10 Question Music Quiz! Good Luck!

[Derek Reaban]

Imagine that you are sitting on stage in the most acoustically perfect hall in the world and sitting beside your favorite orchestral player. It’s just the two of you on stage. You are playing duets or the excerpts of your choice. Your playing has never been better, and your sound is matching perfectly with the other player. Intonation is flawless and the resonance and vibrancy in your sounds is ringing through the hall.

You observe a very strong buzzing sound in your head as the various intervals are locking in (the resultant tones) while you are playing. They are literally strong enough that you could picture a third player on stage (behind you) playing these notes.

Resultant Tones Quiz

You are playing the second trumpet part to Pictures at an Exhibition. Your part enters as the 1st part goes up to the Ab and you play a constant second space Ab as the first part descends. What are the resultant tones that you are hearing?

Interval One

1st Tpt: High Ab
2nd Tpt: 2nd Space Ab
Interval: Octave
Resultant Tone?

Interval Two

1st Tpt: 5th Line F
2nd Tpt: 2nd Space Ab
Interval: Major 6th
Resultant Tone?

Interval Three

1st Tpt: 4th Space Eb
2nd Tpt: 2nd Space Ab
Interval Perfect 5th
Resultant Tone?

Interval Four

1st Tpt: 3rd Line Db
2nd Tpt: 2nd Space Ab
Interval: Perfect 4th
Resultant Tone?

Interval Five

1st Tpt: 2nd Space Ab
2nd Tpt: 2nd Space Ab
Interval: Unison
Resultant Tone?


Take no more than one minute to determine each of these resultant tones (five minutes total). If you don’t know, then move on to the next question.

EXTRA CREDIT

Interval Six

1st Tpt: High A
2nd Tpt: 3rd Space C#
Interval: Minor 6th
Resultant Tone?

Interval Seven

1st Tpt: 5th Line F
2nd Tpt: 4th Line D
Interval: Minor 3rd
Resultant Tone?

Interval Eight

1st Tpt: 4th Space E
2nd Tpt: 3rd Space C
Interval: Major 3rd
Resultant Tone?

Interval Nine

1st Tpt: High A
2nd Tpt: 1st Space F
Interval: Major 10th
Resultant Tone?

Interval Ten

1st Tpt: High Bb
2nd Tpt: 3rd Space C
Interval: Minor 7th
Resultant Tone?

Take no more than one minute to determine each of these resultant tones (5 minutes total).

Give a brief description of how you arrived at your answers.


After enough people have taken this quiz, I will post the answers. I will also provide you with a method to determine the answer that will take AT MOST 10 seconds per interval (any interval) to figure out. In this way you can apply this knowledge the next time you find yourself in this situation!


P.S. I would have gotten, at best, 20-30 percent on this test before discovering this simple little trick, so don’t feel bad if you haven’t gone through this before.
_________________
Derek Reaban
Tempe, Arizona
Tempe Winds / Symphony of the Southwest

[308WIN]

A little spare time on your hands, eh?


Rich

[scarface]

Are those resultant tones audible while one is playing in the concert hall?

[Derek Reaban]

scarface,

Many people can hear resultant tones. For a very small percentage of the poplulation, resultant tones are not audible. These "tones" are actually generated by the mind, filling in the "missing pieces", and creating something that "should" be there. There is a book about this pyscho-acoustic phenomenon by Arthur Benade (I have yet to read the book though).

I've written my impressions about resultant tones in the past. It goes along these lines. Two players are playing exactly in tune.

Scenario One
Both student players (little center to their sounds) on student horns. Resultants are faint, if heard at all.

Scenario Two
A professional player (great center and vibrancy) and a student. Resultant is present, and the student wonders what that buzzing is.

Scenario Three
Two accomplished adult players on professional horns (both with similar vibrant sounds). The resultants and now very present and the pitch of the resultants can be observed.

Scenario Four
Two top-flight players (maximum resonant sounds) on custom horns. Resultants are so strong that an audience full of people hears the resultants so clearly that they are "pulling on their ears".


That's quite a spectrum! This visceral experience is quite amazing and while I will obviously credit the players behind the horns with the majority of producing this sound, the complementary horns also played a part. I was very impressed with the overall experience (the ITG Monette Masterclass with Manny Laureano and Mike Thompson)!

So if you haven't experienced resultant tones while playing with another player, it could be that your sounds are still developing and don't have the full complement of overtones to ring the resultants with sufficient intensity to be heard. Or it could be that you are one of the small percentage of people that can't hear them.
_________________
Derek Reaban
Tempe, Arizona
Tempe Winds / Symphony of the Southwest

[Derek Reaban]

To provide you with the key to quickly and easily find the resultant tone generated by playing two notes simultaneously, you will need a piece of paper to help you better visualize this technique.

Draw two 5-line staves, and label them with a treble clef on the top staff and a bass clef on the bottom staff. You will be drawing in 8 notes: 2nd space C (bass clef), middle C (the ledger line between the two staves), 2nd line G, 3rd space C, 4th space E, top of the staff G, Bb just above the 1st ledger line, High C (2 ledger lines above the staff). Once those are in place you will label them 1 through 8.

In Trumpet Herald format it would be (looks much better on a proper staff):

8 – High C
7 – Bb
6 – G (top of the staff)
5 – 4th Space E
4 – 3rd Space C
3 – 2nd Line G
2 – Low C
1 – Pedal C

Now, visually find the intervals called out in the quiz.

Octave: Between 1 and 2 (C and C), 2 and 4, 4 and 8
Minor 7th: Between 4 and 7 (C and Bb)
Major 6th: Between 3 and 5 (G and E)
Minor 6th: Between 5 and 8 (E and C)
Perfect 5th: Between 2 and 3 (C and G), 4 and 6
Aug. 4th / Dim 5th: Between 5 and 7 (E and Bb)
Perfect 4th: Between 3 and 4 (G and C), 6 and 8
Major 3rd: Between 4 and 5 (C and E)
Minor 3rd: Between 5 and 6 (E and G)
Major 10th: Between 2 and 5 (C and E)

To find the associated resultant for each interval, simply subtract the two numbers from the partial series. For a Major 3rd (5-4) and a Minor 3rd (6-5) the answer is one. This literally means that if you play these notes in the partial series for C major, you will sound the Pedal C (1 on the partial chart). That’s 2 octaves below the bottom note of the Major 3rd and 2 octaves and a fifth below the top note of the Minor 3rd.

Now, the powerful aspect of this is being able to generalize this information quickly (while you are looking at the interval on the page). Let’s say you and your duet partner are playing a Major 3rd between 1st space F and 2nd space A. Think about the above chart (you really have to visualize to make this work quickly). You know that the resultant will be an F, 2 octaves below the bottom note.

So, all the generalized answers would be:

Octave: 2-1 = 1, 4-2 = 2, and 8-4 = 4. The resultant will be the same as the bottom note
Minor 7th: 7-4 = 3. The resultant will be a Perfect 4th below the bottom note
Major 6th: 5-3 = 2. The resultant will be a Perfect 5th below the bottom note
Minor 6th: 8-5 = 3. The resultant will be an Octave and a 4th below the top note
Perfect 5th: 3-2 = 1 and 6-4 = 2. The resultant will be an Octave below the bottom note
Aug. 4th / Dim 5th: 7-5 = 2. The resultant will be an Octave and a 3rd below the bottom note
Perfect 4th: 4-3 = 1 and 8-6 = 2. The resultant will be 2 Octaves below the top note
Major 3rd: 5-4 = 1. The resultant will be 2 octaves below the bottom note
Minor 3rd: 6-5 = 1. The resultant will be 2 octaves and a 5th below the top note.
Major 10th: 5-2 = 3. The resultant will be a 5th above the bottom note.


ANSWERS

Question 1 - 2nd Space Ab
Question 2 – Low Db
Question 3 – Low Ab
Question 4 – Pedal Db
Question 5 – None (by definition, Unisons have a resultant tone of Zero)

Extra Credit

Question 6 – 1st Line E
Question 7 – Pedal Bb
Question 8 – Pedal C
Question 9 – 3rd Space C
Question 10 – Second Line G

A+ = 5 – 10 Correct
A = 4 Correct
B = 1-3 Correct
C = 0 Correct (the average player has probably never even heard of resultant tones)

The point of this is to give you a tool to use when you are playing excerpts or duets with another player. It is amazing what happens when you are already playing well in tune and you know what resultant you should be hearing. It will give you an added dimension to consider when working with another player.

For more discussion and some simple math go to the following post about my experience at the ITG conference and scroll down to the 3rd post.

Thanks to Roger McDuffie for helping me to understand this process to arrive at resultant tones so quickly.



_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2004-07-01 16:46 ]

[hazmat]

Are you sure the resultant tones are just in your mind? I would think that since it's an actual buzzing, and inexperienced players and musicians can hear them then they are an actual physical response. Maybe a resonance somewhere in the head, but not just made up in the mind.

[Derek Reaban]

Matt,

I hear resultant tones VERY clearly. I have had discussions online with different players describing my vivid experiences hearing resultant tones (which are highly dependent on the vibrancy, or ringing quality of the player’s sound sitting beside me). When a poster online has not heard resultant tones for themselves, they then have questioned my personal experience and occasionally the term psycho-acoustic phenomenon is introduced to the discussion (Benade). Are they real or simply imagined?

Well, that’s why I provided the different scenarios related to resultant tones with student to professional players and student to custom horns in the fourth post in this folder. This clearly makes a difference in the perceived amplitude of the resultant tones that both players are hearing (at least in my experience).

They are REAL to me.

I looked briefly on the Internet to see what I could find related to Resultant Tones and psycho-acoustic phenomenon. There were lots of web sites! In addition to Resultant Tones they are known as difference tones, combination tones, and a related term that I hadn’t heard of before called Tartini Tones.

There is a great article called The Power of Hearing by Thomas Duke that briefly mentions Tartini. I have excerpted the following quote (it’s almost at the bottom of the page just above the heading Music to our ears):

[scarface]

I've heard them often as well, but usually in a confined space or a duet setting as Derek's scenario lays out. To rephrase, I wonder how often one would notice these in a concert, and if a pro would use this awareness to refine tuning on-the-fly. For example, add the full orchestra in Pictures...if I were sitting next to ___________, I'd probably be more concerned with laying down a solid foundation. I suppose resultant tones would be icing, aurally speaking - the result of that "BAM!" kind of tuning that no one has to talk about, it just happens.

Some people call it the "thing" - it's either there or not, and you can quickly tell if the person next to you has it. No bending, or adjusting, or looking at the pencil marks on each other's slides, just...KAPOW...like in the old-school comics when someone gets served, oh yeah...

[scarface]

I should point out that fine section players will make that happen for the principal from downbeat to applause; there's no substitute for a great wingman.

[PC]

Hi Derek,

Thanks for your simple rule, I know I will be looking for the right notes now, when playing chords!

I had the priviledge to be introduced early on to this search for the third note by my teacher Ken Cox (former principal of the IPO). We would play duets and stop as often as needed until this clearly audible buzz would happen. I am very thankful for that, as it has ingrained in me a high sense of pitch and just tuning, which is what makes the difference for who will get to play in a decent ensemble!

As for the reality of resultant tones, it kind of reminds me Linear Algebra where any linear combination of solutions is also a solution to a given problem (or something to that effect!). So I would speculate that given 2 waveforms which are correctly related and sufficiently pure, meaning they are linearly related in frequency (a ratio of 2, 3 etc. and not 1.58 or sqrt(3)) and do not comprise too many unrelated harmonics, one would sample them at a given point (for example your ear's membrane) as different linear combinations.

A related experience is also trying to hear as many distinct overtones as possible when playing a long tone (alone). Here also, our ear is in effect taking a Fourier transform of the incominf sound wave, and if the components are in the right proportion and the wave not too jiggly, you can hear more than a few overtones.

Great thread,
Pierre.

[JackD]

I have experienced this before, but I have a question: are 'resultant tones' the result of a combination of overtones in the trumpet's sound?

For example - would this work with a piano playing these notes?

I have heard very clear overtones in clarinets before, so that it sounds as if the instrument is playing two notes. Specifically, I discovered this when listening to the solo clarinet movement in Messiaen's 'Quartet for the End of Time'.

Is this the same thing?

[Derek Reaban]

Jack,

To answer your easiest question first, the piano does not generate "pleasing" resultant tones because it is tuned to Equal Temperament (the notes are all the same, compromised, distance apart). You must be playing instruments capable of using Just Intonation to achieve pure resultant tones (the piano has them, but they’re not in alignment with the chord). I’ve written about this in detail in the post A440, Savings Bonds, and Symphony Trumpet Players.

Your next question is a little more complex. “Are 'resultant tones' the result of a combination of overtones in the trumpet's sound?” I hope others with more knowledge on this topic will contribute, but I believe that the overtone structure of the trumpet simply contributes to the magnitude of the resultant tone that you are hearing. If you are playing a Major 3rd, the math will always tell you that you should be hearing two octaves below the bottom note being played. However, the magnitude or loudness of this resultant tone is highly dependent on the vibrancy or resonance in your sound and that of the other player. In one of the above posts I mention this in the Four separate scenarios with pros and students. The more vibrant your sound, the better chance that you will have in hearing strong resultant tones because the signal that you are working with is just plain and simple: STRONGER!

I think hearing specific overtones in a single player (your clarinet example) is a testament to the ability of that player to really “ring his or her bell off”. That vibrancy or resonance is the carrying power that the best players develop when they are truly in line with where the instrument wants to play (at the resonant center for each note). Any two players or singers can experience resultant tones when they are perfectly in tune (just intonation).

Getting back to your question about overtones in the trumpets sound and hearing resultant tones, I wonder if it’s easier to hear resultant tones when the overtones of the instruments are the same? Would the resultants generated between two fine clarinet players and two fine trumpet players have the same magnitude? What if these players played the same interval with one trumpet and one clarinet? Would the resultant tone be softer? Balance is important in generating resultants, so I’m assuming that the balance is perfect, and the only thing that differs is the overtone structure of the instruments.


Interesting questions!


Pierre,

I’m glad you enjoyed the post! It’s nice to have this “simple rule” to figure out what we are hearing VERY quickly. Sounds like your time with Ken Cox was extremely beneficial to your playing. What a joy to get to sit beside a very fine player and soak in their sound and have them expose you to new and exciting concepts!

I can’t really hear the overtones in a single trumpet sound, but I hear them very clearly in bells. I remember reading a post from Eddie Lewis saying that the sound that we “hear” when a bell is sounded is just the resultant tone, and is not even part of the actual sound being generated (that pycho-acoustic component).

Good stuff!



_________________
Derek Reaban
Tempe, Arizona

[ This Message was edited by: Derek Reaban on 2004-08-04 14:21 ]

[_dcstep]

The tenor player in my rock/funk/soul band and I have been having some amazing revelations with resultant tones. I used to play a Yamaha trumpet and he played a Yamaha tenor and we never heard resultants. Later he went back to Selmer Paris and purchased my Selmer Paris trumpet shortly thereafter. First, one big reason I switched to the Selmer was a very noticeable increase in resonance that I could hear from the playing position.

Anyhoo, the first time we were playing together Selmer-with-Selmer the resultant tones were strong. We both were hearing one single instrument that sounded like neither a tenor or a trumpet and was playing some other note. We both heard it immediately and grinned at the same time. Experimenting with the mics is interesting. If we're each playing into different mics, then we will not hear a resultant note; however, if I turn away from my mic and we both play towards his mic, then the resultant note appears and seems to be amplified (at least from where we're standing). When we're just doing backgrounds I'll do this several times a night.

This is neat stuff.

Dave
_________________
Schilke '60 B1 -- 229 Bach-C/19-350 Blackburn -- Lawler TL Cornet -- Conn V1 Flugel -- Stomvi Master Bb/A/G picc -- GR mpcs
[url=http://www.pitpops.com] The PitPops[/url]
Rocky Mountain Trumpet Fest

[PC]

Hi all,

What dcstep writes confirms my suspicions that resultant tones are real, in a sense where you need nice waveforms to be synchronised so as to have sufficient amplitude buildup at the low frequency.

In an ideal case, if we think of sine waves completely coherent (no phase shift), then it is easy to see why the difference law should work:

Draw a sine wave, then superimpose one with twice the frequency on the original one. Summing up the amplitudes, provided the peaks coincide (that is when the low frequency wave is at max amplitude, so is the double frequency wave), you get reinforcement of the low octave, as we saw in Derek's post.

Now think of a more complex case: a ratio 3 to 2 (visualise it as the equivalent of triplets vs. eight notes (quavers), where the notes coincide only on the beat). You will get "beats" once every 3 peaks of the higher freq. wave and will thus sound the pedal note, if the two waves correspond to low C and G in staff.

Here is an Excel visualisation for those interested:

Column A: type 0, 1 in cell underneath (A2) and drag down to have a column with angles (from 0 to a large enough value so as to have enough wavelengths, or alternately write 0, 10 and drag down less)

Column B: type =SIN(RADIANS(A2)) to get a pure sine wave. Double click the lower right corner of the cell to fill in the column with the right values.

Column C: type =SIN(RADIANS(2*A2-90)) to get double the frequency, adjusted so the "interference" is constructive. Again double click to fill column.

Column D: type =SIN(RADIANS(3*A2-180)) to get triple frequency, again phase adjusted.

Continue likewise...

Column X: type =C2+D2 or any sum of waves you are interested in and again drag down.

Now just select column A and any other column for the waves of interest, plus the "sum" column (Column X above) and make Excel draw the plot!

You will clearly see the resultant frequency appear. The psychology part in hearing this resultant is the ability of our ear to treat the information received in the same way as we do it visually with Excel, thus recognising the significant variations in sound intensity happening at the resulting frequency and equating it with a tone at that frequency.

I hope this is somewhat relevant to the discussion!

Pierre.


[ This Message was edited by: PC on 2004-08-05 06:22 ]

[pipedope]

I know this thread is old but worth revisiting.

I am an electronics tech and ham radio operator, we deal with mixing and resultant frequencies all the time.

Any time you have more than one frequency together in a non-linear system you will have the original frequencies and the sums and differences. These resultant frequencies are of lower amplitude and often need to be filtered and amplified to be usefull.

In our music the non-linear system that allows the mixing is likely to be the ear drum and other parts of the ear. The tones are real but may not be audible in the air.

If an amplified sound system is being used the mixing may happen in the microphone, amplifier, speakers or some combination.
_________________
No substitute for music through the horn!

[WM]

I had some luck producing sine wave graphs with the Excel formulas provided at this link.

http://people.msoe.edu/~taylor/ge100f02/lab1.htm

The chart would suggest that regardless of talent, it would be random chance to produce two notes where the phases would line up to cause a resultant "in the air".

Also, summing sine waves does not produce another well formed sine wave, but can produce waves that look like clipping in digital audio recordings -- which would be consistent with the observations of hearing a "rasp" in the resultant.

This season, I ran into a case in an orchestra where the resultant was so strong with the 2nd player that I had to back off to MF in an FF passage because there was so much "rasp". I characterize the other player's sound as very strong and loud on the primary pitch with few overtones. I believe we were playing octaves.

I don't know WHAT to think at this point!