A List of Allegedly Racist Statements in the Talmud, pt 1

[Some updates to this list are upcoming]

There are lists of offensive quotes from the Talmud that are making the rounds on the internet, and have done so for years. Some of these can be traced back to printed works of the early 20th century, and some of them probably have been passed around by fax during a few decades as well. I have taken as an example of this list one particular example that pretends to be scholarly, and gone to some extent in verifying the quotes. Lists such as this use quote-mining, fabrication and clever rhetorical deceptive approaches.

Many samples from this list appear in the comment sections of newspaper websites whenever Israel, banking or other topics that get anti-Semites' panties in a bunch appears. Some of these have been making the rounds before Internet. Some go back to the 19th century and even 18th centuries. 

This is only an investigation into one particular list of these quotes. The person who reposted that list claims to be a scholar of religion, yet treats the list with surprising gullibility. A really slight bit of criticism 

Sources given without angular parentheses are as given by the source text that I am analyzing. This is relevant, since the text pretends to be a compilation of quotes from actual works. 

"Judaism is not a religion but a Law religionized."

Moses Mendelssohn [1]

How is this, per se, a problem? Rhetorically it is quite clever though - almost all protestants, at the very least, have heard how legalism in religion is a terrible thing. This is a very clever trick for priming your reader to have a negative mindset about the topic from the onset.

Caution: Some of these quotes from Judaism are quite harsh. The quotes from Jewish writings themselves are usually defended as being "taken out of context" or "removed" with apologies, and with the justification that such sentiments were warranted because the Jews were being persecuted. While the sentiment behind such apologies may be genuine, the fact is that Judaism itself is a plan for hegemony, as stated from the very beginning, well attested in the Tenach or Old Testament. [1]

Of course, no one would ever ever really misrepresent any Talmudic (or other Judaic) quote out of context, now, would they? Brushing off the "taken out of context" argument like this is really cheap. Also, "quotes from Judaism" is a very nice way of guilting Jews by association.

Note: The disparaging comments regarding non-Jews are expurgated from various editions of the Talmud. They are, however, found in the Soncino English translation of 1935. (Some of these pages have been reproduced in The Plot Against Christianity.) Where they have been expunged, one can find an "o" indicating "oral tradition," which means they are still taught. Like it or not, Orthodox Judaism fervently teaches ultimate supremacy over "the nations," i.e., Gentiles, whom it pronounces as inferior creatures. [1]

Which particular editions of the Talmud in particular have had the offending passages removed? This claim is often repeated, but no one ever mentions the particular editions that supposedly succumb to the temptation of omitting such material. Given the history of talmud printing, it seems more likely these expurgations are due to Christian censorship rather than Jewish guile.

Of course, derivative works - for instance Cohen's Everyman's Talmud, Steinsaltz' Essential Talmud - are only small samples of the Talmudic text with chapters elucidating the history of the text, its significance in the different branches of Judaism, illustrative examples and some central parts of it. Such books could not be expected to be complete laundry lists - yet even then Cohen's Everyman's Talmud at the very least is quite clear about the troubling aspects of the Talmud. It's more than twelve years since I've read Steinsaltz' introductory work, and I do not recall whether he covered this issue in any great detail. Those editions of the Talmud from which these things have been removed are not issues any person trying to verify these things need to worry about - they are early prints in Aramaic (and Hebrew), by now properly antiquarian editions. Not things you will accidentally run into while researching the Talmud at your regular library.

Further, 'The Plot Against Christianity', linked above, is known to be an antisemitic smear campaign by one Elizabeth Dilling, a famed paranoid anti-semitic writer. I will later give a more detailed review of Dilling's works.

It is true that some of these quotes have been taken out of context. We are attempting here to find them and put them within their context. Also, a number of these contentious remarks are followed by mitigating commentary from another rabbi (such as the quote about heathens studying the Torah at Sanhedrin 59a). Some of these quotes are translated quite differently from version to version, apparently at times softened. According to Prof. Israel Shahak, the original Hebrew passages possess greater vitriol than do the translations. Because of the translation discrepancies, it is difficult to verify these quotes. [1]

"We are attempting here to find them and put them within their context" is a funny way of putting it, since next to none of that kind is actually done, as we will see below - they are just quoted, verbatim from whichever source previously made them up or distorted them – many of them are fabrications in the first place. Why does the author present repetition of mostly slanderous texts as a critical assessment? What kind of ignorant stupidity is this? Alas, the translation discrepancy isn't the big problem in verifying (or debunking) these, the big culprit in that is the way antisemitic sources use distorted names in order to game search engines. If you search for libbre david, it is likely lists similar to this will cover the first several pages of findings - only way down come a few scholars who point out that no such book exists. 

The verses in dark blue are confirmed to be in the Soncino Talmud. We have removed some particularly incendiary remarks because they cannot easily be verified. For more information regarding apparently erroneous Talmudic quotes please see the writings of Judy Andreas. [1]

Funnily enough, quite a few particularly incendiary remarks have not been removed despite being demonstrably false. This kind of pretend-carefulness reeks. As for 'more information regarding apparently erroneous Talmudic quotes, please see the writings of Judy Andreas' – Judy Andreas does not provide particularly much in ways of information regarding apparently erroneous Talmudic quotes, but maybe she has removed that content from her site since the author wrote this? Alas, the author does not give the name of the article or anything, so it is difficult to trace it.

I have retained dark blue verses unaltered in quoted material so as to illustrate the quality of the quality assurance that the person who posted this list assures us has been done. (NB. Yes, I mean 'the quality of the quality assurance', this blog operates on that level of analysis.)

Further, it should be noted that some of these texts, such as the Zohar, Aruch, Yalkut, Tosefta and Soferim, may not be contained within the Talmud proper but are referenced therein, as "commentaries" and "tractates" also considered "sacred texts." While the Soncino Talmud quotes are represented verbatim where possible, at least some of the statements purportedly from these other texts represent paraphrases. The text is frequently deliberately difficult to follow, in Hebrew as well, such that it needs to be simplified.[1]

Except this is also misleading. Zohar and the (Schulchan) Aruch are medieval texts, the Talmud is from late antiquity. The phrasing makes it unclear how the Zohar and the Talmud 'belong together'. Neither the Zohar or the Schulchan Aruch can be 'referenced therein' - although references may appear in the later scholarly apparatus that now comes with many Talmud editions, but these are not part of the Talmud proper. The description of the relation between these texts is confused at best, confusing at worst in the article that I am reviewing.

(Such as concerns the use of terms for non-Jews/Gentiles: "Cutheans," "Samaritans," "Egyptians," "Canaanites," "Karaites" and "Minim," which refers to the "Judeo-Christian heretics," also considered the "Sadducees." "Heathens," of course, and "Goyim," are well-known terms used in the Talmud. "Goyim," referring to Gentiles, is said to mean "unclean.")[1]

"Karaites" are quite explicitly not just a code-word for gentiles or Christians or anything; they were the main schism within medieval Judaism, at time accounting for about 30% of all Jews. The Karaites rejected the non-Biblical parts of Rabbinic Judaism. Regionally, the relations between Rabbinic and Karaite Jews were very bad. Since the Jewish population has been relatively small through history, no huge armed conflicts have erupted between Karaites and Rabbinic Jews (unlike between protestants and catholics). However, the rabbinic Jews of Spain did arrange for the Muslim authorities to come down hard on the Karaites, while the Karaites of Russia schemed in similar fashions to arrange a czarist campaign against the Rabbinic Jews – Abraham Firkovich, for instance, produced fabricated tombstones to demonstrate that the Karaite presence in Crimea predated the crucifixion of Jesus, and thus got the Karaites absolved for the accusation of killing Jesus, unlike the rabbinic Jews. In Greece and parts of Byzans, relations between Karaites and Rabbinical Jews were more cordial, however. "Goyim" further does not mean "unclean", although I am sure it is "said" to do so by some people who are all about misrepresentation. Compare, for instance, genesis 12:2,

 וְאֶֽעֶשְׂךָ לְגֹוי גָּדֹול וַאֲבָרֶכְךָ וַאֲגַדְּלָה שְׁמֶךָ וֶהְיֵה בְּרָכָֽה׃

wə'a`esokha ləgoy gadol wa'avarekkha wə'agadlah šəmkha wehye bərakha

And I will make you into a great people and bless you and and make your name great and make you a blessing. ('Goy' and its transliteration and translation have been emphasized).

The author, who elsewhere pretends to know basically all the relevant languages for biblical studies failed to notice this? How the hell? How the everfucking fucking hell would a scholar of these languages fail to notice this?

Further, some instances of Cuthean and Samaritan certainly are results of later substitutions, but while researching this I came across very many where it's clearly not a result of substitutions - i.e. the Cutheans/Samaritans are contrasted with the Gentiles due to the Cutheans having the same religious obligations as the Jews, whereas the Gentiles do not - however, the Cutheans, according to the rabbinic view of things, do not live up to all their obligations.

It should be also kept in mind that much of the Talmud was orally taught and/or written down long before pogroms and persecution of Jews were common. Thus, the notion that these anti-Gentile statements are a reaction to such persecution is untrue. Indeed, it has been suggested that the persecution was in large part on account of such anti-Gentile sentiments and behavior. For a rebuttal or apology of these quotes, see below.

Although victim-blaming has gotten a bad reputation these days, the author seems to think there's still much to recommend it. The Jews, of course, were the only people in antiquity and medieval times to look down on other nations [hint: no, they were not].

Regarding what is actually in the Talmud, Rabbi Lewis Browne says in Stranger Than Fiction:

"There are in it myths and vagaries, idiotic superstitions and unhappy thoughts, things that are not merely irrational but sometimes even quite offensive. But there is also much profound wisdom buried in it, and much lofty and generous thinking. Not all the rabbis were bitter and hateful -- though, Heaven knows, they all had reason to be. And not all of them were small-minded and bigoted.... Granted there is much chaff in the work, there are also kernels of richest wheat."

Unfortunately, those kernels could take forever to find, by which time many people would starve. Why look for seeds in manure when you can go buy a bag of seeds? There are MANY writings in the world much better than this neurotic nonsense. 

I would like to voice a similar remark regarding the quality of anti-semitic slander lists right now; there are kernels of richest wheat in it (viz. one can learn from it that 'people who are driven by hate are eager to fabricate shit'), but alas, it can take quite a bit of work to figure that out.

You must not make your brother pay interest, interest on money, interest on food, interest on anything on which one may claim interest. You may make a foreigner [Gentile] pay interest but your brother [fellow Jew] you must not make pay interest.Deuteronomy 23: 19-20

Although an accurate biblical quote, this needs to be seen in a context where every other legal system in existence permitted every lender to demand interest from any debtor. What this does is not to make Jewish money-lenders better-off - it puts more restrictions on Jewish money-lenders operating in the countries surrounding Israel than it puts on gentile money-lenders. Roman money-lenders loaned at interest to everyone - Roman, Greek and Jew alike. Thus, the Jewish lender was at a disadvantage. Until Christianity and Islam forbade loans at interest (something the Christians have reneged on, and how!), Jewish lenders were not at an advantage. Jewish debtors to Jewish lenders were at an advantage compared to other debtors (to Jewish or gentile lenders), however. This is a complex topic, but one where it's easy to make soundbites that come off as very damning evidence regarding Judaism, whereas if you actually take a closer look at the issue, it turns out to be much less clear-cut.

To this day, most legal systems permit the taking of interest. The Islamic system is exceptional in forbidding it, and the Jewish system is exceptional in having such a particular restricted weakening of the creditor's rights. But of course, we can still moan and gripe about Jewish bankers and their interest. It is not like any Christian ever has been a banker, nor is it like the stereotype of the Jewish banker still infuriates lots of conspiracy theorists.

For sources regarding interest in the economic system of antiquity, see

A Dictionary of Greek and Roman Antiquities

A Brief History of Interest, Stephen Zarlenga

The decisions of the Talmud are words of the living God. Jehovah himself asks the opinions of earthly rabbis when there are difficult affairs in heaven.

Rabbi Menachen, Comments for the Fifth Book

The source sounds pretty spurious, although I have come across comparable ideas in the Talmud. However, is it really all that problematic an idea? Looking at the name of the book and the rabbi does not help much - Menachem is a very common name for rabbis, and there's any number of books that can be considered 'the fifth book'. (Nowhere is it specified out of how many, or fifth book of what?) Posting such a thing without digging deeper into it is irresponsible.

Jehovah himself in heaven studies the Talmud, standing: he has such respect for that book. Tractate Mechilla/Me'ilah

This is not in tractate Me'ilah. Neither is it in tractate Megillah, in case someone's gotten the spelling totally wrong. Since about a thousand websites tell me it is in Megillah (and about as many claim it is in Me'ilah) (but the Soncino edition, which as per the author's claim is less censored, does not contain this at all), it's fairly difficult to find where it actually is, due to the sheer amount of misinformation online. There are a few indications that it might be a fabrication: talmud is not "a book", secondarily, only the really final generations of talmudists would have called the written work 'the talmud', and finally, the tetragrammaton does not appear particularly often in phrases of this style in the Talmud.

And in addition, since there actually is in existence a standard pagination of the entirety of the Talmud and a standard way of referring to pages in it, one should get suspicious whenever the folio is not given. 

R. Johanan said: A heathen who studies the Torah deserves death, for it is written, Moses commanded us a law for an inheritance; it is our inheritance, not theirs. Then why is this not included in the Noachian laws? -- On the reading morasha [an inheritance] he steals it; on the reading me'orasah [betrothed], he is guilty as one who violates a betrothed maiden, who is stoned. An objection is raised: R. Meir used to say. Whence do we know that even a heathen who studies the Torah is as a High Priest? From the verse, [Ye shall therefore keep my statutes, and my judgments:] which, if man do, he shall live in them. Priests, Levites, and Israelites are not mentioned, but men: hence thou mayest learn that even a heathen who studies the Torah is as a High Priest! -- That refers to their own seven laws.Sanhedrin 59a

Here, the author actually has looked the quote up. One may find today that even orthodox Jews permit gentiles to study the Torah, and clearly then it's not just the seven laws that are included in the later understanding of this. Bear in mind that the Talmud records many opinions that never were accepted as valid rulings. However, orthodox authorities are of the opinion that the non-Jew would benefit the most from studying the seven laws that God gave humanity according to orthodox Judaism. (These 'seven laws' are really 'headings' for about 70 or so more particular laws.)

To communicate anything to a Goy about our religious relations would be equal to the killing of all Jews, for if the Goyim knew what we teach about them, they would kill us openly.

Libbre David 37

Libbre David does not exist, except in lists like these. I have searched for it for a few years ever since first coming across this in a similar list, whenever I have the opportunity. Its non-existence is fortunate for whoever made this shit up, since it means when you google it you'll find a lot of reposts of the same antisemitic smear campaign, and no actual substance. (My guess is that Libbre is taken from Latin - some doofus who has not realized Hebrew isn't closely related to Latin has guessed that 'libbre' probably sounds Hebrew-like and probably means book, and then went and attempted to back-translate 'Book of David' in a highly haphazard way.)

A Jew should and must make a false oath when the Goyim asks if our books contain anything against them.

Szaaloth-Utszabot, The Book of Jore Dia 17

Szaaloth-Utszabot does not exist, Jore Dia is the Yoreh Diah of Schulchan Aruch, and nowhere in it is a commandment like this. (Further, 'Sz' as a sequence appears in two languages in Europe - Polish and Hungarian. Both of these have been spoken by large numbers of Jews, but very little literature of this kind has been written in either one of them. Furthermore, the letter sequence 'aa' is uncommon in Hungarian, where it normally would be written á. The sequence 'aa' is also very uncommon in Polish. In neither of the languages does '-th' really occur all that often at the end of words. The book name 'Szaaloth-Utszabot' is with great likelihood a complete fabrication. Some guesses as to why those who fabricate these names (and distort the names of more standard works) use such names will be presented later (in part two or three of this article).

If you can read Hebrew, Yoreh Deah 17 can be found here. It is a bunch of laws relating to kashrut, i.e. the laws on what foods are okay.

We beg Thee, O Lord, indict Thy wrath on the nations not believing in Thee, and not calling on Thy name. Let down Thy wrath on them and inflict them with Thy wrath. Drive them away in Thy wrath and crush them into pieces. Take away, O Lord, all bone from them. In a moment indict all disbelievers. Destroy in a moment all foes of Thy nation. Draw out with the root, disperse and ruin unworthy nations. Destroy them! Destroy them immediately, in this very moment!Prayer said on the eve of Passover (Pranajtis: Christianus in Talmudae Judeorum, quotations from: Synagoga Judaica)[1]

Regarding Pranaitis, it is well worth reading up on his activities. Wikipedia actually gives a good summary. Pranaitis was an ignorant 19th century antisemite, who made shit up even to get Jews executed on blood libel charges. This knowledge should prompt the author to verify any claims along these lines. If the author lacks this knowledge, that's quite a compelling argument not to listen to the author at all – no one should listen to ignoramuses.

When the Messiah comes every Jew will have 2800 slaves. 

Simeon Haddarsen, fol. 56-D

Again, a work that does not appear to exist! Further, folio numbering generally goes to B - there are only two pages to a folio (since a folio is the designation of the two pages you see when you have a book opened in front of you). Whoever made up this claim is ignorant of how folio numbering works. Some antisemitic retards of course attribute the non-existence of the book to a Jewish conspiracy - Simeon Haddarsen, in their view, has been made to vanish by Jews [2]. Nonfalsifiability is a great trick if people are dumb enough to be convinced by it.

A similar statement is indeed made by Simeon Lakish (also known as Resh Lakish) in the Talmud - and the article I am reviewing does get that reference right. However, the quote seems metaphorical: whoever observes the commandments of tzitzis (fringes) gets boons! A direct quote:

Resh Lakish said: He who is observant of fringes will be privileged to be served by two thousand eight hundred slaves, for it is said, Thus saith the Lord of hosts: In those days it shall come to pass, that ten men shall take hold, out of all the languages of the nations shall even take hold of the skirt of him that is a Jew, saying, We will go with you, etc. [Talmud, Shabbat 32B]

The arithmetic is simple: the talmudists considered there to be seventy languages, the fringes are four, so 10x70x4 = 2800. This does sound more like an exhortation for observance in regard to the commandment of fringes rather than a promise, as weird exaggerations along these lines are often given in order to emphasize the importance of carrying out the commandments - who in their right mind thinks the rabbis meant it literally when they said that whosoever saves a single human life is as though he has saved the entire world – clearly that too is an exaggeration, given that the world has millions upon millions of people in it? Observance of any number of commandments is exhorted by promise of all kinds of rewards – on the other hand, one is also commanded to observe commandments not for the sake of rewards, but for the love of their giver.

On the house of the Goy [Goy means unclean, and is the disparaging term for a non-Jew] one looks as on the fold of cattle.

Tosefta, Tractate Erubin VIII

Nope, not in Eruvin 8. It does not even fit into the context all that much. [Tosefta, tractate Erubin] Also, again, goy does not mean 'unclean', goy means 'nation', and has come to signify 'gentiles' in Hebrew, Aramaic and other Jewish jargons, dialects and languages. It has disparaging significance now and probably had so back then as well, but its use is certainly not disparaging in every context as previously illustrated in this post. The author could have edited this quote so as to exclude the clear fabrication, but clearly the author elected not to do so - is this perhaps because it feels good to accuse the Jews of this kind of shit?

Happy will be the lost of Israel, whom the Holy One, blessed be He, has chosen from amongst the Goyim, of whom the Scriptures say: "Their work is but vanity, it is an illusion at which we must laugh; they will all perish when God visits them in His wrath." At the moment when the Holy One, blessed be He, will exterminate all the Goyim of the world, Israel alone will subsist, even as it is written: "The Lord alone will appear great on that day!...
Zohar, Vayshlah 177b

Although I am not entirely certain that whoever wrote this got the references right, compare Vayshlah 177 (the 'b' is a nice touch that does not belong there, but whoever compiles these lists seems to think references to Judaica are incomplete unless they also have a superfluous letter).

THE REVERSE IS TRUE FOR THE IDOLATROUS PEOPLE OF THE WORLD. In the future, the Holy One, blessed be He, will deal with them gently at first, but afterward with severe Judgment. This is the meaning of the verse: "Hashem shall go forth as a mighty man, He shall stir up ardor like a man of war." First comes "Hashem," the Merciful; then He comes "as a mighty man," not a REAL mighty man; and later, "like a man of war," not a REAL man of war. Finally, Judgment will be given against them, and He will destroy them, as it is written: "He shall cry, indeed, roar, He shall show Himself mighty against His foes" (Yeshayah 42:13), and "Then shall Hashem go out, and fight against those nations, as when He fought in the day of battle" (Zecharyah 14:3), and "Who is this that comes from Edom, with crimsoned garments from Bozra..." (Yeshayah 63:1)

These are not kind words, nor are they particularly tolerant. Notice, however, that this specifies 'the idolatrous people of the world', and not 'all the goyim' (the Hebrew versions I've been able to find all use that wording). That is a significant difference. The text no longer hates on gentiles per se, but idolatrous such. Certainly this is religious intolerance, but no worse than comparable statements that can be found in Christianity or Islam. The Hebrew text itself is clear on the idolatry - it uses the term 'akum', which is used for practitioners of idolatry.

[1] Almost all quotes from http://www.truthbeknown.com/judaismquotes.htm

[2] Other sources are generally named in the text.

A recommended read

This is a great post.

I don't think the author fully has managed to give an exhaustive description of fundamentalism, but he has identified an important component that seldom is discussed, and that needs discussion.

Bullshit Oscillating at 432hz: Pythagoras and Numerological Nonsense

Quite a few A432hz enthusiasts claim that only in A432hz are all the tones of the scale integers.

This is a truth with quite a bit of modification. Some of the advocates of this claim also claim that you can make your music A432hz by tuning it down using software. There are instructions all around the internet, especially on how to use Audacity to achieve such a detuning.

However, what will happen if you use Audacity to those ends is the following set of pitches:
A = 432
Bb = 457.688056763
B = 484.90360487
C = 513.737473681

etc

Of course, the actual pitches will vary a bit from these idealized values, as singers deviate from their exact pitch both intentionally (to make their melody or harmony bit more expressive or more 'in tune' in quite a different sense from the one proposed by the A432hz enthusiasts) or by mistake. The same goes for free pitch instruments such as trombones or violins. Guitarists often have badly intonated guitars, so their pitches may also deviate quite a bit, and intentional and unintentional string bends add to the deviation there. Hammond organs have a peculiar tuning of their own that approximates regular tuning but is ever so slightly off, etc. The organ-builder may have missed by a hundredth of a millimetre the exact length a certain pipe should have been, and thus the tuning may be ever so slightly off, thus making the A:WHATEVER come out as A:WHATEVER±a bit.

Thus, the table of tunings for a small bunch of pitches provided there is only a sort of idealized average. Electronic music might get pretty close, though.

Why do the A432hz people believe that tuning to A432hz gives integer frequencies to most of or indeed to the whole scale? Many of them favor Pythagorean tuning, which is not just a question of readjusting the tuning of the reference pitch, it is a question of calculating the other pitches in other ways relative to the reference pitch. (Which requires tuning every pitch on your instrument differently, individually.)

Tuning down a song with audacity does not achieve that result. However, if you were to build your own instrument in such a way that it does have Pythagorean tuning, the idea regarding the integers will be slightly true, and I will explain why in a bit.

Why they believe that this is only achievable with A432hz is a bit less easy to understand. I have no idea, to be honest. I guess they just don't understand evidence-based thinking?

So, why does A432hz give integer frequencies with Pythagorean tuning? It's not entirely true, but it is true for the keys of C major/A minor^A. Pythagorean tuning consists of tuning a bunch of new intervals by repeatedly tuning a new one up a perfect fifth, and a new one on top of that down a perfect fourth, and repeating that pair of operations (at some points, two perfect fourths will need to be added in sequence for this formulation to work, however). The untempered perfect fifth is a ratio between two frequencies, exactly 3/2. So, 100hz and 150hz are a perfect fifth apart. 

We start by C256, and we immediately obtain G384. We now multiply that by 3/4 (the perfect fourth is 3/4 downwards, 4/3 upwards. Notice that 3/2 * 4/3 = 2) and get 288. We go on to obtain 432 hz, and from there we still add 324hz, 486hz, and 364.5. So, in the [256, 512]-range we have one non-integer, but since octaves correspond to doubling a frequency, that problem does disappear in the 512-1024 range, as well as even in the 432-864 range (which is of interest if we focus on A).

However, why should we construct scales using this method? This method was indeed known to the ancient Greeks but so were other methods, such as those described by Archytas, for instance. It does give very nice fifths, but it sacrifices the consonance of the major and minor thirds significantly. Unlike equal temperament, you either end up with infinitely many pitches or a wolf interval.

Pythagoras* allegedly discovered that having two things that differ by simple ratios - 2/1 or 3/2 and such - produces consonant intervals. Examples given in Pythagorean literature consist of anvils whose masses differ by such a ratio, strings of the same dimensions weighted down by weights differing by such a ratio, etc. Weird enough, the examples given in the early Pythagorean descriptions don't work - they simply do not produce results that correspond to the perfect fifth.

The piece here below that is indented might not interest all readers. It has to do with number theory and pythagorean tuning.

3/2 does produce a very consonant interval. The method I gave above is basically the same as stacking 3/2s on top of each other, and sometimes reducing them by octaves (dividing by powers of two) to keep them in the same octave - 1/1 - 3/2 - 9/4 (9/8) - 27/8 (27/16) - 81/16 (81/64) - 243/32 (243/128), etc.
Now, we need to look a bit at factorization to understand why A432 / C256 have these results.
432 factors to 2 * 2 * 2 * 2 * 3 * 3 * 3. 256 factors to 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2. When multiplying, we simply concatenate the strings of factors. When dividing, we remove some shared factors:
555 / 27 = (3 * 5 * 37) / (3 * 3 * 3) = (5 * 37) / (3 * 3). 55 * 231 = (5 * 11) * (3 * 7 * 11) = (3 * 5 * 7 * 11 * 11)

This does not give us beautiful and easily comprehensible numbers, but this way of illustrating multiplication and divsion may illustrate why certain things work the way they do.

2 * 2 * 2 * 2 * 3 * 3 * 3 can obviously be divided by 3 exactly three times without yielding a non-integer. Multiplying 256 by 3/2 will be doable up to eight times until we've depleted the twos from the factorization. Since we basically alternate between adding a 3 and removing a 2 (by multiplying by 3/2), and adding a 3 and removing two 2s (when multiplying by 3/4), we can basically calculate how long it'll take to deplete the 2s - we're removing an average of one and a half per iteration, and thus we run out on the sixth iteration, which explains why the seventh tone is off by half from an integer.

By 432, we have already depleted a few 2s - we have four left. Thus, if we want to build an A major scale (which is a sequence of five leaps of fifths and one leap of 3/4 down from the starting point, including the notes at both ends, this giving us seven notes) we will deplete our 2s before getting all the way:

A: 432, E:  648, B: 486, F#: 729, C#: 546.75, G#: 820.125, D: 576

Pythagorean tuning takes one very consonant interval, and reiterates it to build a full scale. It is a useful musical scale, and probably the tuning that most medieval European music was composed in. However, it has certain issues that make almost all music composed since the renaissance fit less well with it:

• its thirds give rather dissonant chords.
• it does not form a 'cycle', it forms a 'spiral', alternatively 'it requires an infinite number of notes (or it breaks somewhere)

The first problem is the result of how dissonance works. We recall that the C major chord consists of C,E and G. We know that G is very consonant and therefore ignore that for now. We instead look at E, which is 648hz. This E is at 81/64 the frequency of C. We notice that a very nearby ratio, 80/64 = 5/4 looks fairly simple in comparison. We produce tables of overtones of the relevant notes:

512			1024			1536			2048	2560								3072					3584
E'	640			1280			1920			2560									3200
E		648			1296			1944								2592				3240

The slightly lower E at 5/4 in fact has less dissonance, due to the overtones coinciding perfectly every fourth/fifth overtone for the pair C+E' , whereas the 81/64 overtones nearly never coincide and slightly more often also reach into the dissonant 'critical bandwidth'. This is actually entirely audible as well, we can compare the effect of these chords:

Listen Music Files - Embed Audio Files - Pythagorean vs Just Intonati...

Since music is fundamentally a subjective thing, some may prefer the first chord, some may prefer the second. Personally, I find them useful for different purposes - however, the chord you get on your average guitar is a good enough approximation of both for most purposes.

Turns out the first chord type does not really 'resolve' as well as the second - if you end a song on it, there'll be the kind of feeling lingering that 'hey, this song (or part of a song) hasn't come to a halt yet'. That might be a nice effect at times - but it's not what most classical or even pop music goes for. In medieval music, this kind of chord was not used as a consonance, but a dissonance that had to be resolved, either to a perfect fifth or a perfect fifth and an octave (so, in the key of C, C+G, or C+G+c). For more information on this, see Margo Schulter's monumental website on Pythagorean tuning and medieval harmony.

Thus, if you were to somehow magically retune all the works from basically the renaissance onwards up to this day to a Pythagorean tuning, you'd end up with a lot of songs whose chord progressions do not really work as their composers have intended any longer. But who cares for the intent the composers expressed in their compositions when you have a bunch of new age gurus telling you what to do?

As for the cycle thing, we need to look at the concept of modulation and the circle of fifths. In European music, the ideas of chord progressions and of modulation both have been of quite some importance for some time now. A chord progression is a sequence of chords, and chords are sets of three (or more) notes. Most musicians do not think of the progression Am Dm Am E as fundamentally different from the progression Abm Dbm Abm Eb. They may differ in how easy or difficult they are to play on a given instrument, but essentially they have the same internal structure - in isolation, they sound very similar. This can be achieved in both equal temperament and Pythagorean tuning. But, whereas this is possible when using any note as a starting point in equal temperament, it only is possible for a limited number of starting points in Pythagorean temperament (or, you end up with an infinite number of tones you have to work with). We want a system where if a chord is the Nth chord in one key, it has the same function relative to its key as the Nth chord of the other keys. (Of course, we could probably tolerate just a few keys for which it does not hold true, but it adds complications.)

In more modern European music, it happens that the key is changed during the work. Sometimes, this even happens repeatedly. Thus, we want a large set of workable keys between which we can switch.

Further, sometimes, performers' ranges are sufficiently wide for a given work, but the absolute reach does not go sufficiently high or low. In those cases, it is convenient for musicians to adjust the piece of music so that the range of the singer (or other performer) now corresponds to the adjusted piece's demands. Our voices aren't all created the same, so flexibility in this way is very useful.

As I mentioned, Pythagorean temperament does, to some extent, satisfy these demands. However.
It necessarily contains some breaking point. We've built our scale by adding new tones that are perfect fifths apart, and we notice that the distance to the tone from which we started never is an interval we have seen before. Either we stop somewhere, or we go on forever. If we go on forever, we end up with notes whose names would be monstrosities along the lines of C######## (which would be about a fifth of a semitone sharper than G).

The twelfth tone we add is 3¹²/2⁽¹⁹⁾. As it happens, this is fairly close to 1/1. So we ignore it altogether and close our cycle there, letting the error fall on the last tone. (We could go on, and let the error fall elsewhere, but this is as convenient an ending point as we get - we don't end up with dozens of named notes, nor do we end up with a bunch of notes that are very close to each other.) Thus, the last fifth we have is of the form (3/2) / (531441/524288) - which is a very ugly interval - the first note is a perfect fifth, the other is the wolf as it would be if it were tuned to C:

This error will also be present in any interval that "spans" this fifth. Different chords will sound drastically different, and transposing a song from one key to another may turn a chord in the song from consonant to dissonant or vice versa, ruining the song's structure altogether. (A given chord, say, 'G', will constantly be the same, of course, but what we're interesting in is retaining the structure of the scale such that, say, the chord built from the second tone of any given major key will sound sufficiently similar to every other such chord, so that we can say that it has the same 'function' relative to its key as the other 'second chords'.)

Equal temperament solves this by distributing the error given previously over each fifth, having each fifth just slightly off. The difference is barely perceivable.

Further, the fact that we've now reduced the fifth ever so slightly, adds up to four times more of a reduction of the major third - which nudges it closer to the very consonant major third in the first sound clip. And we end up with a system where each key can be used. The sample compares a pythagorean and an equal temperament fifth C-G, C-G'; the second half leaves out the lower part of the intervals so we can just compare the pitch of the two Gs. The difference is tiny.

Now, I've gone on for quite a bit here about tuning. I don't particularly believe that equal temperament is superior in any musical sense than other tunings, but it has many advantages that explain why it is used. I am fairly convinced, however, that most repertoire since the renaissance on to this day would not work very well in Pythagorean renditions.

Because of a thing in arithmetics - viz. the n:th root of an integer will always be irrational unless that integer is another integer to the n:th power - all the frequencies we obtain, except at most one, will be irrational. Regardless if we tune to A432hz or A440hz (or any other hz whatsoever).

However, our dear A432hz enthusiasts have of course done their maths and picked their tuning system so as to have integers all the way. Yet, they do it wrong. Let's compare some different A432hz tuning tables:

	1, **2		3			4*	5	6	A440/12tet	A432/12tet
a	432		432			432	432	432	440	432
a#	**461.3					464	458.21		466.163761518	457.688056763
b	486		484			480	486		493.883301256	484.90360487
	**518
c	512		514			512	512	518.2	523.251130601	513.737473681
c#	**546.75					544	543.06	540	554.365261954	544.285893555
d	576		576			576	576	576	587.329535835	576.650817001
d#	**615.1					608	610.94		622.253967444	610.940258945
e	648		648			640	648	648	659.255113826	647.268657211
f	**691.2
f	7041		688			672	682.66	691.2	698.456462866	685.75725445
f#	**729					736	724.08		739.988845423	726.534502779
g	768		768			768	768		783.990871963	769.736492473
g#	**820.15					800	814.6		830.60939516	815.507406157

Notice how the different sources do not agree on their pitches? Some of these intervals vary by as much as 22/21 (f=704 and f=672). In part this is because they use entirely different approaches to building their scale - the high 704 is not pythagorean at all despite the claims by the source, it's a way more esoteric interval (11/8). By arbitrarily picking our frequencies in such a manner, I can build an integer-only tuning based on A440, viz. A440, Bb466, B494, C523, C#554, D587, ... and this can be done for any arbitrary starting point in that region. The errors introduced for any interval by rounding the frequency to an integer number of hertz will be just slightly wider than the error of the perfect fifth in the 12-tone equal system. And that is of course an idealized error - singers, brass players, violinists, cellists, and even guitarists will regularly be further off.

Of course, ultimately, the second's length has been arbitrarily decided; we could have divided the day into ten equal hours and each hour into 100 equal units and each of those into another 100 equal units, and a tone at 432hz would now be described as ~373.248alternahz. The division of the day into 24*60*60 is arbitrary. In a world with alternahz instead of hz, other frequencies would be integers. Integer hz frequencies have no magical properties despite the dumb beliefs A432hz enthusiasts have regarding this.

But as I might have said before, if you don't like that priests, ministers, imams or rabbis tell you what music to listen to, you can always listen to new age gurus instead - they even have rituals that make your music 'spiritually permissible' (because what else does reducing its audio quality by an ever so slight amount of resampling artefacts in Audacity amount to, but a superstitious ritual - and unlike rituals by older, more well-established religions, this at least has the veneer of technology to it - but who am I kidding, it's really slightly worse than e-mailing a dozen 'hail Mary' into the digital void). Why turn to evidence-based reason when gurus make stuff so much easier? And the added anxiety from believing that Nazis have made the music you hear in the radio increase aggressiveness among your peers is certainly good for your health as well as well.

By further telling you to prefer Pythagorean tuning over other tuning methods, they're essentially imposing a certain music theory on you - one in which modulation is limited, one in which chord resolutions are much more restrictive, one in which the useful keys are much fewer and you end up having to buy new, expensive guitars because your Gibson or Martin or Taylor or Fender simply cannot be tuned in a Pythagorean fashion*. You get less, but at such a steep expense, who can refuse?


* Pythagorean guitars require complicated frets that are damn expensive to manufacture. You'll end up with one along the lines of the guitar neck pictured in this post. Those are not cheap, I can tell you. But of course, if a new age guru tells us to buy them, who are we to refuse? Who are we, indeed, to refuse?

A) A natural minor and C major contain the same seven notes. These are, when ordered as a series of fifths, F-C-G-D-A-E-B. Ordered as a regular scale they are C D E F G A B (c). (Or A B C D E F G). Think of F-C-... as though each "-" signifies "..., which equals 4/3 or 2/3 of ...", so F, which equals 4/3 of C, which equals 2/3 of G, ...

An Appendix: Dissonance and Consonance

Consonance and dissonance are traits we ascribe to sounds. Thus, our perception of sounds is somewhat important to this classification. Clearly, it's not a fully objective quality.

In the main post, I note how Pythagoras contributed to the understanding of harmony.  Although Pythagoras did not understand what sound was, he did understand that relating the sizes of the objects on which you play (say, different lengths of otherwise identical strings) by simple ratios produced appealing sounds. There have been a number of hypotheses as to why this would be the case. The most widely accepted one these days relates to the harmonics (overtones) described previously.

If we add together two sine waves, a and b, of rather similar frequencies,  the resulting wave will have a complication - its amplitude changes periodically in a wave-like way:

Sum of sin(tx) and sin(ytx), where y is a constant relatively close to 1, and t is an arbitrary constant.
I have no idea what causes the graphical misshap close to the second through.
Picture from wolframalpha.com, try out sin(x) + sin(yx), with y≃1 there for a variety of y.

 The frequency of this "metawave" is the same as the difference between the two frequencies a and b. If the difference is small, it does not sound bad - just like a slight wavering volume, somewhat similar to a vibrato in sound, and if it is large we don't perceive it as dissonant either. The range in which we perceive dissonance is called the critical bandwidth. Empirical research has shown that it covers a range from about a handful hertz to 6/5 of the frequency. However, this might seem to fail to explain dissonances over wider ranges, such as the major seventh (which is roughly 15/8, which clearly is wider than 6/5) or the very dissonant tritone (which is sqrt(2), which is a bit less than (6/5)^2, and thus clearly wider than 6/5).

I previously mentioned overtones. These provide us with the actual explanation! To calculate the dissonance for an interval, we should actually look at the amount of overtones for each of the tones in the interval that come within each others' critical bandwidths. The relative amplitudes of course also contribute, but in ways that is less easy to investigate by just eyeballing a graph.

Let us compare two intervals. A440 and E660 vs. A440 and D#622. (Note: for ease of calculation, I have reduced the usual D# by an ever so slight bit.) These have the following overtone series:

A	440
D#			622
E		660
A	880
d#			1244
e	1320	1320
a	1760
a#			1866
b		1980
c	2200
d#			2488
e	2640	2640
g*	3080		3110
g#		3300
		.
		.
		.

We can see that the A column (the one starting out with 440), and the E column (660) often coincide. Even if they didn't perfectly coincide (say, we replaced 660 with 659 or 661), the numbers would be close, and thus not reach the requisite width to enter into the critical bandwidth until several overtones down the line. However, 622 quickly enters it - 1320 is within the critical bandwidth of 1244 (or rather, they're within each other's range), 1866 is within the critical bandwidth of 1760, etc. Sure, 1980 is within the critical bandwidth of 2200 too, so E will cause some slight dissonance. However, the further up the overtone series we have to go to find critical bandwidth issues, the less dissonant an interval is. Of course, timbre may also affect the dissonance - clarinets and many woodwinds lack even-integer overtones (so, a tone sounding at a 100hz will only have overtones at 300hz, 500hz, etc), and for a good enough analysis, we would have to look into them as well.

Relative amplitude of the overtones is relevant, but we're not going to look at that now. More detailed models for understanding dissonance exist (e.g. 'harmonic entropy'), but this post is mainly meant as an appendix to an upcoming piece of reasoning about scale construction (that is part of a greater piece of reasoning regarding claims made by A432hz enthusiasts). Harmonic entropy has been used by people interested in scale construction, and various predictions made by it seem to have been accurate.

Anyways, this is a very short introduction to the issues of consonance and dissonance, and one where further complications can ensue - instruments where the overtones are not integer multiples of the fundamental, for instance, have their own complications with regards to what intervals are consonant and what intervals are dissonant.