A440, Savings Bonds, and Symphony Trumpet Players

[Derek Reaban]

This is a post that I submitted to TPIN about a year ago. If you haven't seen it before, I think you will enjoy it as it answers two very common questions related to intonation.


A440, Savings Bonds, and Symphony Trumpet Players

Have you ever wondered why the third of a Major chord is supposed to be played low? Or the fifth of a Major chord is supposed to be played high? Well, if you've ever purchased a Savings Bond and have played lip flexibility exercises, I can explain why these statements are true.

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

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 A440 (in Hz) that is used to tune an orchestra can be "doubled" to sound the A, exactly one octave above. This High A has a frequency of 880 Hz. The A an octave below the tuning A has a frequency of 220 Hz. So any time that you multiply a frequency by 2, you are raising the pitch exactly one octave. Multiplying by 4 raises the pitch by two octaves. From the savings bond example above, this doubling makes sense.

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

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 A440, let's use A Major as the key for this example, with 440 Hz as the frequency for the root of the scale. Let's build our table of frequencies (similar to the Savings Bond example) for the ET scale [using that multiplier of 1.05946 (just less than 1.06)] and show the true position of the 3rd and the 5th using the ratios described above:

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


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 A440. You can do the math if you are interested (it's not too difficult).

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 Message was edited by: Derek Reaban on 2003-07-11 12:39 ]

[ This Message was edited by: Derek Reaban on 2003-07-11 15:24 ]

[Derek Reaban]

This message reads much more easily when the numbers in the columns aren't all bunched togther. I removed the tabs and put in spaces, but it still doesn't look right. Any suggestions on how to fix this?
_________________
Derek Reaban
Tempe, Arizona
Tempe Winds / Symphony of the Southwest

[dbacon]

Welcome to the Herald, Derek!

Dave Bacon

[Derek Reaban]

Hey Dave! Thanks. I've done some occasional lurking here, but I thought it was time to post a few things. I've certainly enjoyed many of the different contributors at this site, and thought I'd get involved.

[_Don Herman]

Hi Derek -- great to see you here on TH! Another guy who can add (although I have to use a calculator... ).

As for the spaces, ain't no fix I know. I asked a long time ago, and apparently extra spaces are stripped by the BBS engine, so we're hosed (not hosaphoned, just hosed). You could use underscores _ or periods ... to keep even spacing in tablature.

Welcome! - Don
_________________
Don Herman/Monument, CO
"After silence, that which best expresses the inexpressible, is music." - Aldous Huxley

[Derek Reaban]

Don,

Hello! It's nice to be here. Thanks for your suggestion in modifying the columns in the above posting. It isn't the perfect solution, but it's certainly much more readable now. Looking forward to learning and contributing!

_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2003-07-11 15:37 ]

[fuzzyjon79]

I didn't read your post because it was so long... but I'll take your word for it... I trust you LOL
_________________
J. Fowler
"It takes a big ole' sack of flour, to make a big ole' pan of biscuits!"

[Jarrett Ellis]

I got about 3/4 a way down it before I stopped.. man Im lazy.
-Jarrett
_________________
Eclipse Medium Bell Scratch Gold
Bach 37H Gold Plate
Bach 3c

[rjzeller]

Who cares about the spaces...that's a great post. I will certainly be mindful of that in the future as I continue my eternal quest to further my education and skill on the one instrument it seems has more methods and opinions than it has players.

[Derek Reaban]

As a follow-up to this post, for those wondering why this information is important, you might want to consider this story. I was playing 2nd trumpet for Mahler Symphony No. 2, and during a sectional many years ago, we were rehearsing the IVth movement (Urlich) chorale. There were six of us at this sectional, and we alternated playing the different parts for the chorale to begin locking in our intonation (three parts I believe).

The first time we played through this chorale, I discovered that even though my part remained on a printed low Ab for several bars, the other parts changed around me, forcing my Ab to need to be lowered considerably! What a realization! We had to play this section several times before I discovered how low that Ab had to go (I was the first to really impact the section intonation that had been literally flawless). I think we were all amazed at how much I had to move that note.

I believe that the Ab was the minor third in the first triad and then became the fifth of the next chord, and then changed again (it's been a long time since I've played the part though, and I really can't remember the details that clearly). However, with a working knowledge of which direction I should have moved that "static" note based on the quality of the chord, I would have been much more capable of maintaining that "flawless" intonation that we had been enjoying during the rehearsal.

Years of wondering why I had to fix the only note that remained the same in this chorale led me to write this little essay (after discovering the Chris Leuba paper and trying to understand what he was telling me). As far as I know, this essay is about the most user friendly approach to this topic that I have seen.

For me, understanding this information, and then applying this knowledge through diligent practice (drone pitches) has allowed me to play in tune in a section with much greater consistency.

I seem to remember a quote from the recording session of the Antiphonal Music of Gabrieli album that impressed upon me the importance of this concept. I think the players from the Chicago Symphony and the Philadelphia Orchestra were so impressed with how in tune the players were from the Cleveland Orchestra, that they asked them what their secret was. Their response was, "We play just as out of tune as everybody else. We just fix it faster!".
_________________
Derek Reaban
Tempe, Arizona
Tempe Winds / Symphony of the Southwest

[Jarrett Ellis]

Haha, thats a good point...
-J
_________________
Eclipse Medium Bell Scratch Gold
Bach 37H Gold Plate
Bach 3c

[Derek Reaban]

It has been a long time since I have thought about the Mahler 2 story, and I realized that I’ve never really gone back to fully understand the intonation challenges in this chorale (since I wrote my post on A440, Savings Bond, and Symphony Trumpet Players). This may interest some of you, but I’m really putting it together for myself to finish my thinking on this topic.

I found all the individual trumpet parts and have spelled out the notes and chords that are important for this example (since music theory isn’t my strong point, maybe someone like Eric Bolvin could help me with nomenclature if it is in error):

1st Tpt:.....Db........C........C.........Db.........Bb..........Bb.........Ab
2nd Tpt:....Bb........Ab.......Ab.......Ab.........Ab..........G
3rd Tpt:.....F..........F.........F..........F............Eb

................Bbmin...Fmin....Fmin....DbMaj.....Ebsus.....EbMaj....AbMaj

Now let’s look at that Ab in the 2nd trumpet part. Ab is the minor third in a Fmin triad. We haven’t discussed minor thirds in the above post, but the intonation tendencies can be derived from the major 3rd and perfect 5th. The relationship between the major 3rd and perfect 5th is always a minor third (think about an E and G in a C major triad). So if a major third is 14 cents lower than ET and a perfect fifth is 2 cents higher than ET, a minor third must be 16 cents higher than ET in Just Intonation. The math is 702 cents – 386 cents = 316 cents for Just Intonation compared with 700 cents – 400 cents = 300 cents for ET.

In the DbMaj chord Ab is the 5th so it must be 2 cents higher than ET. This means that the Ab needs to be lowered by 14 cents from the previous chord when the Ab was acting as a minor third. No wonder I had to lower this pitch so much to keep it in tune with the rest of the section!

Continuing with this exercise, the Ab becomes the 4th of the Ebsus chord. We haven’t developed the intonation tendency for a perfect 4th, but it’s easy to derive from the perfect 5th and the octave. 1200 cents – 702 cents = 498 cents. So a perfect 4th needs to be 2 cents lower than ET (i.e. 500 cents – 498 cents = 2 cents). This Ab needs to come down another 4 cents from when it was the 5th in the DbMaj chord. That’s 18 cents of intonation movement from where it began in the Fmin chord!

Now for the details of why I had such a difficult time figuring this out during the sectional. The parts are printed for Trumpet in F, so that Ab is really a Db when played on C trumpet. I don’t know about you, but I always extend my third valve slide when playing that note. When I was playing this sectional, I didn’t know what the intonation tendency was for a minor third. In fact, I think all I really knew at that point was that thirds should be LOW and fifths should be HIGH. In this example, the third (minor) had to be HIGH and the fifth had to be 14 cents LOWER than the third. Talk about blowing my mind!

Now that I’ve had time to contemplate this example, I can really see why I was doomed to maintain great section intonation without a working knowledge of the theory in combination with studying the parts (score).

I have an audition coming up in September, and if I’m fortunate enough to advance to the finals, I know that there will be some section playing required. I can see why a chorale passage like Mahler 2 would be used to evaluate the intonation capabilities of a section player. On the audition that I’m playing, Bartok Concerto for Orchestra is on the list. There is a beautiful chorale in the 2nd movement that I know has many challenging intonation tendencies for the 2nd trumpet. After thinking through the Mahler 2 example, I’m certainly going to study the Bartok part so that I know it backwards and forwards. This could mean the difference between being selected for the position versus being given comments on how to improve for next time.

If anyone has the chords for that section in the Bartok, I would love to see them so I can figure out how the 2nd trumpet part fits in.

I hope this was of interest to some of you. It certainly cleared up why I failed in playing this part so miserably the first time I saw it many years ago!


_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2003-07-14 14:34 ]

[HorneyMikey]

Re: Savings Bonds and trumpet players.


What is the difference between a Savings Bond and a professional trumpet player?

The Savings Bond will eventually mature and make money...



Ba-Dum; Crash

[LaserBoy]

I love the savings bond explanation! What a great way to introduce the topic! If I could add my two cents, that special number 1.059463... is the twelfth root of 2. In case anyone was dying of curiosity.

Nerd Power!
--LaserBoy
_________________
Do not look into laser with remaining eye!

[LaserBoy]

I love the savings bond explanation! What a great way to introduce the topic! If I could add my two cents, that special number 1.059463... is the twelfth root of 2. In case anyone was dying of curiosity.

Nerd Power!
--LaserBoy
_________________
Do not look into laser with remaining eye!

[vic]

Derek, it dawned on me that pianos (and guitars pretty much) don't have the opportunity to make pitch adjustments to get the intervals right. If I understand you, the piano is tuned according to Equal Temperament, which means this compromise is not what we want to hear, correct? If the piano were to play your Mahler triads, it seems to me they would not come out as the ideal trumpet section would play them.
Is the other approach, Just Intonation, utilized by the string sections, too? It just is puzzling to me that pianos, organs, and guitars have to just keep pressing on "as is".


[ This Message was edited by: vic on 2003-07-21 17:43 ]

[Derek Reaban]

Vic,

Equal Temperament is certainly a compromise. I believe that someone more knowledgeable than me could provide some additional discussion. I remember reading once that a piano was tuned to Just Intonation for a certain key (let’s say C major for example). The related keys (F and G) had only one “bad” note each in them. The other keys had what were known as “wolf” tones. The spacing of the intervals for Just Intonation on a keyboard instrument allows the instrument to only play in one key really well, 2 keys with minor problems, and the rest are just really terrible. The spacing of the scale for Just Intonation corrupts the proper spacing for the other scales.

String players certainly strive for playing with Just Intonation just as we do. I love to play in small ensembles (brass quintet) because it’s possible to really focus on great intonation! There are relationships that need to be considered when changing keys in an ensemble when playing with Just Intonation. Steve Colley discusses this in his TuneUp program.

If you have a small metronome with the A440 pitch, try setting it to the tone generator and then hold it up close to your ear. Play an E above that A. You should be able to find a ringing resultant tone (an “A” 1 octave down). Now try playing a C#. You will find the A 2 octaves down. Your ear will naturally guide you to where this note is most clearly in tune (the resultant gives some immediate feedback telling you how close you are). Now let that C# float sharp. The C# is 14 cents higher than Just Intonation in ET (the piano scale). You will start hearing lot’s of weird things happen to that great resultant tone. This is the compromise that ET sets up in a piano. Of course we’ve all gotten used to the sound of ET, so hearing a great brass section playing in Just Intonation is really a powerful experience!


_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2003-07-23 12:10 ]

[Derek Reaban]

I had a lesson yesterday and we were working on the intonation for the Carmen Prelude. There is a section towards the end that spells out an F diminished triad (F, G#, B). My instructor wanted to be sure that this line was in tune with itself, so we began working to assure that the minor third was an appropriately "wide" interval for the F-G# and the G#-B. He would maintain the starting pitch and then I would play the line to arrive on the next note (i.e. from F to G#). I knew what resultants to listen for based on the studying that I have done.

This was the really neat part of the lesson, though. He asked me what we should be listening for when we got to the tri-tone (i.e. F-B). I said that I had no idea, but then I stopped and thought about a fantastic article that I had read at Ole's site written by Roger McDuffie. This article flashed back in my mind in vivid detail. I told him that we could figure it out very easily. Then I told him that if you think about the harmonic series, the tri-tone exists between E and Bb for the open scale. Since Bb is the 7th partial and E is the 5th partial, the resultant should be the 2nd partial (i.e. 7-5) which is the low C. Since we were working with the F-B tritone, I said, our resultant should be Db.

That was really cool! I didn't know that I knew that answer, but sometimes I surprise myself. I'm really impressed how easy it is to figure out the resultants "immediately" as opposed to going home to figure it out, based on this simple subtraction concept in this article.

Check out Roger's article at:

http://www-it.hive.no/oj/musikk/trompet/tpin/pitch_on_horns.html

Enjoy!


EDIT: Add reference to 10 Question Music Quiz
_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2004-07-13 16:38 ]

[Derek Reaban]

I've been having some conversations with TPINer Roger McDuffie about intonation and he gave me this paragraph to consider:

Consider this. In an ensemble that goes from an F major chord to an Ab major chord, (1) should a person holding the note C throughout lower his pitch a lot for the second chord, (2) should everybody else play their pitches appropriately higher, or (3) should/would there be a compromise? I think the real world solution is that everybody listens and everybody reacts, thus making everybody (especially the person holding the C) adjust to some extent.

I would agree with point (3). However, it seems to me that there are really two different aspects of intonation that must occur simultaneously. The first component of just intonation involves the "section" being in tune with each other and understanding the different relationships for the intervals in the chords that they are playing. The second component is the intonation of the melody line that must be in tune with itself.

There is a great story about a guest conductor working with the LA Philharmonic. He isolated a line in the trombone section, and listened to the players very critically (the section intonation was flawless). After several moments of consideration, he said, "Gentlemen, the melody was out of tune. Please try it again." This time the relationships in the melody line were played so that it had intonation integrity by itself and the rest of the section adapted accordingly. The result was stunning, and this "fine tuning" made all the difference between good and great.

I had forgotten about this story and thought is fit nicely with this topic!

I am making progress on analyzing the chords in the 2nd Movement to Bartok's Concerto for Orchestra. When I have completed that project, I will post that information here (Hopefully in the next week or so).



_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2003-07-28 12:06 ]

[Derek Reaban]

After some searching, I was able to borrow a score for Bartok’s Concerto for Orchestra. While I have not performed this piece with an orchestra, I have worked up both the 1st and 2nd trumpet parts for different auditions, and have listened to this work extensively (Chicago and Montreal recordings). The 2nd trumpet part in the chorale in the 2nd movement (beginning in Bar 123) has always proven challenging for me when I play the part with CDs. While I have always been extremely close in aligning the intonation for this part, there are always at least two places that I feel are just slightly off the mark.

Well, after doing this analysis and then playing through the part with the Montreal recording, all I have to say is, “Holy Cow Mackerel! This stuff really works!” For the first time in my life I feel like I have the magic combination to make this part really work!

I had to mark my part to provide myself with a mental queue on how each note “fits” intonation wise along with the understanding of exactly how “low” or “high” each note needs to be. After you read through this, you’ll be able to mark your part to arrive at the same results that I have experienced.

This chorale is orchestrated for 2 trumpets, 2 trombones, and tuba. I have provided the individual notes for each part and the associated chords that these notes spell. For nomenclature, a C Major chord with a Major 7th would be notated CM7, and a c# minor chord with a minor 7th would be notated c#m7. I have also shown the degree of the chord for each of the notes in the 2nd trumpet part.


BAR 123
Number..1..........2.........3...........4.........5..........6..........7
Tpt1.......F#........E.........F#........G#.......A..........E..........F#
Tpt2.......D#.......C#......D#........D#.......E..........C#.......D#
Tbn1......B..........G#......B..........B..........C#........A.........B
Tbn2......F#...................F#........D#........C#.......G#.......F#
Tuba......B..........C#......B..........G#........F#........C#.......B
Chord....B..........c#.......B..........g#........f#M7.....AM7......B

2nd Tpt..M3rd......R........M3rd.....P5th......M7th.....M3rd....M3rd
Degree
Of Chord


BAR 129
Number..8.........9........10.......11........12..............13..........14..........15
Tpt1.......G#.......A#.....B.........C.........B...............A#.........G#..........G#
Tpt2.......D#.......F#......F#.......E.........E...............C#..........D#..........D#
Tbn1.......B........C#......B.........G........G...............F#..........C#...........B
Tbn2.......D#......C#.....D.........C.........E...............D#..........D#..........D#
Tuba.......G#......F#......G.........C........C#.............D#..........G#..........G#
Chord.....g#.......F#......GM7.....C........C#dimM7....d#m7.....G#sus......G#

2nd Tpt...P5th.....R........M7th....M3rd....m3rd..........m7th......P5th.........P5th
Degree
Of Chord


BAR 135
Number..16..........17.......18.......19......20.........21..........22
Tpt1........C#.........B........A#......G#.....F#........G#.........E
Tpt2........E...........D#......C#......B........D#.......D#.........C#
Tbn1.......G#.........G#......F#.......B........B.........B...........G#
Tbn2.......C#.........C#......C#......E........F#........E............E
Tuba.......C#.........E.........F#......G#......B.........E.............A
Chord.....c#..........c#M9...F#......E.........B.........EM7........AM7

2nd Tpt....m3rd.....M9th....P5th....P5th....M3rd.....m3rd.....M3rd
Degree
Of Chord


Number....23.........24.........25
Tpt1.........F#.........G#.........A
Tpt2.........E...........E............E
Tbn1........C#........C#.........C#
Tbn2........E...........E............E
Tuba........A...........G#........G#
Chord......f#m7.....c#.........AM7

2nd Tpt....m7th......m3rd.....P5th
Degree
Of Chord


BAR 141
Number....26.........27.........28.........29
Tpt1.........G#........G#........F#.........F#
Tpt2.........E...........D#........C#........C#
Tbn1........B...........B..........C#.........C#
Tbn2........E...........G#........A#.........A#
Tuba........C#........E............F#.........F#
Chord......c#M7.....EM7........F#.........F#

2nd Tpt....m3rd.....M7th.......P5th.......P5th
Degree
Of Chord


Now to make sense of where to put each note (i.e. low or high with respect to equal temperament), this intonation key for every interval is very important. I have derived some of these relationships in the above posts, but this table is just very handy to have.

m2nd.....12 cents wide (i.e. larger than ET)
M2nd......4 cents wide
m3rd.....16 cents wide
M3rd.....14 cents narrow (i.e. smaller than ET)
P4th......2 cents narrow
P5th......2 cents wide
m6th.....14 cents wide
M6th.....16 cents narrow
m7th.....18 cents wide
M7th.....12 cents narrow

Make sure you start the chorale with the first D# being low enough. Now look at the 3rd and 4th chords. The D# remains the same in the 2nd trumpet part, but it moves from being the M3rd in the B Maj chord to being the P5th in the g# minor chord, so it has to go from 14 cents lower than ET to 2 cents higher than ET. The same thing happens between the 7th and 8th chords. So kick the slide out quite a bit for the major 3rd and pull it in for the P5th.

Now look at the E in the 11th and 12th chords. It acts as the Major 3rd and then as the minor 3rd. The M3rd needs to be 14 cents lower than ET and the m3rd needs to be 16 cents higher than ET. That’s 30 cents of movement to be in alignment for just intonation (50 cents is a quarter step)! This was one of the problem areas that I just couldn’t get right. Kick the 1st slide out and then pull it in all the way to make the E align properly with each chord.

The D# in the 20th and 21st chords acts as the Major 3rd and then the Major 7th. The M3rd is 14 cents lower than ET and the M7th is 12 cents lower than ET, so there is hardly any movement once you are low enough for the 20th chord.

Finally, the E in chords 23, 24, 25, and 26 was my other big problem area. In chord 23 the E is the m7th requiring it to be 18 cents higher than ET. In chord 24 the E is the m3rd requiring it to be 16 cents higher than ET (requiring little change once the E in chord 23 is high enough). The E then becomes the P5th requiring it to be 2 cents higher than ET for chord 25. And then the E becomes the m3rd again in chord 26 (16 cents higher than ET). So the arrows to mark in your part are up, up, down, up for this one note! And it really works!

I hope you’ve enjoyed reading about this little intonation exercise. It’s amazing how much work I had to do to get this chorale to be perfect, but I’m so glad that I did. If I’m fortunate enough to play this in the final round of my upcoming audition, I know that I’m going to nail it! I’m glad that I can share it with the group!


_________________
Derek Reaban
Tempe, Arizona