Equal vs. Meantone Temperaments

 Equal vs. Meantone Temperaments 

For many years I have puzzled over the best way to explain the dilemmas posed by Equal Temperament. I discovered some of them for myself in 1986 when I put together the Spinet kit I bought from John Storrs of Chichester (https://www.friendsofsquarepianos.co.uk/spinets-harpsichords-and-clavichords-for-sale/john-storrs-spinet/). These problems barely show on a piano and, in any case, I think one gets used to equal temperament. There is something about the twang of a spinet/harpsichord that shows up the shimmering 'beats' of the discordant thirds. 

By equal temperament I refer to the practice of (a) defining an octave as an exact doubling of the frequency of a note, and of (b) dividing the octave into twelve equal semitones.

Equal Tempered Semitone (ets) = 12√2 =1.059463094359

Multiplier	If fundamental  =440 Hz	True ratios (Pythagorean) intervals frequencies	Error as ratio (or 1/ratio)
1.0000	A440	Fundamental
1.0000*ets=> 1.0595	A♯, B♭	Semitone
1*ets^2=> 1.1225	B	Pythagorean tone 9/8 = 1.125)
1*ets^3=> 1.1892	C	Minor third = 6/5 = 1.20000	0.991(1.0091)
1*ets^4=> 1.2599	C♯, D♭	Major third = 5/4 = 1.25000	1.0079
1*ets^5=> 1.3348	D	Fourth = 4/3 = 1.33333	1.0011
1*ets^6=> 1.4142	D♯, E♭
1*ets^7=> 1.4983	E	Fifth = 3/2 = 1.500000	0.9989 (1.0011)
1*ets^8=> 1.5874	F
1*ets^9=> 1.6818	F♯, G♭
1*ets^10=> 1.7818	G
1*ets^11=> 1.8877	G♯, A♭
1*ets^12=> 2.0000	A	Octave = 2/1 = 2.00000

From the table we can see that the 'harmony' of the equal tempered fifth and fourth are fairly close to the true harmonies of the Pythagorean scale. Nevertheless, the equal tempered fifth is too small by a small amount, and the equal tempered fourth is too big by a similar and therefore a partially compensating amount. 

However, the equal tempered major third is too big while the equal tempered minor third is too small; and these errors are 8 to 9 times greater than the errors of the fourths and fifths. Once again these errors tend to cancel out, but not completely. Thus, a major plus a minor third equals a fifth, where the error is 8 times smaller than that of the major third.

An interesting question arrises as to how the instrument tuner in 1600 or 1700, or even 1950 tuned without the aid of an electronic tuner. You might naively start (as I did ) by tuning A4 using a tuning fork at A4=440 Hz.  Then, by tuning by ear up a fifth and down a fourth, you get to B. Repeating the process you get to C♯ etc, till you get back to A (jumping an octave when you need or want.)  But you crash into the shattering fact that it is a different A; it is 1.36% too sharp. The gap between the two is called a comma. It you narrow the  fifth slightly, from a true fifth to make it an equal tempered fifth, you can spread the comma round the octave. That takes skill, listening for, and counting, the 'beats' per second that appear and speed up as you depart from perfect harmony. The same thing happens if you simply go up a fifth over, and over, 12 times, for (3/2)12 =129.746 while 7 octaves (27 =128); 129.746/128=1.01364; you get exactly the same discrepancy or comma.

I learned an easier method for tuning to the so-called "quarter comma" meantone temperament, which gives very sweet-sounding harmonies for rennaissance and baroque pieces in 'unadventurous' keys (e.g.  from 2 flats to 2 sharps).  Remember, octaves are always perfect. Tune your A4 to the A440Hz tuning fork (or signal), then A3 (220 Hz) to that.  To A3, tune a perfect third to F3. Go back to A3 , and from it tune D3 almost a pure fifth below but leave it slightly sharp; about 2 'beats' per second. From that tune up a fourth again nearly perfect to get a G3, again a tad sharp; and from that down to C3 again leaving it slightly sharp.  The instructions are to "spread the error evenly" between these 3 intervals: A3 ➞D3, D3, ➞G3, G3 ➞C3 . Check it is tolerable with the chord C3 + F3 + A3. All the other intervals are tuned as pure major thirds (5/4), or octaves (2/1):  C➞E, D➞F♯, D➞B♭,  B♭➞B♭, E➞G♯, G♮➞E♭, A➞C♯,➞C♯, G➞B♮➞B♮.  (See Figure below.)

Having set 'the temperament', use pure octaves to complete the instrument. Electronic tuners are quicker and easier, and are available free for most smart mobile phones. I use a Table of deviations from equal temperament (in cents) given me by a friend 40 years ago. After considerable searching I found a very comprehensive table online at (http://www.instrument-tuner.com/TemperamentTables.html)

Michel de Montainge

Michel de Montaigne 

Michel de Montaigne is an old friend of mine. 

We were introduced back in 1960. My aunt Kay and her husband Henry had invited my brother and me to spend a fortnight with them on Scotland's 'Outer Isles'. We walked a lot during the day, enjoying the peaty hinterland of Lewis and the sandy 'machair' sward and the 'crottle'-covered, rocky, hills of Harris. After evening dinner, Henry amused himself with 'Les Essais de Montaigne'  in their archaic French of 1588. ***  I was thrilled to note how easy and succinct I found the old French.

Henry told us of this distinguished gentleman, active in the politics of Bordeaux, removing to his ancestral village and castle de Montaigne when the plague came to the city; troubled in later life from kidney stones, touring Europe from Spa to Spa searching for a cure. (Here Henry digressed to give his own opinion that drinking pure water was the best cure, and he described the terrible operation by which Samuel Pepys was successfully 'cut of the stone' in 1658.) He was called back from Italy when elected, in his absence, to be Mayor of Bordeaux, at the age of 48.

Montaigne, as a young boy, was brought up speaking Latin as first language. By strict order of his father, the servants and also his mother had to speak only Latin when around the boy. 

As a young man he enjoyed a deep comradeship with a brilliant poet and politician some four years his senior, with whom he delighted to converse. However, Boétie died (perhaps of the plague) at the age of 32 leaving Montaigne bereft. It is suggested that the honest, intimate, discursive, introspective as well as curious, wandering and learned writing of these essays evolved as a replacement for his conversations with Boétie. 

Inclined to solitude...

"In the year of Christ 1571, at the age of thirty-eight, on the last day of February, his birthday, Michael de Montaigne, long weary of the servitude of the court and of public employments,........." [he retired to his tower with his books and his pen.]

The mature man showed fairness, inquisitiveness, great learning of the Latin and Greek writers who had come his way, and that rare ability called 'common sense'.  He was liked and trusted by both of the warring religious factions. He talked to seamen about Brazil, and to farmers about crops, 

But when I step back and consider Montaigne as a man and an intellect, I realise that there are also: Voltaire, Rousseau, Euler, Leibnitz, Newton, Locke, Hume, Adam Smith, JS Mill, John Keats, Walter Scott, not to mention Shakespeare. I suppose I would like to think I resemble Montaigne. Other heros (friends, companions) will appeal to other readers.

(*** Footnote. C.f. I'ay veu plusieurs de mon temps conuaincus par leur conscience retenir  . .X . . .(1588); with J'ai vu, de mon temps, nombre de gens, auxquels leur conscience reprochant de s'être approprié  . .X . . . (1907). I was thrilled to note how easy and succinct I found the old French, as long as you realise that v and u are inter-changeable, as also i and j.)

Starmer's Mission

 Starmer's Mission

The idea here came to me while witnessing the bizarre process of the media trying (over the weekend 7 - 8th Feb) to squeeze a political crisis out of the Epstein-Mandelson affair, and from listening two days later to "The Rest is Politics" podcast from the gifted Campbell-Stewart duo titled "Is it 'Game Over' for Starmer?" (9th Feb 2026). And (to be fair to myself) from my own posting of 14th Jan on the previously manufactured 'crisis' over ID cards [1].  It is not primarily my own idea, but I am endorsing it.

My impression is that Sir Keir Starmer's mission is to bring straight-forward honesty into politics. His earlier successful career as a lawyer shows a consistent attraction to justice and decency. His calling (i.e. the force that drives him) was not concerned with I.D, cards, nor with fiscal deficits.  His stance is moral.  It is barely political. If forced to take sides, he would perhaps say that slightly too large a fraction of the wealth generated in the country goes to the bosses, the gamblers and the cheats, and that advantage is constantly being taken of the disadvantaged and ill-educated working class. Political finesse is not his forte; nor is finance. His forte is justice. 

However, like all of us, he may be touched to some extent by ambition. I mean bankers and gamblers want to get rich, scientists want to the the first to discover truths, authors want to be read. And politicians (doubtless) would like to see themselves in command of a well-ordered outfit bringing justice, wealth and happiness to the nation.  

It is embarrassing to see philosophers like Newton and Leibnitz scrabbling over priorities. Similarly, I wince when a chancellor says " 'I' have decided to increase the duty on petrol by a penny." How pompous!  Why not say "We have decided......"?  For it should be the Cabinet that decides, and takes responsibility. 

It should not be part of Starmer's duty to decide on Corporation Tax rates, or allowances for 3-children families. If he wants to dictate to the cabinet on such issues he is indeed touched with madness. He does not need to reply, just because the press ask him for his ideas, or get cajoled into taking the blame for other people's mistakes. Other things being equal, stability is of course good, and dithering bad.  

As Rory Stewart says in the podcast (c. minute 25), what the country surely wants is for Wes Streeting to use all his drive and initiative to sort the NHS, Peter Kyle to take charge of business 'Growth', and Rachel Reeves to balance public spending against taxation. Perhaps there is also a role for younger members of the team like Allin-Kahn. 

There is an important two-way communications job for the whips. To avoid a back-bench revolutions, it is not sufficient for whips merely to explain Government policy to the benches. The back-benchers should know, discuss, and own those policies, and explain their views to the cabinet. 

Starmer's rôle is to be 'honest' and 'decent'. And (copying Neson) to remind his team that "England expects that everyone will do their duty". It would be a bold move to abolish 'Prime Ministers Question Time', but that may be what is needed. The Prime Minister deflecting each question to his appropriate lieutenant.

References:
[1]  https://occidentis.blogspot.com/2026/01/digital-id-cards-and-bbc.html 

My Fracas

 My Fracas

I am constantly being reminded of my recent 'fracas'. When I heave myself out of my chair I  realise I have a bruised left hip, and a sprained left wrist. When I look in the shaving mirror I find I have a small bump above my left eye.

I assure you that I had done absolutely nothing wrong. But I was being pursued by two rather boisterous youngsters. And the mood was deteriorating. I thought I had thrown them off,  but no; there they were again close on my heels. So I decided to tackle the front one and go for his legs. I threw myself with maximum force at that target, missed his legs but, flailing my arms rather futilely in his general direction, I fell to the ground and bumped my head.

Frustrated and rather sore, I found myself half awake and on my bedroom floor, in the narrow gap between my bed and the wall. I had dragged the duvet off the bed and had banged my head on the radiator.  Oh dear! I crawled back onto the bed, pulled the duvet back on top and went straight back to sleep. 

Normally, when awake, I am well controlled and rather passive, looking before leaping and thinking twice before making any irreversible decision. It is while asleep that I get into trouble. Not every night by any means, but five or six times a year. Twice I broke the great brass bedside lamp in Josie's guest room. Several times I kicked my companion. Once I hit her in the face with my fist. Nothing to do with her; my assailants are un-named younger males. She is unfortunately in the wrong place at the wrong time, looming over when I believe I am being attacked. My father has not figured in my dreams for decades. He died 40 years ago.

You might be inclined to suggest I try psychoanalysis. To see who these anonymous demons really are. I am disinclined. Ever since reading Jones' Life and Work of Sigmund Freud back in 1970, I have preferred not to tinker with my mind, in case the 'motivator' sprung out like a watch-spring and I could not coax back in. 

One day I may go too far. One day I might submit to analysis.