The mathematics of phonation: Why the current trend in tenors’ career management is unsustainable

A4=440 Hertz, 440 vocal fold oscillations per second. The ear drum receives 440 little pressures every second. That is in fact a simplification of the process. In fact every fundamental pitch includes an unlimited number of harmonics that (not considering the vocal tract as a filter) would be progressively weaker than the fundamental. However when the vocal tract shaping (vowels) is added to the mix, the strength of the fundamental in relationship to its harmonics can change dramatically. There are pitch-vowel combinations that suppress the strength of the fundamental completely. But because every fundamental pitch has a unique set of harmonics, the brain reconstructs the fundamental through the many harmonics that can only come from that specific fundamental. Instead of hearing a bunch of harmonics, the brain recognizes them as the fundamental that produces them. That is why producing strong harmonics is a requirement for a well produced voice. High harmonics are also responsible for being heard with (not over or through) an orchestra. The human ear is sensitive to the frequency band between 2000 and 3500 Hertz. Sounds in that frequency range make everything else around sound further away by comparison. The orchestra can produce sounds in that frequency range but not very strongly. The human vocal tract has the ability to reinforce the acoustic envelope on every pitch while the conglomorate of orchestral instruments can only do that below the 500 Herzt level. Simply put, the orchestra is not very strong above 500 Hertz, and the human ear is more sensitive above 2000 Hertz. If a singer is able to produce strong harmonics above the 500 Hertz level, the human ear will hear it more strongly than the accompanying instruments and very powerfully when there are strong harmonics between 2000 and 3500 Hertz.


Some people have such keen hearing that sometimes they hear certain overtones more than others and do not hear the synthesis that gives the illusion of the fundamental. Such people often have a hard time matching pitch, particularly when the pitch is given from a mechanical source like a piano. Some people who are called tone-deaf are rather too sensitive to pitch. But I disgress…
Let us now consider the pitch A440! How do we get strong harmonics on that pitch. A strong source tone (which means strong harmonics with it) depends on the strength and volume of compressed air released during every open phase of the vibration cycle. At 440 Hertz the glottis releases 440 pulses of air every second. Each pulse disturbs the air molecules all around us and reaches the ear drum. Consider that when a violin’s A string (A 440) vibrates, its halves and thirds and quarters and fifths, etc also vibrate releasing harmonics at the level of the multiples (i.e. the halves vibrate twice as fast, thirds three times as fast, etc). All vibrating bodies behave in the same manner. So sound waves a the multiples are also sent through the vocal tract. Certain waves are suppressed by the shape of the vocal tract and others are encouraged. Proper shaping of the vocal tract yields strong high harmonics that makes the voice heard better.
Here is the difficult part. Compression of the air raises the air pressure below the glottis higher than that above the glottis. When the glottis then opens, the air is released with force. This is the law of equilibrium (Newton’s first law of motion). We hear this when a cars tire is punctured and compressed air is violently released, or we see it when we release a water valve or see a water dam break. The force with which the pulses of air are released through the glottis determine in part the sound pressure that is felt by the ear drum. If we consider also the principle of momentum, mass x velocity= momentum (second law of motion), the force with which the air molecules strike the ear drum depend not only on the velocity that the compressed air is released but also the mass of the air pulse. How much air is released is as important and in my opinion more important than the velocity. Consider the weight of a 2009 Pontiac Vibe (curb weight 3284 lbs) and that of the 2006 Hummer H1 (curb weight 7847 lbs). Consider both cars going at 50 miles per hour!
Not going into the subtlety of Force vs. Momentum (consider the rate of acceleration to be 1 not changing the rate of velocity), let us for this level of argument equate Force with Momentum, and for the sake of simplicity we will use pound-miles per hour squared as the unit force. The Vibe would deliver a force of 16420 units against the Hummer’s 39235 units. For that reason in a head-on collision between the two vehicles the Vibe would be destroyed. By that same principle the amount of air released during every cycle (mass) is as important as its velocity when it disturbs the air molecules around us. The pressure that causes velocity is based not so much on how much folds press against each other (although that is one mode accomplishing pressure) but basically on the length of time that it takes for the pulse to be released (i.e. how long does the subglottal air have to be pressurized before the pressure is released).
As said earlier, how deep the vocal folds are determines how long it takes the air to travel through them. This determines the length of every cycle and therefore the number of vibrations that would result during one second (the unit of time used to determine pitch, vibrations per second). It is assumed that complete closure of the vocal folds is necessary to build adequate pressure. Let us illustrate this in a concrete way:
Let us take the pitch G2=200 Hz (200 vibrations per second). And let us give an arbitrary value to the ideal fold depth, 2 cm, whereby the folds would remain open for exactly 50 percent of the cycle. In this way the maximum amount of air would be released during the open phase and the folds would be closed completely for the close phase. Let us take this hypothesis as given!
One cycle of G2 would occur during 1/200 of a second. The air would be compressed during one half of that time 1/400 of a second, the time it took to travel through the gradually opening 2cm of fold tissue. The velocity of the air during G2 would be 2cm/400s or 800cm/second. If the velocity of air remained the same but the distance traveled were less (fold depth shallower), say 1.5cm the cycles would be faster. If a car running at 20mph has to travel 15miles how much time does it take?
20/60min=15/z, the solution for this problem is 20z=60 x 15 or 20z=900 or z=900/20 or 45 minutes. 3 quarters of the distance traveled at the same velocity occurs in 3 quarters of the time.
Likewise, 2cm/400s=1.5cm/z or 2cmz=1.5 x 400 cms or 2cmz=600cms or z=300s. Therefore the speed of a half cycle when the folds are 1.5cm deep is 300s, or 3/4 the amount of time necessary to sustain G2. For a singer to sing the same pitch with thinner folds than appropriate, they have to press the folds together to slow down the time of vibration. In this case the amount of air released per cycle would be the same but an extraneous element would have to keep the folds closed for longer. That would have to be the interarytenoids, which would get tired very fast if they had to work that much harder during every cycle of vibration. This strategy has no additional benefit other than giving the singer a more squeezed quality (this is a very popular sound quality among opera singers). Since the time would be constant, the amount of pressure would be the same. Fatigue would be only difference.
If the depth were kept the same but greater pressure applied (the squeezed quality that has become so popular), the cycle would be slowed down unless the open phase were shortened. In that scenario, the interarytenoids would have to become even more active to accelerate the closing of the open folds (shortening the open phase). The shortening of the open phase would yield less air release. While the velocity of the air would be the same as a normal cycle, the amount of air released (mass) would be less and the force (momentum) would be less than a 50% close quotient. In essence, the sound pressure (volume) as we perceive it would be less, but the quality of the tone would be darker.
This is the strategy of lyric tenors who wish to sing more dramatic repertoire. The increased subglottic pressure unfortunately causes a continued cycle of tension in the larynx which causes hyperactivity throughout the system. This leads gradually to dysfunction, particularly a chronic thickening of the folds that leads to difficulty in the top voice. This is precisely what we have seen happen to many lyric tenors who gradually thicken their voices to take on more dramatic repertoire. They lose quality gradually, first by difficulty producing top notes and eventually by vibrational imbalances in the passaggio and ultimately irregularities in the middle range.
In some ways, I did this very thing to the extreme by singing as a baritone with a voice that is naturally a tenor voice. Becoming a tenor has been a healing of sorts. I have regained the lean quality I used to have in my earlier years and the voice is gaining in flexibility and brilliance. It has been fascinating experiencing first big changes and then subtler ones. Experiencing the reverse process gives me some clues as to how gently and subtly structural deconstruction could occur. My change to tenor was timely. I believe I had begun to experience some structural irregularities. In my case, fold depth had reached far beyond the level that would be natural for 50% close quotient. Instead, my fold depth was so deep that breathy phonation what the only way to keep pitch. If the folds had come together completely, pitch would go down. The opposition of fold depth that was too thick was balanced by very little resistance (breathy phonation). This would explain why under electroglottograh I had close quotients near 80 percent in the top voice and my voice did not sound pressed.
I should add that if we take vocal tract inertia into account, skewed pulse (the sluggishness of the supraglottal air) would increase close quotient requiring the folds to remain open long to maintain cycle length. CQ would actually be slightly lower than 50% I am not sure whether the electro glottograph would register the actual fold closure or fold closure including skewed pulse time.
A thickened lyric voice is not the same as a naturally heavier voice. Lyric tenors thicken their voices because of the economics of opera not because of the nature of the instrument. It is true that the laryngeal muscles over time become stronger with correct use and will be able to handle greater air pressure without losing structural balance (i.e. the muscular balance between vocalis and CT that determines fold depth and length for a given pitch). In this manner, a lyric voice may be able to do spinto repertoire over time. The amount of time it takes before a voice is ready to make louder sounds without losing structure depends on the individual voice, how it has been trained, and how it develops over time.
I often listen to the roundtable interview with Domingo and Pavarotti in September 1993 on the night of their 25th anniversary Met performances. Pavarotti described his voice as “a full lyric voice…not even spinto.” Domingo described his own voice as “…not a naturally dramatic voice”.
This was in response to a question about what roles they would consider off limts for their voices.
Pavarotti canceled a Cavaradossi early in his career and waited 14 years to do it because Di Stefano told him that Tosca is an opera that could break his voice. He was careful about which roles he took on and when. His voice grew in strength over the years and he was able to make stronger sounds in his middle voice, which made it possible to sing spinto repertoire. The basic quality of his instrument did not change. Domingo was more adventurous but knew the risks he was taking. He was aware that his voice could have suffered and considered himself lucky that he sang as long as he had.
The difference with young lyric tenors today is that they have no sense of danger. They think they can do anything and general managers and agents are very willing to counsel them to take on heavier roles. Why? The standard repertoire has been reduced to a very small number of works that tenors must be willing to do if they are to sustain a profitable career. The Rossini tenors have been saved because they have a niche repertoire that features their ability to sing faster and higher. Lyric tenors need much more than Faust and Rodolfo, Romeo, Nemorino, Edgardo and Duca. They need repertoire that features their special skills. Don Jose and Cavaradossi, Pinkerton, Calaf and most of the Verdian repertoire belong to spintos. However the training of spinto voices has always been more time-consuming and no one wants to wait for a true spinto to develop. With the shrinking repertoire and Hollywood physical aesthetics, age is also on the side of the young lyrics who are presentable earlier.
Since conservatories are churning out young tenors every year, who are able to sing two or three arias at auditions convincingly, without knowing how well they handle entire roles, agents and general managers are willing to sign any young person with charisma, only to find that their tender muscular structure cannot handle a professional schedule. We routinely hear of cancellations and young tenors jumping in last minute to take over for colleagues who are suddenly indisposed. Because many of these young singers have been signed for roles several years in advance, it is in the monetary interest of those in charge of such decisions to keep them in the system. A young tenor’s fame is an advertising tool. If he has made a name for himself, he becomes very important to the system whether he is able to deliver convincingly or not.
The problem is that singers who routinely have to cancel cause the system a great deal of money. The cancellations of Rolando Villazon at the Metropolitan alone caused a game of tenor-musical-chairs unlike anything seen on the world stage. It was unclear who would take on his roles and when. The negotiations could not have been a money-saver for the Metropolitan. So why does this practice of engaging young tenors who are not ready continue? Young tenors cost less at the onset and there are plenty of them when readiness is not a factor.
The biggest problem however is that those who are hiring cannot tell whether a young tenor who sounds very good is truly ready to hold his own on the world stage. And it is not profitable to consider that a famous lyric tenor should not sing Cavaradossi. If he is famous, he will fill seats. I am getting the feeling that young tenors have little choice. They either take on the heavy roles or someone else will and they will be out of favor.
General managers are truly afraid that they cannot sustain opera if they have to compete with movies, television, broadway and pop culture altogether. The chosen solution has been driven by money and whatever superficial characteristics will capture the imagination of the paying public. Part of the strategy after the success of the Three Tenors has been to democratize opera. The business-makers of opera realized that it was possible to capture a larger crowd. By the second ThreeTenors concert it was obvious that they got it wrong. In Los Angeles in July of 1994, a show more based on superficial Hollywood glitz took the humanity of the original concert out and replaced it with a Miss America-like empty pageantry that was supposed to thrill the world. Even empty pageantry was sellable because the Three Tenors could sing at a level higher than most others out there, even if the quality of their once-thrilling talents had been diminished somewhat.
Many thought they could replace the aging Three Tenors. It is not impossible! But one must consider how long it took the three tenors to establish their careers. Pavarotti and Domingo had to make their names while Corelli and Tucker and other world class tenors were still singing. Carreras, Domingo and Pavarotti have been around so long that it became a goal to replace them with whoever could give a fair impression of their quality rather than finding some bona fide new tenors who are unique and who are able to sing as well. Since the time factor has been ridiculously shortened, and youth is everything now, it is not possible to find young tenors who are truly ready, whose voices are not as fragile as thin glass.
Same is true of other voice types, but tenors are more difficult to train and so there are fewer viable ones even given the low expectations of the current system.
Singing is as physically demanding on the larynx as an olympic sport on the rest of the body. Getting someone off the street to run a marathon is poor strategy for an olympic team. Thinking that some young singer who has not sung more than 6 years total is ready to take on the world stage is ludicrous. Furthermore no olympic organization is going to pit a welterweight boxer against a heavyweight. The welterweight would be in danger of dying in the ring. The strategy of having lyric tenors speedily grow to sing Puccini’s most dramatic roles and even Wagner’s is unconscionable. People then have the nerve to wonder why quality and longevity are rare in the world of opera!
We cannot have it both ways. Either we have slow growth or quick burnout. There are plenty of singers for all repertoire. The system as it has developed in the last thirty years promotes neither the healthy growth of singers nor longevity of career. Just like the excesses of the banking system, without some control opera as a viable art form will meet a dismal end.
© 08/02/2009

10 thoughts on “The mathematics of phonation: Why the current trend in tenors’ career management is unsustainable

  1. I agree that the natural sound is the best in any instrument but it can be really hard to figure out what that sound is. So often people grow up listening to the radio or around people and want to imitate the sounds of others rather than find out what they really sound like. We don't always like our voices and we can be envious of the things others can do that we can not or can not easily.

    It is hard to listen to a work that you really like and go “I am not ready to sing that” or “I will never have the right voice like that” or “I can sing that piece but not as written”. The popular voices today are heavy while the average voice is not that heavy especially in the higher fachs. Any voice properly done is usually capable of volume (at least in the proper tessitura) but weight, drama, and spinto are something completely different. We have a lot of people who trade the natural beauty and power of their voices in for a completely different and mass market sound. Pushing chest where it does not go and producing excessive grain or raspiness are popular styles these days and frequently they just produce really really bad voices.

    I love to sing with the radio and I am always trying to modify songs to fit what I feel like I can sing. I have an unusual voice and no real training so that can make things hard. A lot of artists I just won't sing because the sound is to heavy or I modify it completely. I also often change the notes so I can sing comfortably off key. I like the heavier lyric sound because it is fun and challenging to sing but it is also tiring for me and I often end up dropping weight so I can keep singing. Still, it is fun being in a car and hearing a mass of sound around you.

  2. Wow. Are some companies more guilty than others of this?

    Another thing that occurs to me is that human ears can hear up to around 15000 Hertz. Are these upper parts of the envelope above 3.5 Khertz important only for the fact that they allow the ear to hear the upper harmonics? I am thinking that the highest note on a piano always seems to have a much simpler tonal quality than the lowest one, although I had never thought of this before reading this post.

  3. Dear Pagan Topologist,

    The business of opera has a major disconnect with the art of opera at the current juncture. It is the hope of all artists that this will change. Opera became bigger business than artists can handle. CEO types with mass media credentials are sought after to handle the day to day money-related operations of companies.

    A singer becomes a commodity. When a singer's name has appeared enough on the billboards, people will buy tickets often based on mere familiarity with the singer's name. The bona fide opera fan can tell you details about each singer like a baseball fan knows the batting average of most MLB players. They are more discriminant in their reasoning for buying tickets. The average opera fan today is unfortunately not so bona fide. They often go out of social responsibility or curiosity or because opera is a certain status symbol. Now such fans could become bona fide if what they see thrill them. But they rarely do because the business model is catering to the kind of opera-goer that comes and goes rather than the bona fide fan (as it used to be). The reasoning is that the curious (because of the three tenor phenomenon that gave rise to Bocelli, three Irish tenors, 10 tenors, Three mo' tenors, etc) outnumber the bona fide fans. Even if they go away, more will come. Unfortunately, the less than exciting performances particularly from a vocal standpoint is driving away the bona fide fans as well.

    I am in this for the long haul. There was a time I would give my right arm to go to every Met performance I could go to, but now I chose carefully because I can tell by the cast whether it is worth my while or not.

    As to the acoustic part of your comment, the human voice has strong acoustic energy even near the 5Khz mark. And the harmonics go ad infinitum, so certain adjustments of the vocal tract could even produce some errant strong signals at the limits of human hearing. But for singing nothing is that important beyond 4khz. I do not know how much the weak signals above 4khz help in discerning pitch. It is logical that all harmonics perceived would add to the complexity of the tone and therefore to clearer recognition of the fundamental by amalgamation.

    You are correct about the piano. And that can also be observed with violins and cellos. The resonating cavities of string instrument and the sounding board of the piano have limits. They do not resonate higher notes well. For that reason we need so many violins in the orchestra. High wind instruments sound clearly near the human ear, even though the sound envelope is relatively un-complex. Brass instrument because of valves do a better job of resonating throughout the range. Trumpets are powerful even at their highest notes.

    Thanks for your commentary.

  4. I am sorry if I sound like a besserwisser, but I cannot help but to correct you on the physics issues. Here it goes…

    The math arguments you start up with are used to perform the calculations related to the pitch G2 (in the paragraph starting “One cycle of G2…”), right? To tell you the truth, I cannot really understand the calculations of that paragraph. What I do know is that your reasoning on Newton’s second law (what you call the principle of momentum) is not correct. You cannot equate force with momentum! Force is associated with a temporal change of momentum, not with the absolute value of momentum itself. It is when you want to change momentum that you need to apply force, or to be precise “impulse”, that is, force x time. Assume, for instance, that a 1 kg object at rest is accelerated up to the speed of 1 m/s. That is, we want to change the momentum (mass x velocity) from zero to 1 kg m/s. A force of 1 N accomplishes that acceleration in one second. A 2 N force need only half a second and so on. Any nonzero force (in the right direction) can accomplish that momentum given enough time, and you can accomplish that acceleration arbitrarily fast using enough force.

    To really describe in a quantitative manner the glottal oscillations using the mechanical laws is quite complicated and I do not think (with all respect) that your calculations really makes much sense. That does not mean that your ideas about fold depth are wrong, of course! The reasoning you make after the math sounds convincing.


  5. Dear Martin,

    You are a Besserwisser and I appreciate your input. My physics are weak, certainly but I do think I understand the principles that I want to use. So I will explain what my logic was and you can correct me. And yes the mechanics are much more complex that I make them, but I do think the principles apply.

    I was going by the following equations of motion:

    velocity= distance/time (direction is implied)

    momentum=mass (x) velocity

    Force= mass (x) Acceleration

    Acceleration = velocity/time


    Force = mass (x) velocity/time OR
    Force = mass (x) distance/time/time

    If the second time unit is a constant (1) then the absolute value of momentum and force are the same.

    I understand that momentum and force are not the same thing. I did not want to change the second time value, which would make things much more complex.

    I could be quite wrong here. You know these equations far better than I due, being a physicist. I welcome your guidance to understand the principles better.


  6. The “real” formulation of the equations of motion involves time derivatives rather than division by time. Writing the equations like you do using division with time is only correct under very special circumstances, and it is easy to be misled. For instance, you draw the conclusion in your blog that The Vibe would deliver a force of 16420 units against the Hummer's 39235 units which is, well, not correct. There is no particular force associated with a movement.

    The “calculus-free” formulation of Newton’s law that I was alluding to in my previous comment is

    Force (x) time interval = change in (mass (x) velocity)

    which should be interpreted as follows: applying a particular force to a system during a certain time interval causes the momentum (mass (x) velocity) to change with corresponding amount (force (x) time interval). The “change in” part in the equation above is crucial!

    Using the notation f = force, dt = time interval, d(mv) = change in momentum, we can shorten the formula to

    f dt = d(mv)

    or, by “dividing” by dt, f = d(mv)/dt, which is the calculus version of Newton’s second law, involving the time derivative.

    Cool that a singer and voice teacher is interested in this stuff…


  7. Dear Martin,

    It has been a bit of time since calculus, but this formulation is easy enough for me to reconsider the formula and make corrections.

    I am hugely interested in this because there are practical implications to the mechanics of singing at that level. If I can understand the physics of it over time I know that I can find the practical applications.

    This is where I think the scientists fall short. They are not singers and do not know how singers think. The science must be more “alive”. There is a place between absolute mathematics and the performers experience that can be articulated. Because of this, I will keep working on my physics and mathematics and calculus.

    You will be patient with me if it does not roll of the tip of my tongue as easily as vocal concepts do.

    Thank you again for your input.


  8. Thank you Martin for commenting on the physics part, and JRL – cudos for making a valiant attempt. 🙂 Indeed, this is a very important area. Even if the dynamics of voice production are extremely complex, we can, and must, keep trying to grasp the fundamental physical (and physiological) principles of singing. My calculus and physics are rusty too, but I'm with you.

    On imitating voice qualities, I've made the mistake in the past of imitating singers like Giacomini, and suffering the immediate consequences. Luckily, being a Swede and tenor, I've been sufficiently marinated in Bjoerling – much closer to my own voice type – to have a role model that I can return to after such failed excursions. 🙂

    (Nothing against Giacomini – he was a marvellous Othello, but I'm a lyric tenor.)

    What I finally learned from both Bjoerling and Pavarotti was how careful they were never to sing out of their fach. If they did venture into potentially dangerous repertoire, they always sang in their own voice, relying on legato rather than force, and never pushing the voice off limits. It's curious that many rate Bjoerling's and Pavarotti's renditions of Calaf (indeed, they were both exceptional) without considering the fact that both sang the part *only* in the studio or in concert (Nessun dorma) – never on stage.
    At least I know this to be true for Bjoerling.

    As much as I love listening to Jussi's Nessun dorma, I find that few people today realise that Calaf requires something more in line with Corelli or Martinucci in order to tackle the whole opera.

    I don't think one should underestimate the problem that lyrics tend to sound better than dramatic tenors on recordings, even in fairly dramatic operas. They are certainly easier to record. (:

    /Ulf W

  9. So many excellent points Ulf. I am running, but will comment later. I am very sure that Pavarotti did not do Calaf on stage. More later.

    I welcome anyone who wishes to join in on resurrecting any knowledge of physics I had in my youth. Misery loves company (respect to Martin and the other physicists here)!

  10. Hmm, I meant to write “It's curious that many rate Bjoerling's and Pavarotti's renditions of Calaf *among the best* …”, but perhaps you figured that out by yourselves. (:

    My maestro Bengt used to say that Pavarotti had a light lyric voice, but heroic diction. That's why he could pull off singing heroic roles on recordings.

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