The purpose of this blog is to shed a bit of light on an issue that gets
brought up time and time again: “Does More Speakers Equal More
Volume”. At first, I thought this was a pretty open and shut case, but
after some time I realized there is more to this problem than the
surface arguments. For those who are not familiar with the debate, I
will summarize the two most popular sides:
Side A: adding an
extra speaker gives a +3 dB boost over using a single speaker because
two speakers push more air than one speaker
Side B: no matter
many speakers you run, the law of conservation of energy must hold, and
there should be no increase in output just by adding speakers to the
By the way, I was firmly entrenched with Side B, I took
enough physics in school to know that you can’t violate conservation
laws. What I have found is that both sides aren’t really right, not
because they aren’t both partially right. Each side has a bit of truth,
but both sides simplify the problem too much to be conclusive.
order to understand what is going on here, we are going to have to get a
bit technical. We are going to have to know what harmonics are, some
basic wave mechanics, and even get into idealized thought experiments.
So let’s get started. What is Sound?
is basically air moving, usually a systematic increase and decrease of
air pressure. In our musical world, we can think of an object moving
which causes the air to move around it. Think of an object moving back
and forth, when the object moves forward it compresses the air in front
of it and then when it moves back it rarefies the air in front of it.
This creates a series of higher and lower pressures, which can be
represented as a sine wave.
For the sake of saving time, here is a link with pictures and a better explanation of the basics of sound.
the simplest sound can be represented as a sine wave that has an
amplitude (how loud it is, aka: it’s intensity) and a wavelength (this
corresponds to the frequency). But real world noises, including most
instruments we play, don’t produce ‘simple sounds’. Real sounds we
encounter are not made of one single frequency, instead they are made by
multiple frequencies interacting with one another.
When we talk
about more complicated sounds, with multiple frequencies interacting,
then we refer to the lowest frequency in the sound as the ‘fundamental’
while the higher frequencies are called ‘harmonics’. Technically even
the fundamental is a harmonic, it’s called the ‘first harmonic’; the
next highest frequency is called the ‘second harmonic’; the third
highest frequency is called the ‘third harmonic’, etc. Here is another
link with graphics:
each one of these harmonics can be considered a sine wave in and of
itself, it just so happens to interact with other frequencies and when
you add these sine waves together you get waves that look nothing like a
sine wave. Here is a link with some good pictures so you can get the
do I mention all of this? Well it is important for you to know about a
simple sound of just a pure sine wave as we will be using that in our
thought experiments. It is also important for you to understand that
real sounds (like notes on a guitar) are made of multiple sine waves
summed together because that will be important in appreciating real
world complexities. We will also be dealing with waves interfering with
one another, so it is important to understand how waves can interact
with each other.
The Near Field
briefly, let us review what a near field is because it is an important
concept in our thought experiments. The near field represents an ideal
listening environment, absent of any surfaces or imperfections that may
reflect or interfere at all with our listening experience. When we
invoke the near field in a thought experiment, then will consider no
reflections or wave interference at all. You can think of a near field
as floating in the sky, far away from any object, so that all we hear is
sound from the source signal exactly as it is reproduced.
important to realize a near field is impossible to achieve. In real
life you will have to deal with the environment and it’s impact on
sound, but the near field is a useful concept to help us simplify a
thought experiment so that we pay attention to the concept we are trying
to understand without compounding complications.
All of our calculations are going to assume a near field.
is important to note a few things about how humans hear noise. First
off, we can hear a very large range of loudness/volume (represented by
the amplitude of the sound wave) and we call this range of loudness that
we hear ‘Dynamic Range’. For example: if we are at a very loud rock
concert that pushes the limits of the loudness our ears can handle, we
can also distinguish a noise with an amplitude 100,000 times less than
the rock concert (not at the same time though). That is pretty
impressive, and it also leads to problems in representing numbers
because they can end up becoming HUGE when we make comparisons. Because
the dynamic range of our hearing is so big, we actually adopt a
logarithmic scale called the decibel scale. The benefit of doing this
is that numbers in decibel form remain small and easy to manipulate even
when differences in sound are huge.
thing to briefly mention is that humans also have quite a large range
of wavelengths or frequencies of sound we can hear. Humans can hear
sounds from 20 Hz to 20,000 Hz. Just as importantly, a human’s ears do
not treat all frequencies equally. Some frequencies (around ~1,000 Hz)
sound louder to us than all the other frequencies, because our ear is
more sensitive in this range.
is a confusing topic surrounding sound that we will try to address and
clear up. Sound power represents the amount of sound energy a source
radiates directly, the sound power does not take into consideration
environmental factors at all. Sound pressure on the other hand is how
the radiated sound interacts with the environment and is indicative of
what our ears actually hear.
To give a more tangible example, let
us compare sound to heat. A heater has a certain rating for output
(btu or watt) that represents the intensity of the energy output of the
device. But if someone is standing in a room with the heat source and
you ask them “How hot is it in here?” their answer will most likely not
match the output of the heat device exactly.
The reason for this
difference in device output and perceived temperature has much to do
with heat’s interaction with the environment.
- Heat will be the most intense nearest the heating device, as you approach the heating device you will feel more heat. -
Heat distributes itself throughout the room, when you first turn on a
heating device in a cold room it will take some time for the whole room
to warm up. - The shape of the room will come into play as to
where the hotter air will end up, for example: a room with tall ceilings
will take longer to heat because heat rises. - The size of the
room has to be considered when figuring out how long the room will take
to heat up and what temperature the room will reach.
factors all determine the temperature of the room and explain how a room
may feel cold even though you have a heater with immense output.
sound power can be compared to the output of the heater and sound
pressure is analogous to room temperature. When we measure sound power
output of a speaker it may not match entirely with the perceived volume
in a room.
- Loudness should be the most intense the closer
you are to the speaker, as you approach the speaker you will hear a more
intense sound level (given a near field environment). - Sound
will be absorbed and reflected when it comes into contact with a
barrier. The degree of absorption and reflection has to do with the
material the sound comes into contact with. - Lower frequencies of sound need a larger barrier to reflect them due to their longer wavelength. - When sound gets reflected inside a room, waveforms can sum or cancel to create areas of higher or lower intensity. -
Reflections can also cancel out sound of particular frequencies (comb
filtering) or create areas of constant pressure (standing waves).
factors (and others) determine the quality and loudness of how a sound
source will be perceived in a room and they help explain how a speaker
cab may sound louder/more bass heavy/worse in one room than another.
Sound pressure levels are very sensitive to environmental factors. This
is important to note because later we will use math to calculate a
special case dealing with sound pressure (called coherent summing).
Does More Speakers = More Volume? Pt 2 (Mutual Cou
In the previous blog, background information was discussed to clarify certain attributes of sound that will play a role in discussing this subject. Here is a link if you are interested in some background:
Without further ado, let's dive into the matter. Calculating Sound Output From a Loudspeaker and Amplifier (1x12 math)
An amplifier’s output is rated in watts (which is a unit of energy conversion). A speaker’s loudness is called the sensitivity of the speaker, it is rated in decibels and the reference is taken at 1 meter with 1 watt of power supplied to the speaker. So we will use these calculations to determine nominal output:
Nominal Output = (10*LOG10(P1)) + sensitivity
Where P1 is the amp’s nominal output. So a 50 watt amp paired with a speaker of 100 dB sensitivity would be plugged in as such:
Nominal Output = (10*LOG10(50)) + 100 = 116.99 dB
This would be the nominal output in a 1x12 cabinet.
Adding Incoherent Acoustic Signals (2x12 math)
Now, these are the calculations that Side B will use to show how their side of the argument works. When adding acoustic signals from different sources that don’t share attributes we will use the formula:
Where S1 is first source, S2 is the second source. You can actually keep extending this formula to include as many sources as you like.
Now, because power is distributed evenly among speakers we have to recalculate our sources (we’ll assume the same setup as before, 50 watt head with 100 dB speakers), but we’ll adjust for how many watts each speaker will ‘see’ (one half of 50 watts is 25 watts).
Nominal Output = (10*LOG10(25)) + 100 = 113.979
So let’s plug this into our function
Incoherent Output = 10*LOG10(10^(113.979/10)+10^(113.979/10) = 116.99 dB
Using this math, a 50 watt head with 100 dB speakers puts out the same amount of sound with a 1x12 cab as it does a 2x12 cab.
There is a problem though. There is an interaction going on between these two speakers we are not accounting for. The phenomenon of coherent signal summing is NOT considered in this example. “When the two signals being summed are of the same frequency… the sum will be the voltage sum of the two signals. This is referred to as coherent summing of the signals, and it causes some unusual effects that are not intuitively obvious.”
Now, by being more rigorous and realizing that both signals are the same and if you apply the concepts of coherent signal summing, then we use another formula for adding the signals and this is where Side A get’s their argument from:
Coherent Output = S1 + 20 log10(N)
Where S1 is the signal output and N is the number of speakers producing the signal. Plugging in the information for a 50 watt head with a 100 dB speaker we get:
Coherent Output = 113.979 + 20 log(2) = 120 dB
Interesting, using this math gives us +3 dB compared to using a single 100 dB speaker alone with a 50 watt head. If we take this result seriously, then more speakers does equal more volume when completely coherent signals are summed.
But there is still a problem. There are stipulations to coherent signal summing, one of the stipulations states that coherent signals are signals that are completely ‘in-phase’ with each other. What does that mean?
Two signals that are in phase are perfectly synced up, their crests will add up to a larger crest, their troughs will add up to a deeper trough. This is considered constructive interference, and this is what summing a coherent signal assumes.
But what if the two signals are not perfectly synced up? Well, you get a phenomenon known as ‘destructive interference’, this is where the crests and the troughs of the waves are not perfectly lined up and can actually start canceling one another when summed together and the result is a final wave that is actually less intense than the two input waves.
So, in the real world it would be incredibly hard for two sources to be perfectly in phase with each other for these reasons:
- Individual sound sources must be displaced some distance away from one another - Sounds have frequencies that have inherent wavelength - Most real sounds are made of a mix of these wavelengths - Most musical sounds will produce notes of different frequencies (musicians will play different notes)
When we consider these points we come to a conclusion: in order for two signals from two sources to be perfectly in phase, the distance between the sources must be a constant multiple of the wavelength of the note being produced. In other words, if the distance from output source A to output source B is not equal to the wavelength of the note being produced then the signals won’t sum perfectly.
That is not all, since the distance between the speakers in a 2x12 is fixed (and by necessity they can only get so close together, so they will be displaced by a minimum of ~12” signals will only be perfectly in phase for frequencies of particular wavelengths. Since real sounds (like notes from a guitar) are made of many wavelengths, this makes it impossible for pure coherent signal summing to occur. Beyond that, while certain frequencies may coherently sum to sound louder, other frequencies will destructively sum to and cancel each other out.
There is also another problem with coherent signal summing, we have not taken into consideration the location of the listener. All of the formulas for coherent summing assume the listener is directly in line, in front and on axis with the signal sources; in other words we have been assuming the most ideal position for signal summing. Once the listener displaces himself from the ideal listening position more phasing issues arise because sound from the closer source will arrive at the listener before sounds from the further source. So the +3 dB bump is an actual phenomenon, you just have to keep in mind that the listener must be in an optimal location in order to perceive it and it is only active in particular frequency ranges.
So coherent signal summing only works for certain frequencies (so simple sounds would work best) and only if the listener is in the optimal position. Where does that leave us?
http://www.recordingeq.com/EQ/req1001/mmi.htm * the concepts in this section are equivocal to the microphone placing in the above link. Most of the concepts in this section were brought up in sound engineering books I have read as well. For example: Sound and Recording - Francis Rumsey, Tim McCormick 
So with all this complicated stuff going on, what is the result? The result is mutual coupling; I will quote another source:
Let's say you have a 12" speaker that produces 100db/w. If all specs stay the same, but you double the surface area of the cone, the speaker will now have a 103 dB/w sensitivity. Each time you double the cone surface area you get a 3 dB increase in SPL, which is a mechanical/acoustical transfer of air. When adding another speaker to the 12" equation, you are trying to merge both speakers into one. Essentially adding the surface area together for the 3db gain.
The problem here is that the cones of the speakers are physically not capable of being close enough to reproduce the 20hz-20khz frequency range as a single system. The problem introduced here is that as the speaker centers separate, the benefit of speaker coupling tapers off from the high end. Since two 12" speakers can only be physically 12" in distance from each other - cone center to center - , this presents a limitation of a wavelength at a certain frequency. The 3db effect starts at a 1/2 wave distance, and is pretty much full at a 1/4 wave. So for 12", you get a 550hz half wave, to 275hz quarter wave. This is the best performance you will get from two 12" speakers in proximity in relation to speaker coupling 3db gain. As the speakers gain distance you will start lowering the frequency of this effect.
To reiterate, the speakers need to be within 1/4 to 1/2 wavelength for that frequency range to get any gain. This means, if a wavelength of 200 hz is 67 inches, by four (quarter wave) and two (half wave) gives you 17 to 34 inches respectively, which is the distance both speakers need to be from each other (cone center to center) to get a gain at that frequency. So once you get to 400hz, that gap is half the distance. So the speaker cone centers would need to be 8 to 16" from each other. Get to 800Hz and you need the cone centers to be 4 to 8" close, which is physically impossible with 12" speakers.
So we can safely say that doubling speakers only has a benefit in the low end frequencies. Now let's take the same math to a 4x12, since the speaker centers have to match for all 4 speakers, this means that the furthest centers are to be used for that calculation. So for a 4x12, we are talking about 300Hz or less being a realistic region for speaker coupling gain. Adding another 4x12 stacked on top, would mean that the very top left speaker and very bottom right speaker is now the new length to use for the formula to work, which means roughly 50-60inches in distance. At this point the only frequency range getting advantage is ~100hz and below. So there is still an advantage, but the window closes quickly as you add distance to the speakers. Now let's say you have two 4x12 cabs, and you position them angled in, the more you angle the cabs the less you will get that gain. The speakers have to be directed in the same way and be even, so you can't have one closer than the other. This means an angled cab won't have as much gain in the low end as a straight cab.
A concept brought up in the above quote, but not addressed so far in this blog, is the fact that signals don’t need to be 100% coherent in order to constructively interfere. Signals that are closely in phase can start to constructively sum when the signals are as far apart as half a wavelength and they can sum to almost a full +3 dB by the time the signals are a quarter wavelength apart.
Mutual coupling is the result of all the imperfections mentioned before. To summarize
- Signal summing will only happen within a certain ratio of wavelength - Most of the signal summing will happen below a low end threshold - This threshold frequency is determined by furthest distance between drivers in the cabinet - Frequencies higher than this threshold frequency will have destructive interference introduced in the form of comb filtering - The frequencies where signal summing occurs are frequencies that our ear is not most sensitive, while the frequencies that our ear is most sensitive to is subject to destructive interference - The amount of summing perceived will depends on the location of the listener
So what does this mean to us? It means if you add more speakers, you should get more low end at the expense of some (most likely not overly noticeable) distortions in your higher frequencies. The more speakers you add, the lower the threshold bump becomes effective (reducing the range of frequencies that benefit from the bump) and the more phase cancellation you introduce in the higher frequencies. It is also worth noting that this ‘low end bump’ you get from mutual coupling deals with frequency ranges that are quite pertinent for guitar playing as long as you don’t introduce too many speakers.
Another interesting tidbit: mutual coupling benefits closed back cabs much more than open back cabs. Open back cabs suffer from phase cancellation of low notes (due to it’s limited baffling)*, so extra speakers in a open back cab won’t get as much ‘low end bump’ because mutual coupling especially effects the low end.
So once again, where does this leave us? Well, most of the math we ran through earlier gives us no real quantitative idea of how much of a overall dB bump we get when using more speakers because they just don’t apply to mutual coupling, unless I can come across more math we will just have to live with the qualitative results we obtain from the Mutual Coupling section above. Most likely, depending on where you are standing and what phasing issues apply to you, you may notice anything from a +0 dB boost to a +3 dB boost in certain frequency ranges (keep in mind, obtaining a full 3 dB in practice is difficult to do as it represents the max increase that can be observed).
One thing to note: even in the most ideal situations all we are getting from doubling the speakers is a +3 dB boost overall. When you introduce an amp and speaker into any sort of room (you know, the opposite of a near field) and the sound starts interacting with that environment, you can get MUCH more dramatic comb filtering, standing waves, and increased SPL’s that just swallow any type of boost using an extra speaker may enable**.
Tubes. archaic, obscure, 'shrouded in veils'. such terms as 'cathode bias', 'grid screens' and 'plate voltage' are a sampling of the commonly used, dense venacular; these terms mean something to someone... evidently, at some point in time, we learned how to make things loud via these little glass packages. i believe a bit background and a little history could be very enlightening.
so, where do tubes come from? how/why do they work? well the stage was set for the development of the vacuum tube with a number of incidents, here are a few:
-in the 1600's Robert Boyle's discovery of electro-magnetic forces in action across a vacuum. this had profound effects on theories over the propagation of electro-magnetic forces, but it also eventually lead people to discover that creating a 'perfect-vacuum' would create an environment free of charge particles that would interfere with an electron beam.
-Vacuum tube experiments with "cathode rays" (aka: electrons) start reaching the point of understanding (evidently, the better the vacuum, the more dramatic the effect). tubes like X-Ray tubes and Crooke's Tubes were being designed with greater understanding and for purposeful use. but by the late 1800's, an advanced understanding of even electrons was yet proposed.
-by 1900 we discovered what the electron was, and what it was doing flying around inside of vacuum tubes. thanks to guys like gauss, maxwell, faraday, volta, coulomb and many others, we had a pretty good 'classical' model, we had the math and experimental data to manipulate electron flow.
the ancestor of modern audio style vacuum tubes would have been called a 'cathode tube' back in the day. the tube would have come in various shapes, and would have had most of the air removed from it or may have even been filled with certain gases for experimentation. Two plates would have been set up on either side of the tube and voltage would be applied to the plates and causes a flow of electrons from one plate to another (cathode to anode). later a number of people (including edison) discovered thermionic emission, which produced a more steady and economical source of electrons.
up to ~1900, scientific aparatus was designed to explore the properties of everything electro-magnetic. along the way a quite a few devices were invented that benefited humanity greatly, but the stage was set for more intelligent design.
since electricity was already in use, certain needs became apparent. for example, AC current won the 'Current War' and long range transmission of electricity was now possible; but the need for a self contained rectification system became quite necessary since DC current was necessary in for widespread home use (rectification converts an AC current into a DC current). also, audio encoded onto a media and played back from and AC signal is already in use and the need for amplification of these weak signals was needed (this was necessary for both radio transmission, telephone/wire transmission, and disc/can audio media amplification)
armed with our more complete understanding of electro-magnetism, the first diode vacuum tube purposely built to allow flow of current in one direction, which became useful in the rectification process and in radio wave detection. it worked by passing an electrical current through a piece of filament until it reached it saturation point and thermionic emission was induced. these extra electrons around the fillament (the cathode) are then attracted to a [relatively] positively charged metal plate (the anode).
soon after the triode was invented because of the need for amplication devices. the device was very similar to the diode, but the added component of a 'grid' screen/electrode. the purpose of the grid component was sit between the cathode and anode so as to disrupt the electron flow between the components and effectively control the current passing between the cathode to anode. the controlling effect of the grid was acheived by running a current to the grid itself: as you alternate the current run to the grid you sympathetically divert the flow of the electrons from the cathode and divide them between the grid and the anode.
the HYDRAULIC ANALOGY link in the 'explanation of voltage, current and resistance' offers a great tanglble description of a triode.
the learning curve was brutal, progress was rapid. amplification was noisy and dirty. it broke alot too on account of the high plate voltages(tubes didn't even use emulsifiers back then!). for example, back in the day engineers didn't take into account 'biasing' the grid voltage on a tube and they'd get poor audio quality due to the fact that the tube nominally worked on the bottom of it's operational range and performance was not linear at such levels. later tube circuits would bias the tube to start operation closer to the center of it's 'linear' operating range so that you could get hi fidelity at even now volumes.
many such improvements were made, and people started adding electrodes while improving triode and diode design. so while people were dabbling with pin sets and optimization of triodes, engineers were also adding another couple 'grid' electrodes called a 'screen grid' and a 'suppressor grid' and soon pentode (5 electrode) tubes entered the market. around this time RCA started making tubes for commercial use (~1920)
we got more crafty, learned how to focus beams (beam tetrodes like the 6L6 in 1936), couple multiple tubes into 1 tube (like the dual triode 12AX7 is 2 triode tubes in one package ~1945)
an amplifier is an electrical circuit that takes an alternating current (AC) as input signal and usually increases the amplitude/voltage of the AC signal. amplifiers are used for MANY applications, but the type of amplifier we will be focusing on is a guitar amplifier. this is important to note mainly because:
-there are many principles of amplifiers that are important for an electrical engineering student to understand but will be beyond what the average guitarist will need to know.
-narrowing the application of the amplifier will allow the talk to be more specific.
a guitar amplifier has to be able to accept a very weak passive AC signal that is produced by a guitar pickup and needs to be able amplify that signal many times to an AC signal powerful enough to drive a loudspeaker cabinet. a guitar amplifier does this by dividing the amplification process into 2 main sections: the preamp and the power amp (aka power section).
-the preamplifier section specializes in amplifying low-level passive signals that are generated by pickups or microphones.
-preamps are usually run in a 'class a' operation with 'dual triode' tubes like 12AX7's or 12AT7's or transistors/FETS/opamps performing the amplification. as a side note, it is possible to see pentode tubes used in a preamp section (think EF86's).
-the signal is actually sent through a series of amplifying stages to amplify the signal in small sequential steps, this allows for particular tubes/transistors to perform particular roles in processing the input signal (for example, the second preamp tube is often used as a 'driver' tube, or a tube can be used to buffer send/receives to stuff like reverb tanks or effects loops).
-the 'volume' or 'gain' knob is most often wired in after the first gain stage of an amplifier (this is why it sounds different when running you amp at low volume with the volume knob on the guitar all the way up; as compared to turning up the amp and turning the volume knob on your guitar down).
an important section in the preamp is known as the 'Tone Stack'; you may recognize the interface for the tone stack as the EQ knobs (bass, mids, treble). the job of the tone stack is easy: to rid the signal of unwanted frequencies while being able to adjust frequency response in a pertinent ranges. the reason a tone stack is desirable or needed is quite a bit more complicated, but it has much to do with undesirable frequencies/noise being amplified with the input signal and how your ear/brain hears sounds differently at low volumes and high volumes. the actual implementation of a tone stack consists of a web of capacitors and resistors wired to potentiometers or rheostats.
************************************************** ******* POWER AMP -the roll of a power amp is to take a line level signal (usually provided by a preamplifier) and to amplify that signal to a level that can be sent to a loudspeaker for conversion to an audio signal
-power amplifiers in guitar amps can come in many different modes of operation, but the most common is some type of class A/B push/pull configuration. when an amp is called 'class a' or 'class A/B', then this is the part of the circuit they are referring to.
-the signal from the preamp may need to be prepared for the power section, so at
times a dual triode tube must be used as a phase inverter (PI) for class A/B
push/pull amps. -depending on the mode of operation, the power section may amplify the signal many different ways; but power sections will generally provide amplification in 'one stage' as opposed to the multiple stages you see in preamps. in tubes amps, much more powerful beam pentodes and kinkless tetrodes are commonly used for power amplification (but a triode in the power section can be used, like the 12BH7).
-the 'master volume' is normally wired into the circuit after the preamp and before the power amp. this is why you can't get power tube saturation with the master down.
after the power section of a tube amp the signal is usually sent to a transformer to convert the final signal from the amplification into something appropriate to drive a loudspeaker (though there are transformer-less tube amp designs out there). solid state amps usually lack this 'output transformer' because the impedance of the output AC signal from the transistors in the power section is low enough impedance to be properly 'loaded' with a speaker cabinet. most guitar amplifiers have certain impedance requirements from the speaker attached to the amplifier:
-tube amplifiers usually require a 'matching impedance', meaning the outputs of the amplifier are label and a speaker cabinet with corresponding impedance must be matched to the speaker jack.
-solid state (SS) amplifiers use impedance bridging instead of impedance matching (because of the lack of the output transformer), so usually they require a minimum impedance from a speaker cabinet, this means that any speaker cabinet above the minimum setting may be used. solid state amplifier output is usually dependent upon the impedance of the cabinet (example: 100 watts @ 16 ohm, 200 watts @ 8 ohm).
sometimes speakers are included with a guitar amplifier (a combo) and sometimes the speaker and head components are separated (piggy back or stack). the benefits and merits of these setups will be covered in the 'speakers and cabinets' section.