I come to a discussion about aspect ratio as a photographer who generally doesn’t think about specific aspect ratios in my images. I look at an image and crop according to the dynamics of the photograph (with the possible exception of square images, which I like to be exactly square). It is when a photographer is talking about cropping an image and wondering which fixed aspect ratio would suit it best, as if non-standard ratios were not a possibility, that I find myself wondering why.
Is there something inherently satisfying about the standard ratios, which make other ratios look unsatisfying in some way?
Ratios and musical sounds
This got me thinking about some very basic mathematics, and to help me I thought about a phenomenon in sound, which I know about through being a musician, called the harmonic series. If a single note is played with a frequency f, a series of additional notes will automatically sound, and these “overtones” or “harmonics” are at the frequences 2f, 3f, 4f, 5f, 6f, and so on. The relative amplitude or volume of these harmonics gives what most people would call the “tone” of the instrument. What becomes interesting for anyone looking at ratios is that the relative simplicity of mathematical ratios give rise to simple and pleasing musical ratios.
The most significant facts about the harmonic series:
The notes of the harmonic series, as each of them is added, first form a standard chord (major triad), then a dominant seventh, and then a ninth. These are the most common chords in music.
The most simple frequency ratios (as found in the harmonic series) produce the simplest musical intervals; In fact, the simplest ratio has the most consonant (blended) sound and the most complex ratio has the most dissonant (clashing) sound. This gives rise to a hierarchy of dissonace. This is as follows;
· Octave (simplest ratio) 2:1
· (Perfect) 5th (next simplest) 3:2
· (Perfect) 4th 4:3
· Major 3rd 5:4
· Minor 3rd 6:5
· (Diatonic) semitone 16:15
The most clashing interval in music, with the most complex ratio (see above) is called a semitone; this can be found on a keyboard, for example, by playing two adjacent white notes where there is no black note in between.
3. Mathematical ratios in musical pitch therefore have an aesthetic corollary: simple ratios have consonant sounds; complex ratios have dissonant sounds.
Photographic image aspect ratios
Where does this leave photographers and different aspect ratios, and whether or not there is a mathematical basis for the aesthetics of different proportions in their images? Do the most satisfying and commonly used sizes reflect simple ratios? Is there a hierarchy of satisfyingness as in the musical intervals? Or do the dynamics of individual images totally over-ride the simple dimensions of the image? What are the dimensions of an image – do they represent the actual edge of a reality we are asked to look at, or are we looking at an arbitrary section of reality, where borders are an inconvenience because we cannot have an infinite image? Of course, these considerations occupy the minds and eyes of photographers who might, for example, be very careful that edge details do not “leak” out of the image, or who commonly like to ensure a scarcely noticeable vignette which darkens the outer area of the picture to “draw the eye in” to the main part of the picture; but the whole varied array of things that a photograph can be make this a complex subject where rules for some photos don’t seem to fit others.
Back to ratios! Here is a chart of commonly used photo sizes. I have included a column for “mathematical simplicity” to mirror that of the harmonic series in music – the lower the number the simpler the ratio. where this is not clear I have put a question mark.
Looking at most sizes above we can see how exactly the same ratios are present as in the harmonic series, which seems to prove the case relating mathematics to aesthetics; but is any hierarchy of ratios which puts the 1:1 ratio as most simple/easy on the eye and the 5:4/10:8 the least simple? I don't think this has been answered….
Next we look at the three ratios at the bottom: The “A” series, the golden ratio, and US letter size. With the A series there is a simple ratio (√2:1) – which of course is why A series paper folded into two produces A series dimensions; and with the golden section the ratio of the height (y) to the width (x) is the same as the sum of the height and width (y+x) to the longer of the two sides (y:x = y+x:y); US letter size is purely historic and related to the convenience of making paper (something about the size of a vatman’s arms!).
The above will give some justification to the view that the standard photo sizes/aspect ratio have a sound base in mathematics, by linking mathematical simplicity with aesthetics, and the harmonic series thought of as a musical example of a similar link.
A few thoughts
But…. not quite: there are some loose ends to be tied up…
1) There doesn’t seem to be a hierarchy of simplicity as there is with musical intervals, linked to the degree of mathematical simplicity of their ratios
2) There are plenty of other simple ratios which generally do not generate satisfying aspect ratios; notably, 2:1, or √3:1 etc. etc. Could any aspect ratio be given some kind of ratio or other to justify its proportions?
3) Photographs have their inner dynamics: we are not just looking at blank rectangles
4) Purely from an aesthetic point of view, does a photo even need to be rectangular/square?
5) Thought: photographers love to speak of the golden section and find it all over the place not only in photo dimensions, but within images – why do none of the standardized sizes available from printers conform to this ratio?
6) Not mentioned above is the 16:9 ratio: the maths of this ratio require an extremely mathematical brain - therefore a compex ratio - but it has become standard in the moving image world.
The convenience factor
I have noticed that photographers are increasingly making images in A4, A3 etc. This highlights the factor not yet considered, which I believe trumps the others: the convenience of standardization. For example, buying an A4 frame, or other standard size, is very cheap compared to having a custom size made, and there are plenty of other circumstances where having a small range of set sizes makes life a lot simpler and more practicable. If the convenience rationale is indeed true, or even partially true, then photographs may sometimes be being put into a standardized straitjacket totally unconnected to aesthetics.
Conclusion
I’m not sure if there is a clear conclusion from all this, but I will take from these thoughts a self-justification for not paying too much attention to set aspect ratios, and I won’t start asking myself whether such and such an image would be better as a 5:4 or 6:5; but I’m ready to admit that there is some mathematical justification for these set ratios.
I hope the link to musical intervals may have been of interest, whether or not you have an interest in the physical basis of sounds.
Finally, I’m certain that there is a wide range of views on all this, and my intention is only to stimulate discussion!