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You're probably asking yourself, "How does a savings bond have anything to do with the third or fifth of a major chord"? Let's look at the savings bond example first and then tie it to musical terms.

When you purchase a US Savings Bond, you pay for one half of the face value of the bond. So, for example, if you were to purchase a $1,000 US Savings Bond, you would pay $500 for the bond. The guarantee behind the savings bond is that if you hold the bond for approximately 12 years, the savings bond will be worth the face value shown on the front of the bond. So the question then becomes, why does it take 12 years to reach the face value of the bond?

For many years the US Government has guaranteed that Savings Bonds would return an interest rate of 6% per year for the life of the bond (sometimes it's somewhat higher or somewhat lower). At this rate, it would take approximately 12 years for the original amount paid for the bond to "double" bringing the savings bond up to face value. Let's see how this works. Beginning with the amount paid for the bond, $500 in our example, and multiplying this amount by 6% per year, let's see what happens after 12 years (this would be a good time to get your calculator). Remember: 6% is the same as 0.06, so $500 x 0.06 is $30. Using the multiplier of 1.06 makes the math a little easier, and just saves a step in adding this interest amount to the original value:

................Beginning..................Ending

Year..........Value........Multiplier...Value

1..............$500.00.....1.06.........$530.00

2..............$530.00.....1.06.........$561.80

3..............$561.80.....1.06.........$595.51

4..............$595.51.....1.06.........$631.24

5..............$631.24.....1.06.........$669.11

6..............$669.11.....1.06.........$709.26

7..............$709.26.....1.06.........$751.82

8..............$751.82.....1.06.........$796.92

9..............$796.92.....1.06.........$844.74

10............$844.74.....1.06.........$895.42

11............$895.42.....1.06.........$949.15

12............$949.15.....1.06.........$1,006.10

Year..........Value........Multiplier...Value

1..............$500.00.....1.06.........$530.00

2..............$530.00.....1.06.........$561.80

3..............$561.80.....1.06.........$595.51

4..............$595.51.....1.06.........$631.24

5..............$631.24.....1.06.........$669.11

6..............$669.11.....1.06.........$709.26

7..............$709.26.....1.06.........$751.82

8..............$751.82.....1.06.........$796.92

9..............$796.92.....1.06.........$844.74

10............$844.74.....1.06.........$895.42

11............$895.42.....1.06.........$949.15

12............$949.15.....1.06.........$1,006.10

Notice that at the end of the 12 years the original $500 has now doubled (it probably reached exactly $1,000 several days before the 12 year mark). If we wanted the value of the savings bond to be exactly $1,000 at the end of the 12 years the interest rate would be just a little less than 6% (for those of you that are interested, the value is 0.0594630943592953).

Congratulations! You have now made an important discovery (and you didn't even know it)! This Savings Bond multiplier is the same multiplier that is used to develop a musical scale in Equal Temperament (the scale used for the piano). Each "year" in the above example corresponds to a half step (or semi-tone) in a chromatic scale, and instead of dollars, frequencies (in Hz) are used to represent every "pitch" in the scale. The multiplier used for an Equal Temperament scale is actually the value that is just less than 6% value (shown above) - the one that gets you to exactly $1,000 for a savings bond.

Here's something else that will ring true with your general knowledge as a musician. The

As brass players, we know that this Equal Temperament scale (or ET for short) that is used for the piano is an intonation "compromise". This is why phases like, "you need to lower the third" or "that fifth needs to be a little wide" have become common place in our language. The important bit of information that is usually omitted from the phrase "you need to lower the third" is "with respect to what?" Of course the third must be lowered with respect to ET, but how much?

Before we answer that question, lets look at something else that all brass players are very familiar with. If you have played lip flexibilities from Arban, Schlossberg, Irons, Remington, Collins, etc. you have clearly been developing the understanding needed to wrap your mind around this next section. Let's list the harmonic structure (or partials) for the open combination, along with its position in the harmonic structure.

Partial.....Note Description

1..............Pedal C

2..............Low C

3..............2nd Line G

4..............Third Space C

5..............Fourth Space E

6..............Top of the Staff G

7..............Bb above the staff

8..............High C

9..............D above High C

10............E above High C

1..............Pedal C

2..............Low C

3..............2nd Line G

4..............Third Space C

5..............Fourth Space E

6..............Top of the Staff G

7..............Bb above the staff

8..............High C

9..............D above High C

10............E above High C

Each of these harmonics or partials can be used to show the theoretical position of every note in a major or minor scale by using a ratio factor. The first thing to notice is that Pedal C is the starting point for all of the naturally occurring partials in this harmonic structure (or close enough for this example). Also notice that if you multiply the frequency of the Pedal C (the 1st Harmonic) by 2 (from our Savings Bond example), you will arrive at Low C (the 2nd Harmonic)(one octave above). Multiply by 2 again and you get to the 4th harmonic (or third space C). Multiply by 2 again and you get the 8th harmonic (or High C).

I'll just develop the two notes that we are most interested in from our original example (the third and the fifth), but I'll give you the secret to arriving at all of the pitches in a Major scale (this is called Just Intonation). Notice that in the harmonic structure shown above that from third space C to fourth space E is the interval of a Major third. Also notice that from Low C to 2nd Line G is a Perfect fifth. We've been talking about "multiplying" by 2 to move from one octave to the next. We can also multiply by numbers less than 2 and greater than 1 to get to each of the degrees of the major scale. Look at the Perfect 5th. This interval occurs between the 2nd and 3rd harmonics. If we use this ratio of 3:2 (or 1.5) as our multiplier, we can get to the frequency for a Perfect fifth by multiplying this against the frequency of the root of the scale. Similarly, look at the Major third. This interval occurs between the 4th and 5th harmonics. If we use this ratio of 5:4 (or 1.25) as our multiplier, we can get to the frequency for a Major third by multiplying it against the frequency of the root of the scale. The important thing about these ratios is that they hold true for all Perfect fifths and Major thirds.

Now, let's show why the third and the fifth pitches in a major scale need to be altered with respect to ET to be "in tune". Since everyone is familiar with

Note.......Equal Temp....ET
Multiplier....Cents

A............440.000.........1.05946..........0

(A#).......466.162.........1.05946..........100

B............493.880.........1.05946..........200

(C).........523.247.........1.05946..........300

C#.........554.359.........1.05946..........400

D............587.321.........1.05946..........500

(D#).......622.243.........1.05946..........600

E............659.242.........1.05946..........700

(F)..........698.440.................................800

Note.........Just Major..........Multiplier

A..............440.000.............(1:1) 1.00

C#...........550.000.............(5:4) 1.25

E..............660.000.............(3:2) 1.50

A............440.000.........1.05946..........0

(A#).......466.162.........1.05946..........100

B............493.880.........1.05946..........200

(C).........523.247.........1.05946..........300

C#.........554.359.........1.05946..........400

D............587.321.........1.05946..........500

(D#).......622.243.........1.05946..........600

E............659.242.........1.05946..........700

(F)..........698.440.................................800

Note.........Just Major..........Multiplier

A..............440.000.............(1:1) 1.00

C#...........550.000.............(5:4) 1.25

E..............660.000.............(3:2) 1.50

At this point it's important to know that each half step (or semi-tone) is 100 cents apart in ET. I've shown this above and the third is 400 cents above the root, and the fifth is 700 cents above the root. An octave is comprised of 1200 cents. This is just some nomenclature for ET.

To determine where the third of the scale should be located (i.e. how many cents sharp or flat) with respect to ET, we have to apply another ratio. Looking at the distance between C and C# in ET, the two pitches are separated by 31.112 Hz (554.359 - 523.247). Using Just Major intonation, the C# has a frequency of 550 Hz. This C# is lower than the ET C# that has a frequency of 554.359 Hz by 4.359 Hz. Here's where the ratio is determined. Since 4.359 Hz divided by 31.112 Hz equals 14%, we can say that the 3rd of the major scale should be 14% or 14 cents lower than ET. The ET third is 400 cents above the root while the Just Major third is only 386 cents above the root. This interval would be considered narrower than ET.

To determine where the fifth of the scale should be located (i.e. how many cents sharp or flat) with respect to ET is similar to the example above for the third. Looking at the distance between E and F in ET, the two pitches are separated by 39.198 Hz (698.440 - 659.242). Using Just Major intonation, the E has a frequency of 660 Hz. This E is higher than the ET E that has a frequency of 659.242 Hz by 0.758 Hz. Here's where the ratio is determined. Since 0.758 Hz divided by 39.198 Hz is 1.9% (or approximately 2%), we can say that the 5th of the major scale should be 2% or 2 cents higher than ET. The ET fifth is 700 cents above the root while the Just Major fifth is 702 cents above the root. This interval would be considered wider than ET.

If you made it this far, Congratulations! You should receive continuing education units or maybe a certificate with a gold seal! However, since you've come this far, there's one other compelling reason why Just Intonation is more "in-tune" than ET. It has to do with resultant tones. The resultant tone is just the frequency difference between two notes that are played simultaneously.

In the example above, lets calculate the resultant tones for a Major triad in the key of A using Just Major Intonation. When you play an A and a C# together the resultant tone is 110 Hz (550 Hz - 440 Hz). When you play an A and an E together the resultant tone is 220 Hz (660 Hz - 440 Hz). Now if you multiply 220 Hz by 2 (remember the savings bond example), you know that this is an A. Similarly, the 110 Hz resultant is also an A, but an octave below the other resultant. By playing the third of the chord 14 cents low, and the fifth of the chord 2 cents high, the resultants sounding align perfectly to enhance the quality of the Major triad in just intonation. I think you can see that the resultants for ET would be less than desirable and would clash with original

Finally, I have to credit the paper by Christopher Leuba "A Study of Musical Intonation" for all of the musical frequency relationships I have cited above. I just chose to say it in another way.

Of course this extremely long example would be the information for those of you that like to get "under the hood". What about those of you that just want to drive the car?

I'm working on some tables that will show the relationship between all of the intervals in every key. This, of course, would be extremely important when working out the intonation of difficult intervals in the standard orchestral literature (for instance D-F in C major), duets, Concertos for two trumpets, etc. I'm sure that working with a drone pitch and practicing to hear these resultants will bring a heightened sense of intonation, and I know that a product called Tune-Up Boot Camp is available to work on these intonation concepts.

I put this together as a resource for my trumpet instructor who has the Leuba paper but got lost in the details. I'm hoping that this example will lead to answers for the "rest" of us.

The Christopher Leuba paper "A Study of Musical Intonation" is available from the author at: 4800 NE 70th, Seattle, Washington 98115.

I hope that this glimpse of how theory can be related to actual playing will give those of you who choose to invest the time a better sense of intonation. If nothing else, you can say you've learned something!

Thanks,

Derek Reaban

Tempe, Arizona

This text was first submitted to TPIN. Later it appeared on Trumpet Herald