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This is part 3 of my blog series.
Part 1: Introduction and the major scale
Part 2: More about intervals, and the minor scale
I recommend to read the these entries first, if you haven't done it yet.
Let's recapitulate the last blog, which was about triads: Beginning with the first note – the root – a triad also consists of a third interval which is either major or minor, and a perfect fifth as the overarching interval. Another way to view a triad is the stacking of both kinds of thirds on top of each other, because two thirds of each types will add up to a fifth interval. Each major scale – and therefore, each major triad – is related to a minor scale, and therefore minor triad, and vice versa.
The Tonic And The Sound Of Scales
Scales provide a selection of tones. The word tone has Greek origins, one meaning of "tonos" is tension. All notes of a scale are not only having a tension between each other, but also a recognizable relationship to the first note of the scale. This allows the first note of the scale to function as tonic center, as root of the entire scale. That means, the first degree of a scale acts as point of rest, where any tension is resolved.
If a composer wants to write a victory song, he will most likely use a major scale because the tonic triad features a major third, providing a bright sound. A song which is about worries and troubles will more likely use a minor scale, because it has a minor third in the tonic triad, creating a duller sound, which supports a sad melody.
However, using the right rhythm and melody, a piece in minor can still be festive and vivid. The sad ballad "Yesterday" on the other hand is written in a major key.
Today, almost any popular piece is written in either major or minor. That wasn't always so.
Five Notes To Begin With
The starting point for our short lesson in the history of music theory is a scale of five notes. To get it, we stack fifths and push all notes into the same octave. An example: We start with an arbitrary pitch and go to its fifth. That means we multiply the pitch frequency by 1.5 and get another note. Not we calculate the fifth the note we just created. It is 1.5x1.5 = 2.25 as high as the pitch we started with. This pitch already exceeds the octave, so we shift the new note down an octave by dividing its frequency by two: 2.25 / 2 = 1.125 and have another note for our scale. This one is by the way also the whole tone step.
We continue to stack fifths, but stop before we get any step within our scale which is smaller than a whole tone. We will end up with five notes.
Even with no developed mathematical concept of harmony, that scale was used by many cultures, though sometimes with a different tuning. As the fifth creates the best harmonic note to another one, we just build up a series of notes which are sounding best to one another.
Depending on the note which is used as the first degree, a pentatonic (five-note) scale can already have a major or minor characteristic as it contains one triad of each type. While common pentatonic scales have no semi-tone steps, they include steps larger than a whole tone, but smaller than a two whole tones.
To play around with this scale, use the white-key notes and leave out F and B. Or use the black keys only. There are five per octave and form a pentatonic scale.
The History Of Diatonic Scales
The ancient Greeks noticed the steps larger than a whole tone within the then-used pentatonic scale. This encouraged them to create scales which have no steps greater than a whole tone. For this blog, the term "diatonic scale" is used for gamuts which can be projected on the white-key notes. Other usages of the term exist, but shall not be taken into account for this blog.
The Greek built a tetrachord, that is a string instrument with four ("tetra") strings ("chords".) The pitch range of the lowest note to the highest one spanned a perfect fourth. Within this overarching interval, mostly whole tone steps were used, but one semi-tone step was required to get to a fourth interval using four notes. Then they build another tetrachord and tuned each string a perfect fifth higher than in the first one. Now they had eight strings in total.
Such scales cover exactly one octave, meaning the first note appears as prime and as octave. Depending on where the one necessary semi-tone step is placed in the tetrachord, different diatonic scales are possible. Since we use two tetrachords, the final scale will have two semi-tone steps.
Church Modes
Choral composers in the middle ages of Europe discovered the Greek records about scales. They noticed that the different Greek gamuts are actually all the same scale, just with a different note used as the first degree. Sadly, they confused the scale names and assigned then in the wrong order. But the mixed up nomenclature is still effective, and will be used in this blog, too.
Disregarding slight differences in tuning, all church modes can be played with white-key notes on a modern piano. Four different modes were established during the middle ages, two of each similar to major or minor respectively. The authentic church modes do not include natural major, nor natural minor.
The number of these four scales was doubled with a "hypo" modifier, adding three lower notes using the respective gamut. In today's terms, there are just four actually different church scales.
Perfect Math Doesn't Play Along
Even though we use simple math to establish harmony – the harmonic series, with the simple ratios of 1:2:3:4 and so on – we cannot use all those intervals with their actual ratio. The reason is that they don't fully fit together.
The octave with the 1:2 ratio is the simplest interval, simpler even than the fifth, so simple that it doesn't even create a different note. We must intone it with full purity, with the just ratio, even more so as we repeat a musical scale throughout the octaves and don't want them to get out of tune in different octaves.
One has to temper other intervals. Because the perfect fifth is so harmonic and creates stability, many early musicians used the just 2:3 ratio to get a series of actually perfect fifths while having one fifth out of tune to fit the octave. This intonation worked quite well for a long time.
The main drawback were the thirds. Both the major and the minor third were out of tune compared to just intonation.
Modal Scales
Only some hundred years ago, composers had the idea to change the tuning of their instruments to get pure thirds. This however works only for a very limited number of keys. Compromises were made to allow thirds for more keys. To reach this goal, some fifths had to be slightly off, and still only a limited number of triads were usable. The church modes were complemented by two additional scales. We now call them the natural major or minor scale, respectively, but they also have their own modal name.
We now have six modes in total, but seven notes which could be used as the first degree. The missing scale was added quite recently, less than 200 years ago. As its tonic triad is neither major nor minor, that scale is used very seldom. Here is a table describing all seven diatonic scales which are possible when we use only white-key notes:
* To play the scale with only white-key notes, use this note as first degree of the scale.
** Locrian has a minor third and therefore could be considered minor with the difference of a minor second and diminished fifth compared to natural minor. However, a minor scale requires a perfect fifth and no diminished one, so we will treat Locrian as neither major nor minor.
Expanded from the church modes to seven different gamuts, we now use the term modal scales. Except for Locrian, all modal scales have one thing in common: They include the pentatonic scale in either major or minor mode, depending on which variant the respective church mode represents. The only notes which are different, do not appear in the pentatonic scale.
Each modal scale provides its own overall sound depending on the difference to the natural major or minor mode. Before the major/minor dualism was established, the different sounds of the modes were used to create the desired atmosphere. For example, the minor second in Phrygian creates an Oriental vibe, as many scales from those cultures feature a minor second, too.
An Outlook On Functional Harmonic
In today's practice, modal scales beside major and minor are not used very often. Composers use chords to create a certain tension and feeling. Many chords either are triads, or are based on triads. Triads however could also be viewed as small representatives of an entire scale. In this regard, many chords could also be viewed as remnants of modal scales.
Other Scales
Traditional Chinese scales are pentatonic. Traditional music from Java is pentatonic as well, though with a different tuning. The blues scale used today, originating from Africa, is the pentatonic minor scale plus an additional note which can only be roughly approximated on a modern piano. Other scales which are hard to intone on a piano include some Arabic scales. However, playable approximations do exist.
We should not forget that a scale provides only the tonal material. It is up to the composer to sculpt something out of it, to arrange the material to express himself. If we allow some simplification, a scale can be viewed figuratively as a color palette which is used by a painter to create an image.
Next blog: How intervals sound and how we get even more color for our notes
 
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I'm giving you a 4/5 because this is a relatively well written blog on an interesting subject that is worth knowing. I enjoyed reading this.
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Why not just drop him the extra star? 
But whoa, are you a music professor? Great stuff faths 
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1/5
did not help; made my brain implode.
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your Country52797 Posts
On February 17 2013 07:46 MasterOfPuppets wrote:
5/5
did not help; made my brain implode.
fixed for you, also you stole what I was going to say
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Thank you for writing this!
Looking forward to the continuation.
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On February 17 2013 07:40 Aerisky wrote:Why not just drop him the extra star? But whoa, are you a music professor? Great stuff faths
I am no professor  and didn't study music, instead I picked multimedia technology. I began to take piano lessons less than two years ago, but I was exposed to some instruments (accordion, recorder) since my childhood.
I always wondered, why does the major scale has any semi-tone step at all? Why are those steps placed where they are and not somewhere else? All musical stuff looked so arbitrary to me. Why does G sound so nice with C, why does G-C creates a concluding effect? I could name notes and intuitively recognize if a chord is a major or minor one, but not say why things are the way they are. Very slowly I learned more, with this blog series I try to describe why many things in music are not arbitrary.
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Hey man, i just wanted to ask a few questions or make some corrections if I may:
If a composer wants to write a victory song, he will most likely use a major scale because the tonic triad features a major third, providing a bright sound. A song which is about worries and troubles will more likely use a minor scale, because it has a minor third in the tonic triad, creating a duller sound, which supports a sad melody.
I know this is a much much later topic, but did you plan on talking about the humors at any point? That seems somewhat important when talking about ethos of keys or modes. Also on ethos, I guess an important distinction to be made for ethos of a scale also has to do with the temperaments or characteristics of keys. Because of non-equal temperament, each key had its own specific flavor if you will, and that often played a very important role in what keys were chosen by a particular composer, e.g. there's a reason why composers write pieces in F-Major specifically during the common practice period.
The main drawback were the thirds. Both the major and the minor third were out of tune compared to just intonation.
Modal Scales
Only some hundred years ago, composers had the idea to change the tuning of their instruments to get pure thirds. To reach this goal, some fifths had to be slightly off, and still only a limited number of triads were usable. The church modes were complemented by two additional scales. We now call them the natural major or minor scale, respectively, but they also have their own modal name.
This seems backwards. Pure intervals are completely in tune, whereas tempered intervals are out of tune, even if generally people would hear this in the reverse. Also thirds, at least in organum and early church music wasn't considered dissonant, though 4ths are dissonant at this time (and all throughout species counterpoint.)
as for tuning instruments to get pure intervals, the opposite actually happened as composers began to use more chromaticism it necessitated retuning intervals so that more keys could be used. Maybe I read this wrong, but it seems like what your saying, and how things went historically are just switched up.
In today's practice, modal scales beside major and minor are not used very often. Composers use so-called function chords to create a certain tension and feeling. Function chords are based on triads. But triads could also be viewed as small representatives of an entire scale. Since most function chords can be expressed with a scale degree, those chords could also be viewed as remnants of modal scales.
Maybe its just me, but what are function chords? I'm not understanding this definition as it seems like it could have a lot of answers to it? Are you talking about the beginnings of functional harmony?
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On February 19 2013 04:36 wo1fwood wrote:Hey man, i just wanted to ask a few questions or make some corrections if I may: Show nested quote +If a composer wants to write a victory song, he will most likely use a major scale because the tonic triad features a major third, providing a bright sound. A song which is about worries and troubles will more likely use a minor scale, because it has a minor third in the tonic triad, creating a duller sound, which supports a sad melody. I know this is a much much later topic, but did you plan on talking about the humors at any point? That seems somewhat important when talking about ethos of keys or modes. Also on ethos, I guess an important distinction to be made for ethos of a scale also has to do with the temperaments or characteristics of keys. Because of non-equal temperament, each key had its own specific flavor if you will, and that often played a very important role in what keys were chosen by a particular composer, e.g. there's a reason why composers write pieces in F-Major specifically during the common practice period.
I am no expert here, so the following is an opinion based on things I know so far: Many instruments were build with a particular key in mind as they resonate especially well on that frequency due to its construction and size. Since particular instruments are often used in particular genres, one began to associate those keys with the genre. As I never saw a list of exact tunings which the important composers had in mind, I don't know how much the key flavors were actually used, beside some things like the wolf interval (which however appeared in pythagorean tunings.)
Even though the tuning is of course important to interpret a piece, it carries less information than the notes. At first I wanted to leave the temperament out completely and just focus on general scale ideas, but since the chromatic scale in today's music only works with the equal temperament, I thought I should mention historic temperaments as well.
Equal temperament – again, this is my opinion only – still carries some flavors even though all intervals are equal in any key. But we still have the resonance of the soundbox and we still have a memory of tradition, expecting certain keys for certain types of music.
On February 19 2013 04:36 wo1fwood wrote:Show nested quote +The main drawback were the thirds. Both the major and the minor third were out of tune compared to just intonation.
Modal Scales
Only some hundred years ago, composers had the idea to change the tuning of their instruments to get pure thirds. To reach this goal, some fifths had to be slightly off, and still only a limited number of triads were usable. The church modes were complemented by two additional scales. We now call them the natural major or minor scale, respectively, but they also have their own modal name. This seems backwards. Pure intervals are completely in tune, whereas tempered intervals are out of tune, even if generally people would hear this in the reverse. Also thirds, at least in organum and early church music wasn't considered dissonant, though 4ths are dissonant at this time (and all throughout species counterpoint.)
I went from pythagorean tuning, with dissonant thirds, right to meantone, forgetting the just intonation. I changed that part now, thank you for bringing that up.
As far as I know, organs at that time (the time where major and minor were established) were tuned just or in meantone, having some just thirds. If I am not mistaken, the fourth was considered dissonant even though when tuned just.
On February 19 2013 04:36 wo1fwood wrote:as for tuning instruments to get pure intervals, the opposite actually happened as composers began to use more chromaticism it necessitated retuning intervals so that more keys could be used. Maybe I read this wrong, but it seems like what your saying, and how things went historically are just switched up.
I intend to go to chromatism and the equal temperament in the blog after the next one. This blog mention the core ideas pythagorean, just and meantone temperament without going into the details. Because otherwise I would have to mention different just intonations for the major and the minor scales, having different whole tone and minor sevenths intervals, and going into different approaches of meantone temperament.
On February 19 2013 04:36 wo1fwood wrote:Show nested quote +In today's practice, modal scales beside major and minor are not used very often. Composers use so-called function chords to create a certain tension and feeling. Function chords are based on triads. But triads could also be viewed as small representatives of an entire scale. Since most function chords can be expressed with a scale degree, those chords could also be viewed as remnants of modal scales. Maybe its just me, but what are function chords? I'm not understanding this definition as it seems like it could have a lot of answers to it? Are you talking about the beginnings of functional harmony?
Yes, I am talking about functional harmony (tonic, dominant, subdominant) to hint a connection between those degrees and modal scales. I am still working on the concept of the blog series for chords as I am not intending to just repeat common explanations how these chords are used, but why they have their effect.
I replace "function chord" with the more common english phrase "functional harmonic" in the blog.
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On February 19 2013 06:30 [F_]aths wrote:+ Show Spoiler +On February 19 2013 04:36 wo1fwood wrote:Hey man, i just wanted to ask a few questions or make some corrections if I may: Show nested quote +If a composer wants to write a victory song, he will most likely use a major scale because the tonic triad features a major third, providing a bright sound. A song which is about worries and troubles will more likely use a minor scale, because it has a minor third in the tonic triad, creating a duller sound, which supports a sad melody. I know this is a much much later topic, but did you plan on talking about the humors at any point? That seems somewhat important when talking about ethos of keys or modes. Also on ethos, I guess an important distinction to be made for ethos of a scale also has to do with the temperaments or characteristics of keys. Because of non-equal temperament, each key had its own specific flavor if you will, and that often played a very important role in what keys were chosen by a particular composer, e.g. there's a reason why composers write pieces in F-Major specifically during the common practice period. I am no expert here, so the following is an opinion based on things I know so far: Many instruments were build with a particular key in mind as they resonate especially well on that frequency due to its construction and size. Since particular instruments are often used in particular genres, one began to associate those keys with the genre. As I never saw a list of exact tunings which the important composers had in mind, I don't know how much the key flavors were actually used, beside some things like the wolf interval (which however appeared in pythagorean tunings.) Even though the tuning is of course important to interpret a piece, it carries less information than the notes. At first I wanted to leave the temperament out completely and just focus on general scale ideas, but since the chromatic scale in today's music only works with the equal temperament, I thought I should mention historic temperaments as well. Equal temperament – again, this is my opinion only – still carries some flavors even though all intervals are equal in any key. But we still have the resonance of the soundbox and we still have a memory of tradition, expecting certain keys for certain types of music. On February 19 2013 04:36 wo1fwood wrote:Show nested quote +The main drawback were the thirds. Both the major and the minor third were out of tune compared to just intonation.
Modal Scales
Only some hundred years ago, composers had the idea to change the tuning of their instruments to get pure thirds. To reach this goal, some fifths had to be slightly off, and still only a limited number of triads were usable. The church modes were complemented by two additional scales. We now call them the natural major or minor scale, respectively, but they also have their own modal name. This seems backwards. Pure intervals are completely in tune, whereas tempered intervals are out of tune, even if generally people would hear this in the reverse. Also thirds, at least in organum and early church music wasn't considered dissonant, though 4ths are dissonant at this time (and all throughout species counterpoint.) I went from pythagorean tuning, with dissonant thirds, right to meantone, forgetting the just intonation. I changed that part now, thank you for bringing that up. As far as I know, organs at that time (the time where major and minor were established) were tuned just or in meantone, having some just thirds. If I am not mistaken, the fourth was considered dissonant even though when tuned just. On February 19 2013 04:36 wo1fwood wrote:as for tuning instruments to get pure intervals, the opposite actually happened as composers began to use more chromaticism it necessitated retuning intervals so that more keys could be used. Maybe I read this wrong, but it seems like what your saying, and how things went historically are just switched up. I intend to go to chromatism and the equal temperament in the blog after the next one. This blog mention the core ideas pythagorean, just and meantone temperament without going into the details. Because otherwise I would have to mention different just intonations for the major and the minor scales, having different whole tone and minor sevenths intervals, and going into different approaches of meantone temperament. On February 19 2013 04:36 wo1fwood wrote:Show nested quote +In today's practice, modal scales beside major and minor are not used very often. Composers use so-called function chords to create a certain tension and feeling. Function chords are based on triads. But triads could also be viewed as small representatives of an entire scale. Since most function chords can be expressed with a scale degree, those chords could also be viewed as remnants of modal scales. Maybe its just me, but what are function chords? I'm not understanding this definition as it seems like it could have a lot of answers to it? Are you talking about the beginnings of functional harmony? Yes, I am talking about functional harmony (tonic, dominant, subdominant) to hint a connection between those degrees and modal scales. I am still working on the concept of the blog series for chords as I am not intending to just repeat common explanations how these chords are used, but why they have their effect. I replace "function chord" with the more common english phrase "functional harmonic" in the blog.
Temperaments: Equal temperament actually fundamentally negates affectations of keys that used to exist in what we might call these 'imperfect' systems. For example Werkmister III gave each key a particular flavor due to the imperfections of the tuning, and by evening out every semitone the way 12-tet does, those affectations are gone now. Its sort of a curiosity, a musicological interest, but 12-tet removed the almost programatic reasons for using specific keys, e.g. C-Major vs F-Major which today sound identical affectation-wise.
As for composers who have used this, its not everyone, but Beethoven (18-19th c. composers especially) is a great example of such a composer who used these specific keys, Symphonies 3, 5 and 6 probably being the most prominent examples (Eroica in E-flat, Pastoral in F-Major, those key choices are not arbitrary). I may be mis-remembering this somewhat, but I'll ask an old prof about it.
4ths: Yea, 4ths are classified as a dissonance due to earlier counterpoint considerations as the P4 is by its nature an unstable interval.
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On February 19 2013 10:43 wo1fwood wrote:Show nested quote +On February 19 2013 06:30 [F_]aths wrote:+ Show Spoiler +On February 19 2013 04:36 wo1fwood wrote:Hey man, i just wanted to ask a few questions or make some corrections if I may: Show nested quote +If a composer wants to write a victory song, he will most likely use a major scale because the tonic triad features a major third, providing a bright sound. A song which is about worries and troubles will more likely use a minor scale, because it has a minor third in the tonic triad, creating a duller sound, which supports a sad melody. I know this is a much much later topic, but did you plan on talking about the humors at any point? That seems somewhat important when talking about ethos of keys or modes. Also on ethos, I guess an important distinction to be made for ethos of a scale also has to do with the temperaments or characteristics of keys. Because of non-equal temperament, each key had its own specific flavor if you will, and that often played a very important role in what keys were chosen by a particular composer, e.g. there's a reason why composers write pieces in F-Major specifically during the common practice period. I am no expert here, so the following is an opinion based on things I know so far: Many instruments were build with a particular key in mind as they resonate especially well on that frequency due to its construction and size. Since particular instruments are often used in particular genres, one began to associate those keys with the genre. As I never saw a list of exact tunings which the important composers had in mind, I don't know how much the key flavors were actually used, beside some things like the wolf interval (which however appeared in pythagorean tunings.) Even though the tuning is of course important to interpret a piece, it carries less information than the notes. At first I wanted to leave the temperament out completely and just focus on general scale ideas, but since the chromatic scale in today's music only works with the equal temperament, I thought I should mention historic temperaments as well. Equal temperament – again, this is my opinion only – still carries some flavors even though all intervals are equal in any key. But we still have the resonance of the soundbox and we still have a memory of tradition, expecting certain keys for certain types of music. On February 19 2013 04:36 wo1fwood wrote:Show nested quote +The main drawback were the thirds. Both the major and the minor third were out of tune compared to just intonation.
Modal Scales
Only some hundred years ago, composers had the idea to change the tuning of their instruments to get pure thirds. To reach this goal, some fifths had to be slightly off, and still only a limited number of triads were usable. The church modes were complemented by two additional scales. We now call them the natural major or minor scale, respectively, but they also have their own modal name. This seems backwards. Pure intervals are completely in tune, whereas tempered intervals are out of tune, even if generally people would hear this in the reverse. Also thirds, at least in organum and early church music wasn't considered dissonant, though 4ths are dissonant at this time (and all throughout species counterpoint.) I went from pythagorean tuning, with dissonant thirds, right to meantone, forgetting the just intonation. I changed that part now, thank you for bringing that up. As far as I know, organs at that time (the time where major and minor were established) were tuned just or in meantone, having some just thirds. If I am not mistaken, the fourth was considered dissonant even though when tuned just. On February 19 2013 04:36 wo1fwood wrote:as for tuning instruments to get pure intervals, the opposite actually happened as composers began to use more chromaticism it necessitated retuning intervals so that more keys could be used. Maybe I read this wrong, but it seems like what your saying, and how things went historically are just switched up. I intend to go to chromatism and the equal temperament in the blog after the next one. This blog mention the core ideas pythagorean, just and meantone temperament without going into the details. Because otherwise I would have to mention different just intonations for the major and the minor scales, having different whole tone and minor sevenths intervals, and going into different approaches of meantone temperament. On February 19 2013 04:36 wo1fwood wrote:Show nested quote +In today's practice, modal scales beside major and minor are not used very often. Composers use so-called function chords to create a certain tension and feeling. Function chords are based on triads. But triads could also be viewed as small representatives of an entire scale. Since most function chords can be expressed with a scale degree, those chords could also be viewed as remnants of modal scales. Maybe its just me, but what are function chords? I'm not understanding this definition as it seems like it could have a lot of answers to it? Are you talking about the beginnings of functional harmony? Yes, I am talking about functional harmony (tonic, dominant, subdominant) to hint a connection between those degrees and modal scales. I am still working on the concept of the blog series for chords as I am not intending to just repeat common explanations how these chords are used, but why they have their effect. I replace "function chord" with the more common english phrase "functional harmonic" in the blog. Temperaments: Equal temperament actually fundamentally negates affectations of keys that used to exist in what we might call these 'imperfect' systems. For example Werkmister III gave each key a particular flavor due to the imperfections of the tuning, and by evening out every semitone the way 12-tet does, those affectations are gone now. Its sort of a curiosity, a musicological interest, but 12-tet removed the almost programatic reasons for using specific keys, e.g. C-Major vs F-Major which today sound identical affectation-wise. As for composers who have used this, its not everyone, but Beethoven (18-19th c. composers especially) is a great example of such a composer who used these specific keys, Symphonies 3, 5 and 6 probably being the most prominent examples (Eroica in E-flat, Pastoral in F-Major, those key choices are not arbitrary). I may be mis-remembering this somewhat, but I'll ask an old prof about it. 4ths: Yea, 4ths are classified as a dissonance due to earlier counterpoint considerations as the P4 is by its nature an unstable interval.
Werckmeister lived roughly 100 years before Beethoven's, I don't know which particular tuning were around at Beethoven's time, it could be that he used specific keys mostly because it was common practice. As Beethoven used quite a lot of chromatic notes, I would guess he had something in mind which was close to equal temperament. The different Werckmeister tempereaments already are an improvement over meantone approaches, getting closer to equal temperament. I would be interested what the professor you mentioned has to say about it.
In equal temperament, any of key flavor coming from different intervals per key is lost of course. But that doesnt mean that the timbre of each note is the same as of each other one. Many wind instrument are made for a specific key. A B-flat clarinet will intone B-flat especially pure. I guess that the key (or to me more specific, the concert pitch) still plays a role when writing a piece with certain instruments in mind.
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EPT![[image loading]](http://i.imgur.com/7YGEwkc.png)
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