Riddle for the week

What are the eight sets of 4 in this picture?

(Click on the image to enlarge it)

Music and mathematics

From Greek times, a relationship has been recognised between musical relationships and the length of strings.

Counting intervals down from the note C:

• A (minor third) should be a string 6/5 = 1.2 times as long
• A♭ (major third) 5/4 = 1.25 times as long
• G (fourth) 4/3 = 1.333333 times as long
• F (fifth) 3/2 = 1.5 times as long
• C (octave) 2 times as long

Intervals can be combined by multiplying ratios, so a minor third and a major third give a fifth: 6/5 × 5/4 = 3/2. A fifth and a fourth give an octave: 4/3 × 3/2 = 2.

What does not work is that 12 fifths gives a string (3/2)¹² = 531441/4096 = 129.7 as long, which is not quite 7 octaves (which would give a string 128 times as long).
This discrepancy (of 1.3%) is the Pythagorean comma.

In equal temperament, each semitone gives a string longer by 1.059463094 (the twelfth root of two). Thus:

• B – 1.059463 times as long
• B♭ – 1.122462 times
• A (minor third) 1.189207 – differs from 6/5 = 1.2 by 0.9%
• A♭ (major third) 1.25992 – differs from 5/4 = 1.25 by 0.8%
• G (fourth) 1.334840 – differs from 4/3 = 1.333333 by 0.1%
• F♯ – 1.414214
  
• F (fifth) 1.498307 – differs from 3/2 = 1.5 by 0.1%
• E – 1.587401
• E♭ – 1.681793
• D – 1.781797
• C♯ – 1.887749
• C (octave) 2

With these ratios, a minor third and a major third give a fifth, a fifth and a fourth gives an octave, 3 major thirds gives an octave, and 12 fifths gives 7 octaves. However, none of the intervals are perfectly harmonious. Has it been worth it? Well, it made the piano possible!

The Naming of Names: a book review

The Naming of Names by Anna Pavord

I recently read The Naming of Names by Anna Pavord. This book tells the story of botany, beginning with the pioneering work by Theophrastus, the Vienna Dioscurides, and other works. The Naming of Names is beautifully illustrated with botanical illustrations such as this:

Blackberry plant, from the Vienna Dioscurides (text is in Greek)

I also found Pavord’s book to be an enjoyable read, with interesting historical snippets (though, I think, a misunderstanding of Augustine), and quotations from ancient writers such as Theophrastus:

“Other [plants] are found in fewer forms, as strykhnos which is a general name covering plants that are quite distinct; one is edible and like a cultivated plant, having a berry-like fruit, and there are two others, of which the one is said to induce sleep, the other to cause madness, or, if it is administered in a larger dose, death.” – Enquiry into Plants, Vol II, Book VII, Ch XV.

English Protestant preacher and botanist William Turner had this to say about the Oleander plant:

“I have sene thys tre in diverse places of Italy but I care not if it never com into England, seying it in all poyntes is lyke a Pharesey, that is beuteus without, and within, a ravenus wolf and murderer.”

Pavord takes her history up to Linnaeus, and his binomial naming system which finally gave standardised names to plants: Glycyrrhiza glabra, Allium sativum, and so forth.

Plants as they truly appear: “Das große Rasenstück” by Albrecht Dürer (1503)

Overall, an enjoyable book, although the beautiful illustrations are its real strength. See also this review by Ursula K Le Guin.

The Naming of Names by Anna Pavord: 3½ stars

Recent reading (April to July)

Over the past few months, I finished the following books (among others – most of the novels and technical books I’ve read are not included). Links go to my reviews. Books marked with ♥ (fiction) or ♦ (non-fiction) were particularly good:

• The Supper of the Lamb: A Culinary Reflection by Robert Farrar Capon ♦
• Pagan Christianity by Frank Viola and George Barna
• 100 Cupboards by N.D. Wilson ♥
• Dandelion Fire by N.D. Wilson ♥
• The Chemistry of the Blood by M. R. De Haan
• Dry Storeroom No. 1 by Richard Fortey
• Shapes by Philip Ball ♦

Older reviews (fiction):

• The Pilgrim’s Progress by John Bunyan ♥
• The Plague by Albert Camus ♥
• Inverted World by Christopher Priest
• Murder Must Advertise by Dorothy L. Sayers ♥
• The Winter Queen by Boris Akunin
• To Say Nothing of the Dog by Connie Willis ♥
• Four Faultless Felons by G. K. Chesterton
• The Dream of Scipio by Iain Pears
• Invisible Cities by Italo Calvino ♥
• The Tower at Stony Wood by Patricia A. McKillip ♥
• Winter Rose by Patricia A. McKillip ♥
• The Dream of Perpetual Motion by Dexter Palmer
• Captain Blood by Rafael Sabatini
• At the Mountains of Madness by H.P. Lovecraft ♥
• The Manual of Detection by Jedediah Berry ♥
• Ukridge by P.G. Wodehouse
• Sovereign by C.J. Sansom
• A Good Man Is Hard to Find and Other Stories by Flannery O’Connor ♥
• Impossible Things by Connie Willis
• Blackout/All Clear by Connie Willis ♥

Older reviews (non-fiction):

• Singing with the Angels by Robert E. Coleman ♦
• Literature and Theology by Ralph C. Wood
• Finding God Beyond Harvard by Kelly Monroe Kullberg ♦
• Christ Plays in Ten Thousand Places by Eugene H. Peterson
• Defending Constantine by Peter J. Leithart ♦
• Notes From The Tilt-A-Whirl by N. D. Wilson
• Only A Theory by Kenneth R. Miller
• The Edge of Evolution by Michael J. Behe
• Surprised by Oxford: A Memoir by Carolyn Weber ♦
• Citrus: A History by Pierre Laszlo
• Field Notes on Science & Nature by Michael R. Canfield ♦
• C. S. Lewis on the Final Frontier: Science and the Supernatural in the Space Trilogy by Sanford Schwartz ♦
• Think by John Piper
• Natural Experiments of History by Jared Diamond and James A. Robinson
• Social Understanding by Jürgen and Christina Klüver
• The Secret Life of Birds by Colin Tudge
• Galileo by Mitch Stokes
• Modern Art and the Death of a Culture by H.R. Rookmaaker
• Terra – Tales of the Earth: Four Events That Changed the World by Richard Hamblyn ♦
• Philosophy, Science and the Sovereignty of God by Vern S. Poythress

The Supper of the Lamb: a book review

The Supper of the Lamb: A Culinary Reflection by Robert Farrar Capon

I recently borrowed the wonderful The Supper of the Lamb: A Culinary Reflection (1967), by Robert Farrar Capon.

I was already in love with this book by the time that I read the unforgettable chapter-long description of how to disassemble an onion (I’ll never think of onions the same way again):

This book is so rich and complex that it’s very difficult to review although some people (such as this woman) have made the attempt. The best I can do is to highlight some of my favourite lines:

On the disassembled onion (p. 15): “Then look. The myth of sphericity is finally dead. The onion, as now displayed, is plainly all vectors, risers and thrusts. Tongues of fire. But the pentecost they mark is that of nature, not grace: the Spirit’s first brooding on the face of the waters. Lift one of the flames; feel its lightness and rigidity, its crispness and strength. Make proof of its membranes. The inner: thin, translucent, easily removed; the outer, however, thinner, almost transparent – and so tightly bonded to the flesh that it protests audibly against separation.”

And again (p. 19): “Man’s real work is to look at the things of the world and to love them for what they are. That is, after all, what God does, and man was not made in God’s image for nothing. The fruits of his attention can be seen in all the arts, crafts, and sciences.”

On reality (p. 21): “One real thing is closer to God than all the diagrams in the world.”

On the right attitude to food (p. 27): “O Lord, refresh our sensibilities. Give us this day our daily taste. Restore to us soups that spoons will not sink in, and sauces which are never the same twice. Raise up among us stews with more gravy than we have bread to blot it with, and casseroles that put starch and substance in our limp modernity. Take away our fear of fat, and make us glad of the oil which ran upon Aaron’s beard. Give us pasta with a hundred fillings, and rice in a thousand variations. Above all, give us grace to live as true men – to fast till we come to a refreshed sense of what we have and then to dine gratefully on all that comes to hand. Drive far from us, O Most Bountiful, all creatures of air and darkness; cast out the demons that possess us; deliver us from the fear of calories and the bondage of nutrition; and set us free once more in our own land, where we shall serve thee as thou hast blessed us – with the dew of heaven, the fatness of the earth, and plenty of corn and wine. Amen.”

On meat cleavers and femininity (p. 71): “A woman with cleaver in mid-swing is no mere woman. She breaks upon the eye of the beholder as an epiphany of power, as mistress of a house in which only trifles may be trifled with – and in which she defines the trifles. A man who has seen women only as gentle arrangers of flowers has not seen all that women have to offer. Unsuspected majesties await him.”

On Divine Providence (p. 85): “Each thing, at every moment, becomes the delight of His hand, the apple of His eye. The bloom of yeast lies upon the grapeskins year after year because He likes it; C₆H₁₂O₆ = 2 C₂H₅OH + 2 CO₂ is a dependable process because, every September, He says, ‘That was nice; do it again.’ ... Let us pause and drink to that.”

On dinner parties (p. 179): “This is the moment of truth – the agonizing minute after the champagne when the ship starts slowly down the ways and slips into its proper element. No true host ever gets through it without holding his breath; but if his guests have been well chosen, and if his other preparations have been what they ought to be, he will be rewarded inwardly with the shout of joy that accompanies every successful launching. As the conversation quickens and the company begins to form in earnest, he will survey the work of his hands and, over soup or fish, catch his wife’s eye to give her the old triumphant word: She floats! By George, she swims!”

For, the host having said grace, “two or three are gathered” in Christ’s name, and He is there among them.

I also loved the connections to Creation and to the Wedding at Cana. Oh, and the book contains a plethora of recipes as well – because this rich and varied universe contains so many things which are good to eat.

The Supper of the Lamb: A Culinary Reflection by Robert Farrar Capon: 4 stars

Pagan Christianity: a book review

Pagan Christianity by Frank Viola and George Barna

I recently read Pagan Christianity by Frank Viola and George Barna, a book which alleges that most modern church practices are pagan in origin. I was astonished at the numerous errors and misrepresentations.

Viola and Barna argue (p. 12) that there were no specialised Christian church buildings before the 4^th century, yet the 3^rd-century Dura-Europos church had many features of later churches: a hall capable of holding about 70 people, a specialised room for the baptismal liturgy, and numerous frescoes.

The Dura-Europos church: plan and reconstructed baptistry

Similarly, the 3^rd-century Megiddo church had a 54-square-metre mosaic floor, as well as inscriptions which suggest a special table for the Communion liturgy.

The Megiddo church: part of the mosaic floor (the inscription reads “Akeptous, she who loves God, offered the table to God Jesus Christ for a memorial” – cf. 1 Cor 11:24)

Larger buildings were simply a natural response to the increasing size of the Christian church. The suggestion (p. 29) that Gothic cathedrals were derived from Plato is also incorrect, and misrepresents the cited sources.

The wildly incorrect “Darth Constantine” theory (p. 18 onwards) has been demolished elsewhere. Viola and Barna also appear (bizarrely) to believe that hymns, pastors, and sermons are un-Biblical. Yet, in Acts 17, Paul clearly gives a prepared sermon, including literary quotations. When the New Testament evidence against the thesis of this book becomes too overwhelming, Viola and Barna argue (p. 235) that Acts and the Pauline Epistles cannot be taken as a model for Church practice today. But if the Old Testament and New Testament do not provide a model, what does?

Somehow, we’re supposed to be convinced that the church has been wrong for 1,900 years, and that only the Anabaptist house-churches are right, but the evidence doesn’t support that thesis. Current churches may have flaws, but the standard Protestant model is closer to the New Testament than the Anabaptist house-churches are.

See here and here for other reviews of this very, very disappointing book.

Pagan Christianity by Frank Viola and George Barna: 1 star

100 Cupboards and Dandelion Fire: a double book review

100 Cupboards and Dandelion Fire by N.D. Wilson

I recently borrowed 100 Cupboards and Dandelion Fire by N.D. Wilson (also author of Notes From The Tilt-A-Whirl). These are the first two books of a children’s trilogy, in which a boy named Henry, while visiting his uncle and aunt in rural Kansas, finds a hundred doorways to other worlds. Some of these worlds are sinister and perilous, and (inevitably) adventure ensues.

Cover art for 100 Cupboards

Although intended for younger readers, I very much enjoyed these books (and look forward to reading the final volume). They are well-written, and reminiscent of Madeleine L’Engle’s A Wrinkle in Time, or Ray Bradbury’s Something Wicked This Way Comes. The second volume opens like this:

“Kansas is not easily impressed. It has seen houses fly and cattle soar. When funnel clouds walk through the wheat, big hail falls behind. As the biggest stones melt, turtles and mice and fish and even men can be seen frozen inside. And Kansas is not surprised.”

Kansas is surprised at some of the things that come out of the cupboards, though. Nevertheless:

“Sometimes standing against evil is more important than defeating it. The greatest heroes stand because it is right to do so, not because they believe they will walk away with their lives.”

Overall, the best children’s fantasy I’ve read in a while. See here and here for other reviews.

100 Cupboards and Dandelion Fire by N.D. Wilson: 4 stars

Sourdough World

In a previous post, I reviewed The Chemistry of the Blood and Other Stirring Messages, an example of the genre of theological or philosophical reflection on science. It inspired this post, and also reminded me of the (very brief) parable in Matthew 13:33:

“[Jesus] told them another parable: ‘The kingdom of heaven is like leaven that a woman took and hid in three measures [about 30 kg] of flour, till it was all leavened.’ ”

According to commentators, this woman was making sourdough bread, by adding some sourdough starter (which can be produced from water, flour, and perhaps a few grapes or sultanas):

Sourdough starter bubbling away (public domain photo by Ibán Yarza)

Such starters contain a mixture of wild yeast, and the kind of bacteria (lactobacillus) also found in yoghurt:

Microscopic view of yeast and bacteria in a one-year-old starter (photo by “Karniecoops” at sourdough.com)

The growth of these microorganisms varies with temperature (and so will the taste of the bread), but at the peak rate, bacterial numbers double every hour.

Bacteria & yeast growth rates, from here (see also this paper). The bacteria causing sourness (red and green) grow best around 33°C (91°F), while the yeast (blue) grows best around 27°C (81°F). Growth rates for bacteria and yeast are the same around 23°C (73°F).

Such exponential growth recalls the story of the man who wanted grains of wheat on a chessboard: 1 on the first square, 2 on the second, then 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... up to 9,223,372,036,854,775,808 on the last square (for a spectacular total of hundreds of billions of tonnes).

The point of the parable is outlined by C.S. Lewis in his wonderful little book Mere Christianity:

“All sorts of people are fond of repeating the Christian statement that ‘God is love.’ But they seem not to notice that the words ‘God is love’ have no real meaning unless God contains at least two Persons. Love is something that one person has for another person... And now, what does it all matter? It matters more than anything else in the world. The whole dance, or drama, or pattern of this three-Personal life is to be played out in each one of us: or (putting it the other way round) each one of us has got to enter that pattern, take his place in that dance. There is no other way to the happiness for which we were made. Good things as well as bad, you know, are caught by a kind of infection... [Christ] came to this world and became a man in order to spread to other men the kind of life He has – by what I call ‘good infection.’ Every Christian is to become a little Christ. The whole purpose of becoming a Christian is simply nothing else. ”

Given exponential growth and time, a little leaven can pass on the good infection to most of the human race – all it takes is regular doubling. And when the world has been leavened, it will come out of the oven as a perfect loaf of bread.

Wonderful sourdough bread (public domain photo by “Tanzania”)

An interesting conference

The 5^th Epistemological Perspectives on Simulation (EPOS) Conference is being held at Trinity University (San Antonio, Texas) on October 10–13 this year.

The conference is accepting papers on philosophical issues arising within modelling and simulation. It will bring together work in both philosophy of science and modelling and simulation practice.

It certainly looks well worth while.

Emily Dickinson #33

“The Moon upon her fluent route
Defiant of a road,
The stars Etruscan argument,
Substantiate a God.
If Aims impel these Astral Ones,
The Ones allowed to know,
Know that which makes them as forgot
As Dawn forgets them now. ” – Emily Dickinson