Lesson 13-6 of the U of Chicago text is called "Uniqueness." In the modern Third Edition of the text, uniqueness appears in Lesson 5-6. Recall that the lessons of the old Chapter 13 appear in various chapters of the new edition. Uniqueness of Parallels (Playfair's Parallel Postulate) now appears in Chapter 5 so that it can be used to prove Triangle Sum. In the old version, Triangle Sum is proved in Chapter 5 using a slightly different proof, since uniqueness doesn't appear until Chapter 13.
It is now spring break in the district I work in more often, though the district on which the blog calendar is based is back from spring break. During this, my "real" spring break, several issues are on my mind now.
First of all, Fawn Nguyen, the Queen of the MTBoS, has her own Easter/April Fool's Day post:
I take a few more steps and look down because I feel something is coming off. I’m wearing flip-flops. It’s broken. No, both straps are broken.
Actually, this isn't an April Fool's joke, but a nightmare.
Meanwhile, I know who's having a real-life nightmare now -- teachers in Kentucky and Oklahoma are now on strike. Hmm, Oklahoma -- so what famous MTBoS bloggers teach in Oklahoma, again?
That's right -- Sarah Carter and her husband Shaun. Actually, Sarah has written about the upcoming strike, but she doesn't mention it in her post on the day of the strike itself:
This whole potential strike business and the frustration of not knowing if I will be teaching next week or marching around the capitol building has left me extremely exhausted. As a result, my motivation to get blog posts written has almost disappeared. This is especially sad because I've been trying out several new activities this week that deserve to be blogged about!
Well, the Carters are now officially on strike. (Oh, and Shaun hasn't posted since February, so there's no mention of a strike on his page at all.)
Indeed, in her most recent post, Sarah Carter gives her weekly Monday Must Reads series. I notice that this week's entry features a fellow Southern California teacher, Janice Mejia:
The fact that there are several Southern California community colleges listed there is a giveaway that Mejia teaches here.
I took a closer look at Mrs. Mejia's Twitter account. Back on Pi Day, she made several tweets about classroom management:
A1: ideal classroom is students collaborating and discussing alternate problem solving approaches. I believe whoever is doing the most talking is doing the most learning
A2: classroom management is about having a set of routines, procedures, and expectations in place to maximize instructional time.
A3: demonstrating respect for each student in your class. Students can honestly make your whole classroom if they know that the teacher cares about them.
A4: avoid confusing classroom management with discipline. It’s not about punishing kids when they do something wrong. Set high expectations and give incentives for positive behavior
A5: I’ll continue setting an example at my school site. And I’ll keep inviting my colleagues to show how my classroom management makes a difference to students
Indeed, spring break is a great time to reevaluate my own classroom management. Right now, I'm very worries that I'm not fulfilling my seven resolutions enough.
For example, I wrote the following back in my February 8th post:
In third period, three seventh graders fail to turn in the homework -- or even produce a blank HW sheet -- and start talking loudly. I write down the names, and the aide suggests that these students receive a Saturday detention, but this doesn't deter the trio. Eventually, the aide calls the office, and security comes to escort the three out of the room.
From my perspective, any call to security constitutes a failure on my part -- a failure to manage the classroom without outside help. So as I reflect on this incident, I ask myself, is there anything I could have done to handle this situation without security being involved?
Well I notice again, what Janice Mejia writes in her A4 tweet:
A4: avoid confusing classroom management with discipline. It’s not about punishing kids when they do something wrong. Set high expectations and give incentives for positive behavior
And we see that my entire response is all about discipline and punishment -- first writing down the names, then assigning Saturday school, and finally calling security. Mejia's A4 suggests that I could have tried another management technique. Writing down the names for the regular teacher is a given, of course. But what could I have done afterward?
Recall that this was a special education class. So perhaps the reason these three students didn't do the work is that they didn't know how to do the math. In other words, what these students needed was help (in this math class), not Saturday school or detention. I need to make more of an effort to help, not punish, special ed students -- just as the special scholar (January 6th post) needed my help.
I wrote that the class was divided into three groups, A, B, and C. I did mention in the February post that the trio was loud, so perhaps the three of them needed to be separated. Then as each group rotated towards me (Learning Centers!) I could help the one troublesome student who in the group I was currently helping. Both the student's proximity to me and the distance from his two friends would allow him to focus on math.
But what should I do if the students refuse to separate -- call security? No -- instead, I should add to my note for the teacher that the students refused to move when asked. Then I find a way to help all three boys at once. The only reason to call security is as a last resort -- if the three students were being so disruptive that it was affecting the other kids in the class. Recall that the misbehavior that started the whole incident was just failure to do the homework -- that's not something so disruptive that security needed to be involved.
Later on in the February 8th post, I wrote:
The problem is that I continue to discuss the incident the remainder of the day. For example, I tell a pair of eighth graders in sixth period that I'm about to write their names for the teacher. One of them tells me that writing down his name means nothing to him, so I tell him about the earlier class and how three students earned Saturday school and a referral to the dean. This, therefore, constitutes a violation of the third resolution [that is, no bringing up past incidents].
Did I violate any resolutions in third period? I can't be sure, but leading up to the security call, I might have argued with the trio (second resolution, no arguing or yelling). The best thing to do is to write the names down and say nothing else until I'm ready to change their seats or help them with the inequality solutions. Then there would have been no incident at all -- hence no past incident to violate the third resolution with.
Moreover, notice that the correct response to the eighth grader's statement about how "writing down his name means nothing to him" is to say nothing at all, regardless of past incidents. It's possible that deep down, he doesn't really want his name written down -- but he said what he said just to get into a power struggle. If I just wrote down the name and ignored his boasting, it's likely he would have become quiet and complied.
I wish to check some of my subsequent days of subbing for more management issues. But the following week, I suddenly decide not to post a "Day in the Life" unless it's a math class. This cuts into my ability to go back and reflect later on -- and only by reflecting on my past problems can I ever hope to improve my management.
And so I've chosen to post more "Day in the Life" for non-math classes. This will be especially the case for middle school classes -- where most of my management problems occur.
This is what I wrote last year about today's lesson:
The Glide Reflection Theorem only works when the preimage and the image have opposite orientation, not the same orientation. If a figure and its image have the same orientation, then we know that the isometry mapping one to the other is either a translation or a rotation. This case may be a bit tricky -- it could be that the easiest way is simply to translate A to A' and see whether this translation maps B to B' -- if not, then a rotation is necessary. But how do we find the center?
We know that the center of rotation is equidistant from A and A'. Thus it lies on the perpendicular bisector ofAA'. For the same reason, the center lies on the perpendicular bisector of BB'. So where these two points intersect is the center of rotation. Notice that if these two perpendicular bisectors are parallel, then the above reasoning constitutes an indirect proof that there is no rotation mapping one to the other -- that is, there is a translation map instead.
Today's worksheet covers all of Lesson 13-6. This means that not only are there questions about the Glide Reflection Theorem, but also about uniqueness in general. A modern form of Euclid's original five postulates are given.
Meanwhile, the Google Doodle today is a scientist -- John Harrison. Indeed, he worked on two concepts that I wrote about in my March 12th post -- time zones and longitude. Notice that latitude is absolute, since the Equator, North Pole, and South Pole are definite places. But longitude is relative and based on an arbitrary meridian through Greenwich. Therefore latitude can be determined by observing celestial objects, but longitude can't be calculated as easily.
Harrison's idea was simple -- to determine your longitude, just find out what time zone you're in. In other words, synchronize a clock to noon at a known longitude (such as Greenwich), travel to the place whose longitude you wish to find, and then compare noon clock time to noon sun time.
Finally, during the spring break I watched a NOVA episode -- "The Great Math Mystery." This episode first came on in 2015 -- and I blogged about it then. It came on again the following year -- always at around this same time of year. (I wonder whether it came on last year -- if it did, it was during one of the long stretches when I didn't post.)
One of the things mentioned during this NOVA episode is math and music. In my April Fool's Day post, I introduced a "music" blog label. So let me repost what I wrote about music in the episode. It's mostly about the 3-limit, with some 5-limit thrown in (as opposed to the 11-limit on Sunday), but it's good for me to reblog it anyway under the "music" label.
So this is what I wrote three years ago about math, music, and NOVA:
I've posted on music theory before on the blog -- after all, I've mentioned that I refer to half of an academic quarter as a "quaver" because this is a British term for an eighth note. But this topic is not about rhythm, but about pitch. Even though I've about to mention Pythagorean music, this topic isn't related to the Pythagorean Theorem at all. It fits better not in an Geometry class, but an Algebra II class, since it involves logarithms. But when I tutored Algebra II students -- especially those who have had some musical experience -- they are often fascinated when I tell them that it was Pythagoras who discovered why certain notes sound good together.
For example, musicians learn that notes are lettered A, B, C, D, E, F, G. But we are told that, for example, the notes C, E, G form what is called a "C chord." But this raises the question, why do we skip the notes D and F, and let the notes C, E, G be the "C chord"? As it turns out, this all started with Pythagoras.
The ancient Greeks played a stringed instrument called a lyre. It was Pythagoras who noticed that, while strings of different lengths produced different notes, the notes sounded best together when the string lengths exhibited a simple whole-number ratio. The simplest nontrivial ratio of whole numbers is 2:1 -- that is, when one string is exactly twice the length of the other. According to NOVA, this ratio is known as an octave. When two notes exactly an octave apart are played together, they end up sounding like the same note, except one's higher and the other's lower. Indeed, the note that's exactly an octave above C is also called C, and any two notes separated by a whole number of octaves are given the same letter. This is true for any stringed instrument -- notice that the twelfth fret of a guitar, the fret that produces a note one octave higher than the original note, is at the midpoint of the string. A violin appears in the episode, played by jazz musician Esperanza Spalding.
The next simple ratio mentioned by Spalding is 3:2 -- that is, the longer string is half again as long as the shorter string. This interval is called a perfect fifth. The note a perfect fifth above C is G. So notice that we already have two of the three notes of the C chord, the C and the G.
The next simple ratio mentioned by Spalding is 4:3 -- that is, the longer string is a third again as long as the shorter string. This interval is called a perfect fourth. The note a perfect fourth about C is F. This gives us the three Pythagorean intervals -- the octave, the perfect fifth, and the perfect fourth.
Musicians are familiar with the circle of fifths. We begin at C and move up a perfect fifth to G. Then we move up another perfect fifth to D, and then to A, then to E, and so on. After twelve steps on the circle of fifths, we return back to C again -- or so we are told. That is, twelve perfect fifths are declared to be equal to a whole number of octaves -- seven, as it turns out.
But what exactly are seven octaves? Notice that one octave is 2:1, two octaves are 4:1, three octaves are 8:1, and so on. Since these are ratios, they must be multiplied, not added, just as performing a dilation with scale factor 2 thrice gives a figure that is eight times the original. So seven octaves are the ratio 128:1.
Now let's try twelve perfect fifths. Each fifth is 3:2, so twelve of them would be 531441:4096, which is not exactly 128:1. Cross multiplying these ratios would give 531441 = 524288, which is false. And we shouldn't be surprised that twelve fifths don't actually equal seven octaves. Since combining musical intervals amounts to multiplying, breaking them up amounts to factoring -- and we can't factor the same interval in two different ways (i.e., as seven octaves and twelve fifths) for the same reason that we can't factor the same number in two different ways -- this would violate the Fundamental Theorem of Arithmetic. The "equation" 531441 = 524288 states that a power of three equals a power of two, which is impossible.
But in music, we declare 531441 and 524288 to be equal. So the ratio 531441:524288 is considered to be equal to 1. In honor of the ancient Greek mathematician, we refer to the ratio 531441:524288 as the Pythagorean comma. And it's because it's the twelfth power of three that is approximately a power of two that our octaves consist of twelve notes -- and why it's the twelfth fret on the guitar that gives us the octave. (Notice that computer scientists have their own "comma." They declare 1024 = 1000 -- that is, a power of two equals a power of ten -- in order to justify the names "kilobyte," "megabyte," etc.)
The NOVA episode mentions only the octave, perfect fourth, and perfect fifth, since these were the only intervals with which Pythagoras was concerned. But notice that these give us the notes C, F, G, while the C chord is actually C, E, G. So where does the note E come from?
This was actually discovered about 500 years or so after Pythagoras, by the musician Didymus. We notice that Pythagoras stopped at the ratio 4:3. The next natural ratio to consider is 5:4. This ratio is now known as a major third, and represents the interval from C to E. Then the full C major chord is the extended ratio 4:5:6.
Notice that there's already an E on the circle of fifths. Declaring E to be a major third above C is the same as setting four perfect fifths to equal the ratio 5:1 -- which, as we know, is impossible. Four fifths would actually be 81:16, so cross-multiplying gives us 81 = 80. The ratio of 81:80 is considered to be another comma -- Didymus's comma.
We are now ready to produce the full C major scale. We start with the three intervals of Pythagoras, the octave, perfect fourth, and perfect fifth -- C, F, G. On each of these three notes, we build a major chord: the C chord (C, E, G), F chord (F, A, C), and G chord (G, B, D). Putting these notes in order gives us the full major scale -- C, D, E, F, G, A, B, C.
I remember taking a piano class the summer after kindergarten. As it was only a beginners' course, the only major scale taught was the C major scale, which was played on the white keys. But I wondered to myself why one couldn't play a major scale beginning on notes other than C. The following December (either for my birthday or Christmas) I received a small electronic keyboard as a gift, and naturally I tried playing other scales. beginning on D, then E -- but none of them sounded like the proper major scale Do, Re, Mi, etc.
But one of these scales sounded almost right -- the scale beginning on G. The scale G, A, B, C, D, E, F, G sounded correct except for the last few notes. But I didn't know how to make the last part sound like the major scale. Disappointed, I started playing around with chromatic scales -- where I included the black keys as well as the white keys. But as I was still tantalized by the G scale, I sometimes started out by playing the part of the G scale that sounded right -- G, A, B, C, D, E -- and then switched to the chromatic scale. The black key between F and G is called F sharp, or F#, so what I played was G, A, B, C, D, E, F, F#, G, and I often played around with this "scale" for awhile.
Then one day, I accidentally skipped the F note, so what I ended up playing was G, A, B, C, D, E, F#, G. And what I played sounded exactly like the major scale that I had been seeking! And I still remember to this day how excited my seven-year-old self was to "discover" the G major scale! After this, I quickly realized that I could make all the other scales (the D scale, E scale, and so on) sound right by including some of the black keys as well as the white keys on my keyboard.
The next ratio to consider is 6:5. This interval is called a minor third. To produce a minor chord, we take the major chord C, E, G and replace the E with the black key just below E, called E flat, often rendered in ASCII as Eb. So the C minor chord is C, Eb, G.
Why do minor chords sound "sadder" than major chords? Recall that a major chord has its notes in the ratio 4:5:6. Well, a minor chord has its notes in the ratio 10:12:15 -- that is, 5:6 is now the lower ratio and 4:5 is now the higher ratio. Since 4:5:6 is simpler than 10:12:15, the 4:5:6 major chord sounds brighter than the 10:12:15 minor chord. Many popular songs on the radio nowadays tend to be about one of two topics -- falling in love and breaking up. The former songs tend to be in major keys, while the latter songs tend to be in minor keys. Over the past few decades, popular hits in minor keys have increased.
The second keyboard that I owned, when I was in the second grade, had several songs built in, including the classic Greensleeves. This song was in the key of A, but I noticed that it was based on A minor rather than A major, as it contained the chord A, C, E (whereas A major would be A, C#, E). I soon learned that there were minor scales as well as major scales, and tried to deduce what the A minor scale was based on the song. Unfortunately, the version of Greensleeves that was built into my keyboard wasn't truly in A minor, but a scale called the Dorian mode. Instead, I was under the misconception that A, B, C, D, E, F#, G, A was the A minor scale -- that is, A minor was just like G major in that it contained F# rather than F natural.
It was not until the third grade until I took private piano lessons. My third grade classroom had a piano, and the teacher was a pianist. So she taught me piano one a week after school until she left on maternity midway through the year, after which she referred me to her mother to continue my study of the piano. This was when I finally learned that the A natural minor scale was actually the notes A, B, C, D, E, F, G, A -- that is, it has no sharps or flats, just like C major. This minor scale is formed by taking the three Pythagorean intervals (octave, fourth, and fifth) and building a minor chord on each note, rather than a major chord.
Here are a few more links discussing major and minor scales:
http://music.stackexchange.com/questions/22236/why-am-i-always-sticking-to-minor-keys
which in turn links to:
http://www.nme.com/blogs/nme-blogs/the-science-of-music-why-do-songs-in-a-minor-key-sound-sad
Later on in the February 8th post, I wrote:
The problem is that I continue to discuss the incident the remainder of the day. For example, I tell a pair of eighth graders in sixth period that I'm about to write their names for the teacher. One of them tells me that writing down his name means nothing to him, so I tell him about the earlier class and how three students earned Saturday school and a referral to the dean. This, therefore, constitutes a violation of the third resolution [that is, no bringing up past incidents].
Did I violate any resolutions in third period? I can't be sure, but leading up to the security call, I might have argued with the trio (second resolution, no arguing or yelling). The best thing to do is to write the names down and say nothing else until I'm ready to change their seats or help them with the inequality solutions. Then there would have been no incident at all -- hence no past incident to violate the third resolution with.
Moreover, notice that the correct response to the eighth grader's statement about how "writing down his name means nothing to him" is to say nothing at all, regardless of past incidents. It's possible that deep down, he doesn't really want his name written down -- but he said what he said just to get into a power struggle. If I just wrote down the name and ignored his boasting, it's likely he would have become quiet and complied.
I wish to check some of my subsequent days of subbing for more management issues. But the following week, I suddenly decide not to post a "Day in the Life" unless it's a math class. This cuts into my ability to go back and reflect later on -- and only by reflecting on my past problems can I ever hope to improve my management.
And so I've chosen to post more "Day in the Life" for non-math classes. This will be especially the case for middle school classes -- where most of my management problems occur.
This is what I wrote last year about today's lesson:
The Glide Reflection Theorem only works when the preimage and the image have opposite orientation, not the same orientation. If a figure and its image have the same orientation, then we know that the isometry mapping one to the other is either a translation or a rotation. This case may be a bit tricky -- it could be that the easiest way is simply to translate A to A' and see whether this translation maps B to B' -- if not, then a rotation is necessary. But how do we find the center?
We know that the center of rotation is equidistant from A and A'. Thus it lies on the perpendicular bisector of
Today's worksheet covers all of Lesson 13-6. This means that not only are there questions about the Glide Reflection Theorem, but also about uniqueness in general. A modern form of Euclid's original five postulates are given.
Meanwhile, the Google Doodle today is a scientist -- John Harrison. Indeed, he worked on two concepts that I wrote about in my March 12th post -- time zones and longitude. Notice that latitude is absolute, since the Equator, North Pole, and South Pole are definite places. But longitude is relative and based on an arbitrary meridian through Greenwich. Therefore latitude can be determined by observing celestial objects, but longitude can't be calculated as easily.
Harrison's idea was simple -- to determine your longitude, just find out what time zone you're in. In other words, synchronize a clock to noon at a known longitude (such as Greenwich), travel to the place whose longitude you wish to find, and then compare noon clock time to noon sun time.
Finally, during the spring break I watched a NOVA episode -- "The Great Math Mystery." This episode first came on in 2015 -- and I blogged about it then. It came on again the following year -- always at around this same time of year. (I wonder whether it came on last year -- if it did, it was during one of the long stretches when I didn't post.)
One of the things mentioned during this NOVA episode is math and music. In my April Fool's Day post, I introduced a "music" blog label. So let me repost what I wrote about music in the episode. It's mostly about the 3-limit, with some 5-limit thrown in (as opposed to the 11-limit on Sunday), but it's good for me to reblog it anyway under the "music" label.
So this is what I wrote three years ago about math, music, and NOVA:
I've posted on music theory before on the blog -- after all, I've mentioned that I refer to half of an academic quarter as a "quaver" because this is a British term for an eighth note. But this topic is not about rhythm, but about pitch. Even though I've about to mention Pythagorean music, this topic isn't related to the Pythagorean Theorem at all. It fits better not in an Geometry class, but an Algebra II class, since it involves logarithms. But when I tutored Algebra II students -- especially those who have had some musical experience -- they are often fascinated when I tell them that it was Pythagoras who discovered why certain notes sound good together.
For example, musicians learn that notes are lettered A, B, C, D, E, F, G. But we are told that, for example, the notes C, E, G form what is called a "C chord." But this raises the question, why do we skip the notes D and F, and let the notes C, E, G be the "C chord"? As it turns out, this all started with Pythagoras.
The ancient Greeks played a stringed instrument called a lyre. It was Pythagoras who noticed that, while strings of different lengths produced different notes, the notes sounded best together when the string lengths exhibited a simple whole-number ratio. The simplest nontrivial ratio of whole numbers is 2:1 -- that is, when one string is exactly twice the length of the other. According to NOVA, this ratio is known as an octave. When two notes exactly an octave apart are played together, they end up sounding like the same note, except one's higher and the other's lower. Indeed, the note that's exactly an octave above C is also called C, and any two notes separated by a whole number of octaves are given the same letter. This is true for any stringed instrument -- notice that the twelfth fret of a guitar, the fret that produces a note one octave higher than the original note, is at the midpoint of the string. A violin appears in the episode, played by jazz musician Esperanza Spalding.
The next simple ratio mentioned by Spalding is 3:2 -- that is, the longer string is half again as long as the shorter string. This interval is called a perfect fifth. The note a perfect fifth above C is G. So notice that we already have two of the three notes of the C chord, the C and the G.
The next simple ratio mentioned by Spalding is 4:3 -- that is, the longer string is a third again as long as the shorter string. This interval is called a perfect fourth. The note a perfect fourth about C is F. This gives us the three Pythagorean intervals -- the octave, the perfect fifth, and the perfect fourth.
Musicians are familiar with the circle of fifths. We begin at C and move up a perfect fifth to G. Then we move up another perfect fifth to D, and then to A, then to E, and so on. After twelve steps on the circle of fifths, we return back to C again -- or so we are told. That is, twelve perfect fifths are declared to be equal to a whole number of octaves -- seven, as it turns out.
But what exactly are seven octaves? Notice that one octave is 2:1, two octaves are 4:1, three octaves are 8:1, and so on. Since these are ratios, they must be multiplied, not added, just as performing a dilation with scale factor 2 thrice gives a figure that is eight times the original. So seven octaves are the ratio 128:1.
Now let's try twelve perfect fifths. Each fifth is 3:2, so twelve of them would be 531441:4096, which is not exactly 128:1. Cross multiplying these ratios would give 531441 = 524288, which is false. And we shouldn't be surprised that twelve fifths don't actually equal seven octaves. Since combining musical intervals amounts to multiplying, breaking them up amounts to factoring -- and we can't factor the same interval in two different ways (i.e., as seven octaves and twelve fifths) for the same reason that we can't factor the same number in two different ways -- this would violate the Fundamental Theorem of Arithmetic. The "equation" 531441 = 524288 states that a power of three equals a power of two, which is impossible.
But in music, we declare 531441 and 524288 to be equal. So the ratio 531441:524288 is considered to be equal to 1. In honor of the ancient Greek mathematician, we refer to the ratio 531441:524288 as the Pythagorean comma. And it's because it's the twelfth power of three that is approximately a power of two that our octaves consist of twelve notes -- and why it's the twelfth fret on the guitar that gives us the octave. (Notice that computer scientists have their own "comma." They declare 1024 = 1000 -- that is, a power of two equals a power of ten -- in order to justify the names "kilobyte," "megabyte," etc.)
The NOVA episode mentions only the octave, perfect fourth, and perfect fifth, since these were the only intervals with which Pythagoras was concerned. But notice that these give us the notes C, F, G, while the C chord is actually C, E, G. So where does the note E come from?
This was actually discovered about 500 years or so after Pythagoras, by the musician Didymus. We notice that Pythagoras stopped at the ratio 4:3. The next natural ratio to consider is 5:4. This ratio is now known as a major third, and represents the interval from C to E. Then the full C major chord is the extended ratio 4:5:6.
Notice that there's already an E on the circle of fifths. Declaring E to be a major third above C is the same as setting four perfect fifths to equal the ratio 5:1 -- which, as we know, is impossible. Four fifths would actually be 81:16, so cross-multiplying gives us 81 = 80. The ratio of 81:80 is considered to be another comma -- Didymus's comma.
We are now ready to produce the full C major scale. We start with the three intervals of Pythagoras, the octave, perfect fourth, and perfect fifth -- C, F, G. On each of these three notes, we build a major chord: the C chord (C, E, G), F chord (F, A, C), and G chord (G, B, D). Putting these notes in order gives us the full major scale -- C, D, E, F, G, A, B, C.
I remember taking a piano class the summer after kindergarten. As it was only a beginners' course, the only major scale taught was the C major scale, which was played on the white keys. But I wondered to myself why one couldn't play a major scale beginning on notes other than C. The following December (either for my birthday or Christmas) I received a small electronic keyboard as a gift, and naturally I tried playing other scales. beginning on D, then E -- but none of them sounded like the proper major scale Do, Re, Mi, etc.
But one of these scales sounded almost right -- the scale beginning on G. The scale G, A, B, C, D, E, F, G sounded correct except for the last few notes. But I didn't know how to make the last part sound like the major scale. Disappointed, I started playing around with chromatic scales -- where I included the black keys as well as the white keys. But as I was still tantalized by the G scale, I sometimes started out by playing the part of the G scale that sounded right -- G, A, B, C, D, E -- and then switched to the chromatic scale. The black key between F and G is called F sharp, or F#, so what I played was G, A, B, C, D, E, F, F#, G, and I often played around with this "scale" for awhile.
Then one day, I accidentally skipped the F note, so what I ended up playing was G, A, B, C, D, E, F#, G. And what I played sounded exactly like the major scale that I had been seeking! And I still remember to this day how excited my seven-year-old self was to "discover" the G major scale! After this, I quickly realized that I could make all the other scales (the D scale, E scale, and so on) sound right by including some of the black keys as well as the white keys on my keyboard.
The next ratio to consider is 6:5. This interval is called a minor third. To produce a minor chord, we take the major chord C, E, G and replace the E with the black key just below E, called E flat, often rendered in ASCII as Eb. So the C minor chord is C, Eb, G.
Why do minor chords sound "sadder" than major chords? Recall that a major chord has its notes in the ratio 4:5:6. Well, a minor chord has its notes in the ratio 10:12:15 -- that is, 5:6 is now the lower ratio and 4:5 is now the higher ratio. Since 4:5:6 is simpler than 10:12:15, the 4:5:6 major chord sounds brighter than the 10:12:15 minor chord. Many popular songs on the radio nowadays tend to be about one of two topics -- falling in love and breaking up. The former songs tend to be in major keys, while the latter songs tend to be in minor keys. Over the past few decades, popular hits in minor keys have increased.
The second keyboard that I owned, when I was in the second grade, had several songs built in, including the classic Greensleeves. This song was in the key of A, but I noticed that it was based on A minor rather than A major, as it contained the chord A, C, E (whereas A major would be A, C#, E). I soon learned that there were minor scales as well as major scales, and tried to deduce what the A minor scale was based on the song. Unfortunately, the version of Greensleeves that was built into my keyboard wasn't truly in A minor, but a scale called the Dorian mode. Instead, I was under the misconception that A, B, C, D, E, F#, G, A was the A minor scale -- that is, A minor was just like G major in that it contained F# rather than F natural.
It was not until the third grade until I took private piano lessons. My third grade classroom had a piano, and the teacher was a pianist. So she taught me piano one a week after school until she left on maternity midway through the year, after which she referred me to her mother to continue my study of the piano. This was when I finally learned that the A natural minor scale was actually the notes A, B, C, D, E, F, G, A -- that is, it has no sharps or flats, just like C major. This minor scale is formed by taking the three Pythagorean intervals (octave, fourth, and fifth) and building a minor chord on each note, rather than a major chord.
Here are a few more links discussing major and minor scales:
http://music.stackexchange.com/questions/22236/why-am-i-always-sticking-to-minor-keys
which in turn links to:
http://www.nme.com/blogs/nme-blogs/the-science-of-music-why-do-songs-in-a-minor-key-sound-sad
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