# Room Ratios

The vast majority of folks attempting to develop a well isolated acoustically correct room will stumble on a few phrases that often make the process of moving forward difficult.

One of these phrases will be “room ratios”. Why even consider the dimensions of a room and how it relates to sound?

Since it does relate to sound, the idea is that these room ratios are very important when in fact, as important as they are, it is simply a small aspect of the build.

The rooms size and shape will determine the modes (wave length and frequency associated with your rooms dimensions).

Small rooms are known to have issues with low frequency since a small room will accumulate longer low frequency waves that will overlap and build up in corners if they cannot locate an exit from the room.
This always presents a challenge to overcome without some understanding of what you are up against.

In some cases, as in using an existing room in your residence, it may not be possible to move any of the existing walls!

So you move forward, knowing that you will have to face this issue down the road, the issue being the existing modes the room might present.

The Golden Ratio

Reference: http://en.wikipedia.org/wiki/Golden_ratio
Golden ratio – Wikipedia, the free encyclopedia

The golden ratio is an irrational mathematical constant, approximately 1.618.

The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years.

According to Mario Livio:
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties.

But the fascination with the Golden Ratio is not confined just to mathematicians.

Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal.

In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics

Room dimensions typically start from the ceiling hard boundary.

If we use the golden ratio then this will produce a room with dimensions based on the ceiling height, then to the width of the room and next the depth of the room.

With an eight foot tall ceiling, based on the Golden ratio we would use 1: 1.62: 2.62, this would be a room with a ceiling height of 8 foot with a width of 13 feet and a depth of 21 feet!

WOW!

That is a lot of room which is certainly good for the music to expand but you may not have this kind of floor space or money to build something so large.

Imagine if you were lucky enough to have a ten foot ceiling, the room would be so large as to become too expensive to build.

Optimized Room Ratios:

Room ratios have been around for decades and rooms in general have been designed to be pleasing visually and sonically by many builders/ architects.

You only have to look around the building you are in right now to recognize this is not something new, it has been around you all your life you just were not aware of this.

Rectangular rooms have been a staple of residential houses since the late 1800’s due to ease of construction when more room is required but a square room means more material and that means more cost per square foot.

The inclusion of room ratios for an acoustical environment is different in that the ratios used have been tested so that the builder has a better starting place than a typical room ratio that a typical builder might use.

Alton Everest ( Master handbook of Acoustics – 4th Edition) presents 3 of the most widely used ratios developed by L.W. Sepmeyer (1965) and M.M. Louden (1971)

```
Height	Width	Length
____________________________________
Sepmeyer	A.	1.00		1.14		1.39
B.	1.00		1.28		1.54
C.	1.00		1.60		2.33

Louden	        A	1.00     	1.4		1.9
B	1.00		1.3		1.9
C	1.00		1.5		2.5
```

It must be noted that when these ratios were being tested, a ten foot tall ceiling height was assumed. In some detail, we will find out why this was an important aspect to this particular testing procedure.

Based on Loudens first ratio “A”, 1.00, 1.4, 1.9 with a ten foot tall ceiling this would produce a room with the interior finished ceiling height of 10 feet with a interior finished width of 14 feet and a depth of 19 feet.

This room will have a volume of 2,660 cubic feet.

Plenty of height for the sound to expand and develop and exceeds the 1500 cubic feet room volume limit determined to be the least amount of volume a quality audio environment should have.

(C.L.S. Gilford, Affiliation: British Broadcasting Corporation,“The Acoustic Design of Talks Studios and Listening Rooms” circa 1979, maintained that a “small” room based on the research done would be a room with a volume of 1500 cubic feet.

Further he states “It is shown that a distinctive characteristic is that, because their dimensions are comparable with the wavelength of low-frequency sound, the sound field is characterized by strong simple standing-wave patterns which cannot be eliminated without eliminating the reverberation itself.

It is shown also that for the audible effects are confined to those associated with simple axial modes and that, by careful adjustment of dimensions, provision of diffusion and the proper distribution of absorbing material, the worst faults can be avoided.”

An interesting thing happens when we look deeper into these ratios, when we look at the single components of the room and not the end result.

The speed of sound at sea-level is considered to be 1,130 feet per second and in order to get the fundamental frequency of the height or width or length we have to use the equation F=1,130/2xD.

The height of ten feet using the above equation will produce: 1,130/20=56.5Hz.

This is important to know since 56.5 Hz relates to the note A1. It actually falls 1.5Hz past the frequency of 55Hz.

The width of 14 feet using the equation F=1,130/2xD (1,130/2×14(28)) = 40.36Hz which closely correlates to 41.20Hz or E1 on a midi keyboard.

The remaining length measurement 19 feet X 2 = 38 produces 1,130/38=29.74Hz, relates closely to 29.14 (A#0/Bb0).

Using the 8 foot ceiling height and Loudens first ratio produces a room 8 feet tall, 11 feet and a few inches wide and 15 feet and a few inches deep.

That is about the size of a typical bedroom or the living room in some homes.

The consideration for having a balanced proportional room is valid and worth the effort to use in any sound related type room.

A few things to consider along the way. The measurements that are obtained from the ratios define the interior side of the wall.

In order to use these measurements, you must determine how much and of what thickness your interior sheathing will be.

This allows you to step out the placement of the sheetrock or MDF/OSB or whatever combination you may use, in order to establish were the actual framing will be placed on the floor of your build.

[needs graph on this stepping out procedure]

To that end, ratios are not scalable…they cannot be modified and expect the same results: http://www.acoustics.salford.ac.uk/acoustics_info/room_sizing/
Room Sizing Tutorial | Acoustics, Audio and Video | University of Salford

# Room Modes

I have a room that measures 11 feet and 6 inchs one way and 11 feet and 6 inchs the other way. It has a ceiling that measures 7 feet and 6 inchs tall. Can I use this as a recording studio mixing room?

If you want to figure Axial room modes by hand it is a simple equation:

c= The Speed of Sound at sea level is considered to move at a static speed of 1,130 feet per second.
W= The measurement in question.
f = The frequency that correlates to the results of the equation.
Then divide that by 2. Reason being is that you have to make a round trip, from one wall to the other.

``` Your room has two sets of measurements that will produce the same results, e.g. it is a square room but anyway, 11'-6" looks like 11.5 to a calculator. ```

1130/11.5=98.26 /2=49.13 or 49Hz

If you do the other wall you get the same results of 49Hz. So these frequencies will support each other meaning that you will get a 49Hz Axial mode from one set of parallel walls and another 49Hz Axial mode from the other set of parallel walls. Two strong and disruptive room modes that support each other and we barely started.

Some will take the “measurement in question” and multiple by two before dividing into “f”, but the results are the same.

For instance, 1130/(11.5X2)=49.13 or 49Hz.

Since we are in this room let’s take a look at the ceiling height 7′-6″ 1130/7.5=150.66 /2=75.33 or 75Hz.

These are the first order Axial Modes,49Hz,49Hz,75Hz. In order to determine other possibilities you have to consider multiples of these frequencies to see if there is continued support or exactly how smooth the room response will be. Will the frequencies be spread out(good) or will they be clumped up(not so good).

If the first order Axial of 49Hz is brought to the third order, 49HzX3, it becomes an third order 147Hz room mode.

Stay with me….

If the first order Axial of 75Hz is brought to the second order, 75HzX2, you get a second order room mode of 150Hz.

You now have a third order 147Hz mode from wall to wall being supported by a second order room mode from floor to ceiling at 150Hz. Only 3 points separate these room modes from being a direct hit, so to speak.

So the square area has proven to be an ineffective area mathematically for our want. Also, this will continue to rule it out in future decisions unless modification of some sort is considered.

You can learn this and more in The Master Handbook of Acoustics

# Small room acoustics for recording or theater playback

A typical small recording room is going to be made in a room available in the home. There are magnified acoustical issues with small rooms that are not issues at all with medium to large rooms, rooms of 1500 cubic feet or larger.

Let us use a room of the dimensions, 12 feet in length, 10 feet in width and 8 feet in height.

The first thing we want to do is to understand how sound is going to be in this room, how it is going to interact with the space as sound becomes the small room acoustics.

Each Frequency has a known length, that means that ever note that you can play, you can determine mathematically how long in feet and inches is the frequency in question.

The simple equation is: 1130 / (measurement in question) / 2= f
The speed of sound is 1130 feet per second.
The “measurement in question” is the length or width or the height of the room in question.

And f is the frequency that presents the issue.

What is the issue?

The issue is that the boundaries of a room present modes also known as standing waves. This is the point in the room where a frequency will seem louder and distort the listeners ability to make valid judgment.

These are our points of interest that will need treatment or require setup of listening position knowing what it is the room presents to us as a listener at the mix position.

Let’s look into the axial modes of a small room 12 feet in length, 10 feet wide and 8 feet in height.

Using only the axial modes we can get an understanding of how complex the sound field will be in this room, which aids in the task of making a more sonically pleasing environment.

1130/12(L)=94.17 Hz /2=47.1Hz (1,0,0)
1130/10(W)=113 Hz /2=56.5Hz (0,1,0)
1130/8(H)=141.25 Hz /2=70.62Hz (0,0,1)

Below 47.1Hz there is no modal support, this means that every frequency below 47.1Hz the room is basically invisible and you have no control of this frequency region.

To continue, we will define all the axial modes in the L,W,H dimensions up to 300Hz.

To do this take each individual frequency, starting at the (L)ength and add the defined frequency to itself, keeping a written record of each resultant number until you get to 300Hz (+/-).

```Length Frequency WaveLength
L=47.1 47.1 24feet(1,0,0) 47.1+47.1=94.2
94.2 12feet(2,0,0) 94.2+47.1=141.3
141.3 8feet(3,0,0) 141.3+47.1=188.4
188.4 6feet(4,0,0) 188.4+47.1=235.5
235.5 4feet 10inches(5,0,0)
235.5+47.1=282.6
282.6 4feet(6,0,0)

Width Frequency WaveLength
W=56.5 56.5 20feet(0,1,0)
56.5+56.5=113
113 10feet(0,2,0)
113+56.5=169.5
169.5 6feet 8inches(0,3,0)
169.5+56.5=226
226 5feet(0,4,0)
226+56.5=282.5
282.5 4feet(0,5,0)

Height Frequency WaveLength
H=70.62 70.62 16feet(0,0,1)
70.62+70.62=141.24
141.24 8feet(0,0,2)
141.24+70.62=211.86
211.86 5feet 4inches(0,0,3)
211.86+70.62=282.48282.48 4feet(0,0,4)
```

These are the problem areas that the room presents in the form of axial modes.

There is an unfortunate area of this room where the axial modes are reinforced.

L,W,H mode(3,0,2): The frequency of 141.3 found in the length of the room and 141.24 found in the height of the room is one area of concern.

L,W,H mode(6,5,4): The frequency of 282.6, 282.5 and 282.48 are all present in the length, width and height respectively. The above graphic is a demonstration of how and where the Axial modes line up in this room.

```Red    lines=Axial Mode 2
Yellow lines=Axial Mode 3
Blue   lines=Axial Mode 4
Black  lines=Axial Mode 5
Green  lines=Axial Mode 6
```

In the above graph of this room you see that axial mode 3 and axial mode 6 in the length measurement of the room are right on top of each other!

Not to leave any stone unturned, there is one more measurement that is the longest of either of the length, width or height and that is the distance from the top corner of the room to the bottom cross corner. This distance is 17 feet 6 5/8 of an inch and represents an oblique mode. The frequency associated with this distance is 32.2 Hz .

Nuances of frequency crossover are to be expected. With this fourth measurement we have yet another frequency added to the mix, sonically and literally.

And if cross over is to be expected we are not let down. Like a musical network of handholding, this 32.2Hz frequency interacts with the mode 2 frequency of 94.2Hz (32.2 X 3 = 96.6Hz) that is found in the (Length) of the room.

32.3Hz X 7 = 225.4Hz and that is a match for the mode 4 frequency of 226Hz that corresponds with the (Width) measurement. That is a close hit musically speaking. 