What is a piano tuner listening to when tuning?

All about beats

A piano tuner will explain that when tuning they are listening to "beats" - being able to recognize and correctly judge the speed of these is one of the key skills developed in the course of learning to tune. So I though it might be useful to put something on the blog about what these "beats" actually are, in physical terms, and why listening to these is an essential part of the process when tuning. To be a tuner you don't need the ability to hear low-flying aircraft, noises that only bats or dogs can normally hear, or strange mystical powers. What you do need to do over time is to train yourself to recognize particular sounds  within the large number of apparently random noises coming out of a piano.

Firstly, very few tuners have perfect (or absolute) pitch, namely the ability to recognize the pitch of a sound ("that's an F#") instantly - this is a rare gift even amongst musicians. It's very rare amongst tuners as well, and not always an advantage!

For the rest of us (and that includes me), to get a piano to pitch, it is necessary to tune the first string either to a tuning fork (the traditional way) or to an electronic tuner. Personally, I use an electronic tuner for the pitch of the first string only - everything is then tuned by ear relative to that first string. For that reason, a tuner needs to be able to recognize different musical intervals and have a good relative sense of pitch. Normally I use middle C as the reference point, but many tuners will use A instead.

The next thing is to get the two unison strings of middle C in tune with the first string.This is where the tuner needs to be able to recognize "beats" in the sound of two strings together, when one is slightly out of tune with the other. It would help at this point to explain why "beats" occur, and what actually causes these sounds in practice.

This graph represents a sound wave as a sine curve:
 

This is a simplified model - of course in practice the sound isn't a perfect sine wave, and the curve is simply a representation of compressions and rarefactions in the air itself - but it does help to illustrate how two sound waves combine together. This second graph shows the effect of two sound waves perfectly in tune with each other - one is coloured green and the other yellow, but you can't see the yellow one because it's covered by the green one. The red curve (of double the amplitude) is the combination of the two.

As you can see, the two sound waves just reinforce each other.

Now watch what happens if the green and yellow sound waves are at a slightly different frequency:

In this particular illustration, the yellow sound wave is at a higher frequency than the green sound wave; the ratio is 16:15 - this isn't typical of the kind of ratio you would normally be listening to as a tuner, but will do very nicely to illustrate the principle. The combination of these two sound waves gives the following result (red line):

You can see that the two waves start off in phase, so the combined sound wave is nearly double the amplitude; however, as the green and yellow waves diverge from each other, they start to cancel each other out, so the red wave goes almost down to nothing, then starts increasing in loudness again.

This graph shows the same pattern - this time I have increased the frequency of the green and yellow waves and the ratio is now 51:50. You can clearly see that the red wave - the result of addition of the green and yellow - shows a clear, regular "beating" pattern which is the consequence of wave interference. This is what a tuner is listening for when attempting to put the two strings in unison.

Here's the same graph except that I have now increased the frequency of the yellow wave again, so it's now in a ratio of 52:50 to the green one. As you can see, the "beat" is now faster - double what it was previously. This illustrates the important point that as two strings get further apart in frequency, the speed of beating increases until the strings are just obviously out of tune. The other thing to note is that the speed of beating tells the tuner how far away the two strings are from each other, but not whether the string being tuned is higher or lower than the reference string - that has to be worked out by listening to the speed of beats changing as the tuning pin is moved.

Obviously, the tuner's objective in this case is to try and get rid of the "beating" sound, which should put the two strings in tune. So this is fine for getting two strings perfectly into unison with each other, but that isn't good enough if we want to tune a whole piano - we need to be able to tune other intervals as well. In practice, a tuner will put in a "scale" or temperament octave - that is tuning every note in an octave around the middle of the piano and then work outwards in octave steps from there. I'm not going to deal with exactly how the scale octave is done now because it involves an understanding of musical temperament (in particular equal temperament, which is nearly always used in piano tuning) and in practice, this process involves putting in "beats" instead of just getting rid of them. I expect to return to these subjects in some future posts, but for now I'll deal with the simpler case of tuning an octave.

The first thing to understand is that two notes an octave apart should have a frequency ratio of 2:1, so the tuner will need to be able to hear the result if two notes are in a perfect octave or slightly out. For this reason, I've set up the graph to show the resulting pattern for an 81:40 ratio - i.e nearly an octave but not quite, to see what results. Although there is some kind of regular change in the wave pattern, there is no clear variation in amplitude (loudness) as with the case of the near-unison we looked at earlier. The bottom line, in fact, is that this does not produce any clearly discernible "beat" which is of use to the tuner; how, then, is it possible to hear when a octave is in tune?

Hearing the harmonics

What saves the day here is a property of a vibrating string fixed at both ends: as well as its "fundamental frequency" (i.e. the soundwave produced when it oscillates along its entire length), it also has separate vibrations through 1/2, 1/3, 1/4, 1/5 of its length and so on. These are called upper harmonics or partials of each note, and help to give the tone of an acoustic piano its richness.

This is called the harmonic series. What this means is that if we look at a particular note (for example the C two octaves blow middle C), it is producing all the following sounds:

The fundamental frequncy: C two octaves below middle C
2nd harmonic: C one octave below middle C
3rd harmonic: G below middle C
4th harmonic: Middle C
5th harmonic: E above middle C
6th harmonic: G above middle C
7th harmonic: Doesn't quite correspond exactly, but close to Bb above middle C
8th harmonic: C above middle C

Yes! Really, all these noises are coming out of your piano when you play just the one note. In fact, the series goes on ad infinitum, but in practice the harmonics above the 8th are not really important for piano tuning and above about the 12th they are very weak indeed, to the point of being virtually undetectable. Also, a bass note will have more audible harmonics than one in the treble - the additional partials get progressively weaker the higher the note.

If you're interested in a longer explanation, there's one here:

How does this help the tuner? Well, if two notes an octave apart are being tuned, the second harmonic of the lower note should be at the same frequency as the note above, and thus produce a recognizable beat, exactly the same as a unison, if the notes are slightly out of tune; thus, the tuner is trying simply to remove the "beat" in exactly the same way as before.

So that's how an octave is tuned - for completeness, I should briefly add that when an octave is tuned on a piano, it's actually slightly wider than a theoretically perfect one. The reason for this is that the strings on a piano aren't "ideal strings", i.e. perfectly flexible; rather they have stiffness and this results in the upper harmonics being slightly sharp of where they ought to be. The technical name for this is "inharmonicity".

But that's enough for now - tying up some of the loose ends should provide topics for future posts!

Yorkshire Piano Makers (2): Waddington and Sons

Here's Stonegate - one of York's most picturesque ancient streets and a popular destination for visitors; but I wonder how many of the people walking past these picturesque shops realize that they were once the site of one of the largest piano factories outside London?

Below is a picture of Stonegate, from nearly the same place, some time around 1890. Across the street is the sign for "Boddy's Star Inn", now known as the "Olde Starre Inne" (there is still a sign in just the same place), but above and to the left of it, there is a sign saying "Waddington". In the 1901 edition of Kelly's directory for York, we find that Nos. 43, 44 and 45 Stonegate were occupied by "Waddington and Son, Pianoforte Manufacturer."

There was a varied list of other activities going on in Stonegate at the time including a cycle maker's, a wire worker, a dancing academy, an electrical engineer, a taxidermist, an antique dealer (then as now!), a fancy repository (roughly a high-class gift shop), a baker, a tailor, a dressmaker, a coal merchant, a watchmaker, a solicitor.... it seems you could get just about anything you might have wanted here at the time. However, such was the popularity of the piano in the period, apart from Waddington's there were two other shops in the street selling them as well - a Mr H Fordham and a Mrs E Bell, both on the opposite side. Both are described as "piano warehouses" so presumably they weren't actually making instruments, though they might well have been doing repairing and tuning work. York was certainly "piano city" at the time because Noyes and Son, piano dealers, could be found in Tower Street, and in Coney Street there was Gray and Sons piano warehouse, as well as a "pianoforte saloon" run by A Ramsden Ltd - though whether of the Wild West variety I cannot tell! (Ramsden crops up a little later in our story, as well).

York in fact has a history of strong connections with keyboard instrument manufacture dating back to the eighteenth century - Thomas Haxby of York (1729-1796) based at 28 Blake Street, produced organs, spinets, and latterly square pianos, which are considered some of the best made during the period. He even merits his own entry in Wikipedia.

It took a bit more work to unearth details about the business of Waddington and Sons, but "The History of Stonegate", by John Ward Knowles, from the York City Archives, offers some interesting insights. An earlier nineteenth-century piano maker in York was a Mr Marsh of Coney Street, who had been a former employee of the famous firm of Broadwood and Sons, but later took to becoming a dealer. Knowles says that "in 1838, William Alfred Waddington came to York and commenced in a small way to manufacture pianos, occupying some workshops behind the Star Inn." From this it can be gathered that he wasn't originally from York - in fact it appears from fragments of information on the internet that he was a native of Everton (Liverpool). Knowles tells us that he died in 1896 at the age of 79, which would put his year of birth as 1816 or 1817.

He was married to Mary Ann Waddington (nee Hunt) who was born in York - but intriguingly she again has a connection to the piano industry. According to Knowles, her (older) brother, Richard Hunt, was working as a hairdresser at No.2 Stonegate in 1840, but by 1843 he had adopted music as a profession and had moved just around the corner to 23 Blake Street, "where he had opened out a musical instrument department and commenced the manufacture of pianos." Whether it was Waddington who inspired Hunt to start off in the piano trade or vice versa is unclear. Apparently in 1851, Hunt produced a "new shaped instrument which appeared externally to resemble a centre table  for a drawing room with  pedestal feet and which he had exhibited at the Great Exhibition of that year." However, by the 1860s he had moved to Scarborough as a hotel owner and was the founder in 1873 of the Scarborough South Cliff Tramway Company, which built the first ever funicular railway in Britain (it still operates today). Possibly he may have handed his residual piano business over to his brother-in-law.

Some more information appears in "A musical place of the first quality - a history of institutional music-making in York, 1550 - 1989", by David Griffiths. He notes that Waddington's took out an advertisement in the York Herald in 1862 thanking patrons for their business over the last fourteen years (implying establishment in 1848) but in the 1920s the firm claimed to have been established in 1838. Whatever the truth of the matter, the firm was reported as employing 135 to 150 people by the 1860s and by 1876 said in an advertisement that it had sold 10,000 instruments. Clearly if these statements are anything like correct then the firm must have been significant within the piano industry - for comparison, the famous London firm of Broadwood and Sons was employing perhaps 500 people at its peak during the late Victorian period. Thus, the claim to have been the "largest piano manufactory outside of London" in 1871 may not have been an enormous exaggeration. Despite this, the firm seems to have attracted very little attention from piano historians - it does not even merit a mention in the list in Alfred Dolge's "Pianos and their Makers" from 1910.

William Alfred Waddington, the firm's founder, seems to have been responsible for at least two patents: No.972 from 1st May, 1854, described as "Improvements in the construction of sounding-boards for pianofortes and other like stringed instruments", and No.3187 of 28th November 1862 "for an invention for Improvements in machinery for cutting wood." However, he also seems to have seen his fair share of legal actions of one kind or another; for example on 19th May 1862 he was granted a "deed of arrangement and composition" - which means an arrangement by a debtor to pay creditors a percentage of the money owed to them (7s 6d in the pound, in this particular case). The main creditor seems to have been the York Union Banking Company respresented by George Dodsworth - this company was established in 1833 (the "Railway King", George Hudson, was one of the original directors) and was eventually absorbed by Barclays in 1902.

Another intriguing snippet of information from the internet concerns a legal dispute in 1879 between William Alfred Waddington and Archibald Ramsden the same who had a "piano saloon" in Coney Street. Unfortunately the details of this action are not available online (only by ordering from the National Archives). However, the said Archibald Ramsden was a performer and impresario, born in 1835, who returned to his native city of Leeds in 1864 to open a shop selling pianos, harmoniums and sheet music (later there was also a shop in London). In fact, there is even a picture of his main showroom in Park Row, Leeds in an advert from the 1870s. Unfortunately this building (a stone's throw from City Square) is now demolished.

Some pianos were manufactured with the name "Archibald Ramsden", but as far as I can determine there is no record of Ramsden's having their own factory. This would not have been unusual at the time - in fact it was commonplace for music shops to have pianos made for sale under their own name by other manufacturers, rather like an "own brand" product. These were known as "stencil" pianos, a system that remained commonplace probably at least until the 1930s. It seems possible that Waddington's were supplying Ramsden with pianos, possibly under the Waddington name or possibly Ramsden's. Whatever the cause of the argument in 1879, it seems at least a distinct possibility that it was piano-related.

I decided to pop into York city centre on a Saturday afternoon to investigate the site of the Stonegate factory. The buildings in Stonegate were subsequently renumbered - I'm not sure exactly when this happened - but 43 and 44 Stonegate are now number 34, whilst 45 Stonegate is now number 32. These buildings are now occupied by the the "White Stuff" clothes shop and by Cath Kidston. The staff in both shops were kind enough to allow me to take photos inside the buildings.

Here's a picture of the front of the two buildings as seen from Stonegate. According to the register of historic buildings in York Library, the property on the right (No.34) was built around 1730; it was originally two houses (hence it originally had two numbers). No.32 on the left was built somewhat later, in the early part of the 19th Century. These two buildings housed the warehouse and shop of Waddington's, and upstairs were music rooms (and presumably also offices).

This is a picture of the tiled staircase in No.32, leading up to the first floor - whether it was the same back when the building was a piano warehouse I don't know, but the tiles certainly look fairly old.

Here's a picture of the first-floor front room at No.34 - possibly this was once used for musical soirees?

This is a picture taken from the rear window of No.32, now Cath Kidston's. The building at the very top right of the photo is the rear extension to the Masonic Hall in Duncombe Place, built in the 1930s and which backs on to the Olde Starre Inne (therefore very likely the site of the original workshop used by Waddington's). There were more buildings over what is now the grassed area in the late nineteenth and early twentieth centuries, so this area was very likely part of the factory at that time.

J. W. Knowles gives us some further information on the history of the firm. He says: "In 1920 the removal of the manufacturing part of the business was considered to be necessary, the premises in Stonegate being old and much cut up into small sections. Therefore a new and up to date factory was planned and commenced at Scarborough and in 1922 the whole of the Stonegate premises were cleared and the two warehouses fronting Stonegate converted into shops, which are to be made into show rooms for the sale of their pianos, and on August 17th two large and artistic posters were affixed to the plate glass windows which had been fixed the previous week."

Until recently, a 36-minute film from the Yorkshire film archive showing operations at the Waddington factory in Scarborough in 1928 was available online (you may still be able to access a copy by contacting them). This factory became part of, or at least was very near, the site of Plaxton's Coachworks in Seamer Road. Part of the building survived in the form of the Mere Social Club (originally the Waddington Works Club); however, this too was demolished in 2012. The Coachworks itself happily survives as a local business, though now it has moved to more modern premises just down the road in Seamer.

Unfortunately, if the directors of Waddington's back in 1920 had had a crystal ball, they might have thought twice about their investment in new manufacturing premises, because the firm succumbed to the onslaught of the wireless, the gramophone and the Great Depression as did most other British piano manufacturers in the early 1930s, and just like Pohlmann's, the subject of my last post. In fact, information on their time in Scarborough is relatively thin, perhaps because they were only based there for a little over 10 years.

The Herne Hill Piano

There is an interesting addendum to this story. Later on, the Waddington factory made pianos under the name "Bremar", and one of these later became something of a celebrity in its own right. Here's a film about it:

Judging by appearance, the piano probably dates from around the 1920s, so it could have been made in either York or Scarborough. Apparently the original Herne Hill piano was retired in late 2016 - however the crowdfunding project for a replacement was so successful that some money has also been raised to help pay for piano lessons for children whose parents can't afford piano lessons.

Yorkshire Piano Makers (1): Pohlmann and Son

I've recently tuned two customers' pianos made by the firm of Pohlmann & Son. Historically, the British piano-making industry was concentrated mainly in London, especially around the area of Camden Town. In his book "Pianos and their Makers" from 1910, Alfred Dolge lists just three piano makers outside London, all of them based in Halifax; one of these firms was Pohlmann's.

 

This is one of the Pohlmann pianos I tuned, dating from around 1905-1910 - pictures reproduced by kind permission of the customer

The origins of the firm seem to be swathed in a certain degree of folklore, because Johannes Pohlman was a noted early maker of square pianos in London between 1768 and 1790. He is often described as one of the "twelve apostles", a group of German and Dutch instrument makers who seem to have emigrated to England around the time of the Seven Years' War (1756 to 1763). Whether there were in fact twelve of them, and exactly who they were, is the subject of some degree of debate, but Johannes Pohlman's instruments are some of the earliest keyboard instruments produced in London and those that survive are of great historical interest.

The Yorkshire firm of Pohlmann's was founded in 1823 by Henry Pohlmann (though Dolge gives the date of establishment as 1832), who evidently grew up in the local area. Local researches suggest that his father was born in Marburg in Hesse, though it is not known why the family originally came to Halifax. Unfortunately information is so scant that it seems impossible to say with certainty that Henry Pohlmann was in any way related to his distinguished predecessor, though the firm seems to have later on claimed, or at least hinted, that this was the case. However, the history of Pohlmann and Sons itself is of great interest, as one of the first and most important firms to base itself outside London.

It does seem that the firm was exceptionally progressive in adopting new innovations. Some information can be discovered from a business directory "Dublin, Cork and South of Ireland", by Stratten and Stratten, 1892, on account of the company's showrooms at 40 Dawson Street, in the centre of Dublin. A picture of the showroom is included:

The Dublin building still stands and today houses the Café en Seine. Does the roof vaulting look at all familiar?

 

The firm's entry in the directory states that they were the first manufacturers in England, apart from Erard's of London, to use the 7¼ octave (88 note) keyboard that is standard on all modern instruments (a great many older pianos have a 7 octave or 85 note keyboard). Also,the firm adopted full cast-iron frames in 1870 and overstringing in 1871, and the article states they were the first English manufacturers to do so. Apparently at this time, the proprietor of the business was George H Pohlmann, and he personally inspected every piano made before it left the factory.

Another clue comes from the official catalogue of the Yorkshire Exhibition of Arts and Manufactures, held in Leeds in 1875. An advertisement for Pohlmann & Son appears and refers to "Prize Medal Upright and Oblique Grand Pianoforte Manufacturers." More significantly, it states that they were the "Only Manufacturers in England of the American Model Upright Iron Overstrung Grand Pianofortes." This perhaps requires a little explanation. Firstly, it may seem a little confusing that the terms "upright" and "grand" appear together, but "Upright Grand" was a term used, particularly in the nineteenth century, by manufacturers - it might be taken to mean a large upright, but essentially it means the same as an upright piano. More importantly, however, it refers to the "American model." This term refers to the system of an overstrung bass with a full cast-iron frame, because the first piano of this type was built by Steinway's of New York in the 1850s. Most German manufacturers were quick to embrace the new technology, but it took much longer to find favour with British piano-makers. Perhaps the German ancestry of Pohlmann's founders and their location outside London helped them resist the conservatism of the rest of the industry. Certainly, their adoption of these innovations by 1875 means that they were early enthusiasts for the new way of building pianos - the way, in fact, all modern pianos are built.

The article refers to the Company's showrooms in Princess Street, Halifax and their factory in Hall Street. The building on Princess Street is now a Turkish restaurant called the Olivetta (which incidentally seems to have some good reviews on Trip Advisor if you are minded to visit):

The location of the "steam factory", as it is described in 1875, took a little more tracking down. The original buildings on the west side of Hall Street were all demolished at some point to make way for a dual carriageway relief road, but some online maps of Halifax around 1890 showed a building in Hall Street marked "piano manufactory", and the building was on the east side of Hall Street, where some of the original structures remain. And indeed, the factory still stands - it is today known as "Rimani House" and is a suite of offices, with Calderdale Carers' Project amongst the tenants:

Back in 1890 there was an iron foundry immediately opposite on the now-demolished west side of the street, which perhaps might have been handy for the manufacture of those cast-iron frames. Nice to think, however, that a piece of piano history still survives in the form of this building.

The firm seems to have gone into decline some time after the First World War, a matter perhaps not helped by the death of Reginald Pohlmann, who served in the Royal Flying Corps along with his brother during the conflict. It seems Pohlmann's stopped making pianos under the onslaught of the gramophone and wireless, ceasing manufacture some time in the 1930s, although I cannot find any record of the exact date when the last piano emerged from the factory doors. Later on the company, like several others of its kind, switched to selling radios, records and eventually television sets until it was taken over by Rediffusion.

The two Pohlmann pianos I have come across recently are robust, well-built instruments which have an extremely pleasant tone for their age (they are both around 100 years old). Although one sometimes has to be careful about manufacturers' claims, at the very least they seem to have been a firm committed to building high-quality pianos, and exceptionally forward-thinking in adopting new methods and technologies.

Although not listed in Dolge's book, there were several other piano manufacturers in Yorkshire around this time - a subject I shall return to in a later post.

Addendum: In a conversation with Dr Alastair Laurence of Broadwood's, he mentioned that he had (at a later date) met two of the brothers who were running the Pohlmann firm when it ceased making pianos.

By the 1930s, the business was being run by three of the Pohlmann brothers and their sister - Henry and Frederick were managing the piano manufacturing side, whilst Arnold and Cissy were looking after the showrooms in Princess Street. Henry Pohlmann was a piano designer who had studied with the famous German firm of Grotrian-Stenweg.

Due to the economic depression and the advance of radios, gramophones and other forms of entertainment, the 1930s were a dreadful period for the British piano industry, and the great majority of established manufacturers ceased trading. The last pianos were made by Pohlmann's in 1933 and the factory was then closed, but all the patents and designs of the firm and the right to use the Pohlmann name were sold to the well-known London company of Danemann's. The designs did not gather dust in a cupboard, because several Danemann models subsequently included string patterns or design features earlier used by Pohlmann's.

Merry Christmas

Just a note to wish customers, colleagues, friends and family all the best for Christmas and the New Year.

As some of you may already know from previous posts, I'm very fond of the humour of the famous pianist Victor Borge, but I couldn't find any specifically Christmas-related sketches by him. However, he is famous for this quote:

"Santa Claus has the right idea - visit people once a year."

Now despite his pianistic talent Mr Borge was not, as far as I know, ever a piano tuner. If he had been, he might have disagreed with this statement - in fact it's a good idea to have your piano tuned at least once a year to keep the tuning in good order, but I have customers who prefer more regular tunings to keep their piano in top condition all the time. So I am always very delighted to visit whenever you need me!

In any case, I thought it might be fun to have a look at Victor Borge's attempt here to conduct the Boston Symphony Orchestra back in 1986 - and I hope you all have an enjoyable festive season.

Feeling the tension (2): Calculating the tension on a piano string

In the last post, I promised that I would explain how to calculate the tension on a piano string, so I'm going to do that now. To do this, we will need to know three things:

• The speaking length of the piano string - that is the length between the capo bar and the upper bridge pin;
• The diameter of the wire - this should ideally be measured with a micrometer to get an accurate reading, since a small error may make a significant difference to the calculation; and,
• The exact pitch of the note (in Hertz or cycles per second).

With the pitch of the note, there are two ways to approach the matter - if you are so minded, you could use a chromatic tuner to ascertain the precise pitch of the note, or you could simply work on the assumption that the piano is at standard pitch (A above Middle C = 440Hz) which very often will be the case (though it is not uncommon for older pianos to be at a lower pitch).

A table of theoretically correct frequencies for each note on a piano can be found here. It should be noted that, on any well-tuned piano, the actual frequencies of notes outside the middle octaves may deviate somewhat from this, because of octave "stretching" - which makes the piano sound much better. Notes in the bass may be slightly flat of the "theoretically correct" frequency, those in the treble slightly sharp.

The Mersenne Equation 

At this point, enter the hero of our story - Marin Mersenne (1588-1648), who was a French monk, theologian, scientist, mathematician and "Renaissance Man", particularly noted for his contribution to acoustic theory.

One of Mersenne's most famous mathematical concepts was the "Mersenne Prime", namely prime numbers with the form:

Where n is a whole number. In fact, the six largest known prime numbers (at the time of writing) are all Mersenne Primes - this is due to the fact that they are easier to test mathematically than other prime numbers.

But on with the acoustic theory - the equation that concerns us is this one:

The Mersenne Equation:

Where F is the fundamental frequency of the note, L is the length of the string, T is the amount of tension on the string, and µ is the mass of the string per unit length. Put another way, this explains what are called Mersenne's Laws, viz., that the frequency is inversely proportional to the length, proportional to the square root of the tension and inversely proportional to the square root of the mass per unit length. This result is called Mersenne's equation (sometimes known as the Mersenne-Taylor equation or the Ideal String equation). The reason for the second alternative name is that the formula assumes that the string is perfectly flexible (zero stiffness) which is not the case in reality, and for that reason it is not quite perfect, but it will still provide an extremely good estimate for piano strings. (As a point of interest, one Galileo Galilei, who was a frequent correspondent of Mersenne's, also worked out the same thing, but Mersenne gets the credit because he demonstrated it experimentally).

We then need a set of units of measurement that will work. I won't go into the reasons for this, but if the following units are used, then the calculation will be correct:

• F (frequency) in Hertz (Hz)
• L (length) in metres (m)
• T (tension) in Newtons (N)
• µ (mass per unit length) in kilogrammes per metre (kg/m)

(There are other combinations of units that will work).

We then need to rearrange the equation above, because we are trying to calculate T, the tension. This gives the following result:

Before we can proceed any further, we also need a formula for µ, the mass per unit length of the string, which is as follows:

Where:

• π is the mathamatical constant pi
• d is the diameter of the wire (in metres)
• ρ is the density of the material from which the wire is made (in kg/m³) 

A sensible value for the density of high-grade steel used in piano wire is 7.85 g/cm³, which is 7,850 kg/m³ in the units we need to use. For the highest bass strings, according to this website, a value of 7.4 g/cm³ (7,400 kg/m³) is appropriate, ranging gradually down to 6.9 g/cm³ (6,900 kg/m³) for the thicker double-wound strings at the very bottom; this is because, although copper (used in the windings) is more dense than steel, the wound strings include a significant amount of air in the column.

Using the calculation in practice

This process can be demonstrated in practice using the middle C string of a Yamaha U1 upright piano, as follows:

I have removed the action of the piano to allow the string to be measured with a rule. In this case I haven't taken out the celeste rail (the piece of felt at the top) - this needs to come out for tuning to allow access to the pins. The top of the speaking length is the capo bar which is a ridge just underneath the pressure bar (the silver-coloured bar with the screws in it). I measured the length of the piano string with a rule and its diameter with a micrometer (seen in photo). Different strings on the same note will always have the same speaking length.

So you can see a little more clearly, here's a picture showing the pins, the pressure bar and the capo bar just below it without my arm in the way. On grand pianos, the end of the speaking length may be on the underside of the frame for some of the strings.

This shows part of the frame below the level of the keyboard. Each string passes across the bridge (the piece of wood sticking up in the middle of the photo), which transfers the vibration of the strings to the soundboard behind it. There are two pins attaching each string to the bridge - the upper one is the bottom end of the speaking length (in most cases there are three strings per note). In this case, the  middle C strings pass over the bridge towards the top right of the photo, behind the bass strings.

The speaking length of the middle C string is 0.665m and the diameter is exactly 1mm (0.001m).

Using the formula:

d² = 0.001² = 0.00001 m²
ρ = 7,850 kg/m³ (remembering, ρ = 7,850 kg/m³ for the steel strings or between 7,400 kg/m (upper bass copper-wound strings) and 6,900 kg/m³ (lower bass strings).

Multiplying up, we get µ = 0.006165 kg/m (i.e. one metre of the string weighs 6.2 grammes).

Then with the formula:

We get:

µ = 0.006165 kg/m
F = 261.626 Hz (the pitch of Middle C in equal temperament when the piano is at standard pitch of A = 440 Hz).
So F² = 68448.2
L = 0.665m
So L² = 0.442225

And the overall equation has the result T = 746 N. The result we get is measured in Newtons, which is a scientific unit of force - as any physicist will tell you, a kilogramme (or a pound) is a unit of mass, not force. However, a kilogramme force can be defined as the downward force exerted by a mass of one kilogramme in the gravitational field at the earth's surface. We can get this figure by dividing our result (746 N) by the physical constant g = 9.81m/s² which is the rate of acceleration of an object in freefall towards the earth.

This gives us our final result of 76.0 kilogrammes force. Assuming the tension on all 218 strings of the piano is roughly the same (as it almost certainly will be on a modern instrument), we can calculate that there will be approximately 16.6 tonnes of total pressure on the cast-iron frame.

One interesting point here is that Samuel Wolfenden, in his "Treatise on the Art of Pianoforte Construction" written in 1916, gives a set of model dimensions for a piano scale, in which the diameter of wire used on the middle C string is 1mm and the length is 0.688m. If this piano were tuned to A = 440 Hz, it would require a higher string tension of 81.4 kilogrammes force, but bearing in mind that Wolfenden was actually aiming for an older pitch standard then in use of A = 435 Hz, it can be seen that the modern U1 uses remarkably similar string dimensions and tensions to those that would have been employed on a high-quality piano from 100 years ago.

There is actually a great deal more that can be said about piano scale design, but as this post has already got fairly long and technical, I'll save that for another occasion.

Feeling the tension (1): why a piano has high-tension strings

A great amount of the weight in a piano, whether upright or grand, comes from the cast-iron frame (as you'll know if you've ever taken a piano apart and rebuilt it!); the essential purpose of this is to withstand the immense tension on the strings. So it is interesting to ask - how much tension is there in the strings of a piano?

The cast-iron frame of an upright piano

The amount of tension on each string of a (modern) piano is commonly the equivalent of the force exerted by a weight of around 75kg (this can vary significantly). An average piano might have 220 strings or thereabouts, so using these calculations the total force is around 16-17 tonnes for a typical instrument. In an article to follow I'll explain how it's possible to calculate the amount of tension on a given string.

However, an obvious question to ask is - why bother having so much tension? Why not have just enough to keep the string taut and allow the pitch to be adjusted, thereby avoiding the need for a heavy cast-iron frame in the first place?

Cranking it up a notch

Early pianos, which were wooden-framed, had much lower string tensions as they couldn't withstand as much force as a cast-iron frame, but despite this on occasion still used to buckle under the strain. The tone and power of these early pianos (which generally had lightweight leather-covered hammers) was fairly modest in comparison to a modern instrument. Over time, piano makers increased the tension on the strings, introduced metal bracings (later cast-iron frames) and replaced smaller leather hammers with heavier felt ones.

Having strings under higher tension confers a number of advantages:

(i) The amount of energy required to get the string vibrating is greater at a higher tension - this is beneficial since, once in motion, the string transfers more energy to the bridge and soundboard, giving improved volume and power (this is also the reason for the change from small leather-covered to more substantial felt hammers).

(ii) The string undergoes more complex vibrations at a higher tension, so there is a brighter and fuller sound; in fact, many early pianos sounded rather like harpsichords - the important difference was that the hammers allowed control over the volume through the strength of the blow on the piano. The timbre we associate with a piano today is a result of the increases in string tension through the 19th Century.

(iii) The pitch of the string is more thermally stable if the string is at a higher tension. Unfortunately, I'm not enough of a physicist to explain to you why this is the case, but it is. Samuel Wolfenden* (see reference below) says that the pitch of some older low-tension pianos from the earlier part of the 19th Century could vary by as much as a semitone(!) between summer and winter (and unevenly between notes, because string tensions weren't designed to be equal on these early insturments).

Generally speaking, when designing a piano, it is better to keep string tensions as similar as possible across all the strings. The main reason for this is that changes in temperature and moisture (the latter mainly affecting the wooden parts of a piano) will tend to make the piano change in pitch more evenly with similar tensions (i.e. the piano will stay in tune with itself), as well as giving a more even spread of tension on the pins and frame.

The 19th Century - The Piano Evolves

The results of these changes can be seen from a comparison of pianos at the beginning of the 19th Century with those at the end.

The picture shows a piano typical of one which might have been built around 1800 - two obvious differences from what would be seen today are the absence of a cast-iron frame and the arrangement of the strings (straight strung, as opposed to the overstrung arrangement used on all modern pianos). There are many other differences - this particular piano has 6¼ octaves as opposed to 7⅓ (88 notes) which is the standard compass today.

It should also be noted that actions in pianos around this period were of several different types, none of which were particularly similar to those used today; the Erard Double Escapement Action, which is the basis for the modern grand action, dates from 1821 whilst Robert Wornum patented the tape check action (the basis of the modern upright action) in 1842 (though it should be noted that similar concepts were employed on actions much earlier than this). A popular type of instrument in 1800 was the square piano, around the size of a large kitchen table (I have written about these in my previous post on the Broadwood Piano Festival); by 1900, the square had long since been eclipsed by the upright.

This picture shows a typical grand piano from just after 1900, which has practically all the features you would expect to see on a modern instrument - notably, the frame is cast-iron, allowing for much higher string tensions, and it is also overstrung (that is the bass strings cross over the tenor ones). It would also be remiss not to mention that the manufacture of wire also greatly improved throughout the century, which was also important in allowing the increased strain on the strings without frequent breakages.

In case you're interested in more technical information about this, there's an article here which has a graph showing changes in tension over time - the values on the left are in kilogrammes per note (I believe the notes shown are trichords on this graph - for the newer pianos at least it looks as if the bichords and monochords are not plotted), so should normally be divided by 3 for the tension on each string individually. These show that on Cristofori's piano of 1726, the tensions varied between 5 and 20kg per note; by 1808 (Streicher) this had increased to 40-80kg per note, and by 1914 (Ibach) was up to 220-260kg per note, where it has roughly remained ever since. Notably also, on the later pianos (Steinway M and Ibach) the tension is relatively much more even across the compass than on the earlier ones, reflecting a better understanding of scale design.

In my next post I'll explain how to calculate the tension of a particular string in any piano and a bit more about piano scale design.

* Reference: Samuel Wolfenden - A Treatise on the Art of Pianoforte Construction (1916)

The fastest piano in the West...

On Sunday 28th August, I attended the inaugural Micklegate Soap Box Derby, which I believe may become a regular annual fixture in future. Apart from anything else, a big well done to all the teams for raising some £50,000 for local charities, including York Community Energy, of which my housemate Tom is a big supporter. If you're interested in seeing a list of the winners, have a look here: http://www.yorkpress.co.uk/news/14710816.York_Soapbox_Challenge__The_award_winners/?ref=arc

I managed to get a few photos:

One of the teams at full pelt on the way down Micklegate

Brilliantly-timed camerawork yet again!

Chocolate or strawberry anyone?

Looks like Ben Hur has popped in for a visit! This impressive soapbox won the novelty prize.

Now, of course this did set me thinking - would it be possible to build a piano into a soapbox? Well, if anyone is up for that please do get in touch - though we have to work out how to get the piano down the big ramp at the start - and prevent it from careering off into the spectators - and perhaps we wouldn't be going all out for speed, especially over the cobbles down Micklegate Hill. Any brilliant ideas on how to do that are welcome....

However, aside from that, I thought it might be fun to look at a few of the weird and wonderful (human-powered) mobile pianos on the internet.

This is a blues and boogie pianist call Mark Lincoln Braun, who loves his piano so much he decided to move it 300 miles across Michigan on a bicycle. You can read all about it here:
http://www.treehugger.com/bikes/mr-bs-joybox-express-modern-minstrels-haul-piano-300-miles-by-bicycle.html

A hugely impressive feat, but unfortunately it's not possible in this case to actually play the piano and entertain passers-by whilst in the process of shifting it about. However, this chap seems to have well and truly solved that problem:

Someone just rode past on a piano...

This gentleman is called Gary Skaggs and his brainwave was to create this piano tricycle which he rides around in San Francisco. He explains a bit more about it here:

One thing that would be interesting to know is how stable the piano-cycle is when going downhill! Usually a piano is more likely to tip backwards as the cast-iron frame makes the back heavier than the front (something to note if you're moving one), so I'm not sure whether you would need any extra weights over the back wheel to keep it stable. It must also be said that he is also doing an excellent job to steer and play at the same time.

Not to be outdone, on this side of the Atlantic, there is a chap called "Rimski" who does the same kind of thing - pictured at Glastonbury in this video - though in this case it's possibly a bit more difficult to see where you're going. On the other hand, there does seem to be a handy bi-directional feature so the piano can reverse if needed. Indeed, which way is reverse, exactly?

A piano that ended up being human-powered in a slightly different way is this one. For some time there was a mystery about how an upright piano came to be at the top of a mountain in California but all is explained in this video:

There is a moral to this story - the pianist reported that the piano "hadn't been tuned for many years" and the keys in the right hand didn't work. If taking your piano up a mountain, I would strongly recommend having it checked by a competent piano technician to make sure that it is in good playing order when it gets to the top - as long as it's still in one piece by then of course.

I haven't yet managed to get a piano onto my bicycle, but I do have my piano tuning kit strapped on the back. Occasionally I need to bring some special equipment as well - I'll leave you with a picture of a string-height jig (used for regulating grand pianos) on the back of the bike, as well as the regular toolbox...