position are played with your first finger. Notes that fall one fret higher than the position are played with your fourth finger. When playing chromatically it’s always easier to play a note that’s out of position and move into position than it is to start in position and move out of position. That’s why it makes the most amount of sense to use form #2 while ascending, and form #3 when descending. A Chord Formula is a Musical Recipe - Lotus Music Lotus Music A Chord Formula is a Musical Recipe Written by Tom Michero A chord formula is musical numbering system that gives us a way of describing a chord without referring to specific notes. This is done by enumerating the intervals that make up a chord. For example, a Major 7th chord is made up of four notes and three intervals.

A chord formula spells out the harmonic structure using the intervals rather than the actual note names. The Numbering System The numbering system that gives us these formulas is based on the intervals that make up the Major scale. The intervals of this scale are the yardstick we use to measure the distances between notes that make up chords. To understand where this numbering system came from lets look a the Chromatic scale. The Chromatic scale is made up of every possible note you can play between an octave on Western instruments. It consists of 12 notes each a half step (also known as a semitone) apart. It’s the scale you get when you play a piano key and then the next 11 keys up the scale without skipping any keys. The Chromatic scale is the 12-note scale that includes every possible note that exists between an octave. There are many scales that can be created from these 12 notes. Over the centuries different cultures with different musical tastes have left us with about a dozen scales that are popular today. The most popular of all is the Major scale. It is made up of only seven of these 12 possible notes. The notes that make up the Major scale are not equally spaced within the octave.

Some of the notes are two semitones apart while others are only one semitone apart. The unequal steps and placement of these notes is what gives this scale a bright sound and a solid finish. If you were to play the Major scale, you would hear a beginning, a middle, and an end. This scale is used in a great majority of the songs we hear today. The Chromatic scale is made up of 12 notes that are spaced a semitone apart. The Major scale is made up of 7 notes with some of them spaced a semitone apart and some space two semitones (whole note) apart. Centuries ago, before the Chromatic scale had been invented, the notes that make up the Major scale were all that anyone really cared about. Instruments existed that could only play these notes. It is not surprising that this scale ended up being the benchmark by which all scales and chords are measured. At some point, the first note of the major scale become known as the root, and the other notes we just numbered according to their position in the scale – second, third, fourth, fifth, sixth, and seventh. This system allowed musicians to tune their instruments without knowing what the pitches were.

In fact, there was really no agreement as to what pitch each note should be until fairly recently. However, everyone knew what the major scale should sound like and so tuned accordingly. Since musicians tend to be creative people, even back then they experimented with different kinds of sound for their music. They created new scales. In order to play these new scales they added strings to their instruments and holes in their flutes. But some of the notes of their new scales did not exist in the Major scale but fell in between the notes numbered, 1-2-3-4-5-6-7. (The number “1″ is the same as the root.) Fortunately, the Chromatic scale helped them arrange and space these notes in a systematic way. Musicians now started paying attention to how many semitones each note was from the root. This meant that the third note of the Major scale is four semitones away from the root. The fifth note of the Major scale is seven semitones from the root. The seventh note of the Major scale is 11 semitones from the root. These harmonic distances are known as intervals. In other words, an interval is the distance between notes.

The Major scale is the scale is made up of intervals by which all others intervals are measured. As the pattern of notes that makes up the Major scale became the yardstick that measures the distance between any two notes. The distance between notes were given names. For instance, distance between the root and third note of the Major scale became called a “third.” The interval of a fifth is the distance between the root and the fifth note of the scale. The seventh note of the scale which is 11 semitones from the root is called a seventh. These names have been expanded upon to delineate them from intervals that were named later. The notes of the Major scale represent the intervals that are used to measure the distances between notes. Scale Formulas If you wanted to convey to someone the intervals that make up the Major scale you could write it this way: 1-2-3-4-5-6-7. This is a scale formula and you use it to “calculate” which notes notes go together. You can overlay this formula over the Chromatic scale to create the Major scale in every key. The Major scale pattern is the same no matter what note the scale begins on.

When you overlay the Major scale pattern over the C Chromatic scale it results in the C Major scale. This same pattern can be overlaid the C# Chromatic scale to create the C# Major scale. This same process can be used to create the Major scale in every key. Measuring Other Scales As mentioned earlier, there are other scales used in music. The Natural Minor scale happens to be one of these. The spacing of its notes creates a musical sound that is often described as a somber and ungrounded. If we compared the notes of the Natural Minor scale to the Major scale it would look like this: The Minor scale patters is different from the Major scale pattern. The 3rd, 5th, and 7th notes of the Minor scale are a semitone lower than their Major scale counterparts. When we use the Major pattern as our yardstick, we find that some of the notes of the Minor scale fall between the Major scale notes. To fix this we need to add some fractions to our yardstick. The intervals of the Major scale are like inch-marks on our ruler. But now we need to add some “in between” marks to measure the intervals of the Natural Minor scale.

The third, sixth, and seventh note of the Minor scale are a semitone less (lower) than their Major scale counterparts. To accommodate these “in-between” intervals someone came up with the idea of using sharps (#) and flat (b) symbols to indicate the difference. The sharp (#) adds a semitone to a note’s pitch while the flat (b) diminishes a note by a semitone. In the diagram below you can see that the third note of the Major scale is two whole steps from the root. Meanwhile the third note of the Natural Minors scale is one-and-a-half steps from the root. The notes of the Minor scale are measured according to the intervals established by the Major scale. Since the sharp (#) adds a semitone to a note and a flat (b) diminishes a note by a semitone, it is possible for an interval to be referred to in two ways. A sharp five (#5) is the same as a flat six (b6). Finally, when we add the sharps and flats to the yardstick, we have a complete “ruler” that we can use to measure the distance between any two notes. Each interval has a name (or two) associated with it.

Intervals that are derived from the Major scale pattern are called major or perfect. The other intervals are referred to as minor, diminished, or augmented. All intervals get there names based on the notes of the Major scale. Even the notes in between are indicated as alterations of the Major intervals. Minor Scale Formula Using our complete ruler we see that the Natural Minor scale has the following intervals: 1-2-b3-4-5-b6-b7. We can now overlay this formula on the Chromatic scale to come up with the notes of the Minor scale in whatever key we want. The diagram below shows C, C#, and D Minor scales. The Natural Minor scale formula can be overlaid the Chromatic scale to create the Minor scale in different keys. Minor Scale Formula Now for a little review. The Major scale pattern follows this formula: 1-2-3-4-5-6-7. The Natural Minor scale pattern follows this formula: 1-2-b3-4-5-b6-b7. Every scale you encounter can be expressed using this formula system. The formula is the same and but the root note is what determines the key.

Chords Now that you know how to work with scale formulas, you can apply the same concept to chords. Chords are in essence small scales. Chords are often made up of three or four notes but can have up to seven. For now, we will just consider the three and four-note chords. Consider this three-note chord known as C major (not to be confused with C Major scale). It is made up of these notes: C-E-G. This triad major chord can be described as a formula just as a scale can. C is the root. The second note of a major chord is always a major third from the root. The third note of the chord is an interval of a fifth from the root. That makes the chord fromula for a major chord this: 1-3-5. When this formula is applied to the Chromatic scale this is the result: Chords are like mini-scales. Formulas can be used to create them. The root may change but the formula will stay the same. The names given to chords are based on the intervals that make them up. Unfortunately, the name of a chord can be confusing if you don’t already know the formula for the chord.

In the above examples we have Cmin7, Cmin(maj7), Cmaj7, and C7. If you didn’t know the formula for these chords, you might not know if the fourth note of these chords is a major or minor seventh interval. What makes this even more confusing is that there is not a grand agreed upon naming system. Therefore, there are several “correct” names for chords and ways to write them. Chord names may vary but the formulas do not. That is why understanding chord formulas and how to apply them is so important. Below are chord formulas for several common chords. Along with them are their names and ways to write them. Notes: * Sometimes you will see a minor(major) chord written this way. Actually, a sharp 7 (#7) would be an octave interval. However, in this context the sharp is reminding you that the seventh is major, like in a major 7th chord. ** Often you will see a diminished 7th formula written this way: 1-b3-b5-6. You could even call this chord a minor 6th flat five (min6b5).

However, the double flatted 7th is technically the more correct way to refer to this interval. The general principle to remember here is that a double flatted note is two semitones lower, 6 = bb7.

Question: The Chromatic Scale is a musical scale with how many pitches or notes?
Answer: 12