Pages 11

PREFACE POUR VENIR A LA CONNAISANCE DE LA SIGNIFICATION DU JEU DES CHARTES

[...]

Page 15

(Le Jeu de la Paume est de telle nature comme nous avons dit car il est fondé sur l'Arithmétique et sur la Géométrie) : Pareillement l'ancien Jeu des Chartes lequel finit son nombre et gagne la grande victoire à trente et un (31), en nous démontrant les accords parfaits et les Harmonies de Musique composée à quatre (4) parties : lesquelles nous représentent les tempéraments des quatre Eléments dont toutes choses naturelles sont faites et composées par certaines proportions et harmonies : ainsi que nous déclarerons (par la grâce de

Page 16

Dieu) en ce petit traité, en montrant par les nombres que nous pratiquons au jeu des Chartes; les *Convenances*(*?) et Harmonies, que les quatre Eléments ont entre eux, pour faire la génération et composition de chaque choses naturelle et aussi pour la conserver en son être.
Mais avant d'entrer en ce discours, il convient de connaître certains nombres et leurs proportions dont proviennent les Convenances* (*?) parfaites de Musique : desquelles les Musiciens avaient accoutumer d'user anciennement.
Les nombres du jeu des Chartes nous sont représentés par les images et autres figures qui sont peintes en chacune Charte dont on a coutume d'user en ce jeu.


*convenances : qualité de ce qui convient, qui est conforme.
En liturgie sacrée, on parle des critères de convenance d'une musique

Première partie
Pages 17 -20

Dans cette Première partie , Gosselin déclare 7 proportions (en vérité 14 en comptant les 'sous proportions à chaque fois)

Déclarer les Nombres et leurs proportions dont dépend la signification du jeu des Chartes


[Proportion double et sous double]
Il convient savoir que quand quelque plus grand nombre est deux fois justement aussi grand qu'un autre plus petit nombre, le dit plus grand nombre, a proportion double à ce plus petit nombre. Et le dit plus petit nombre a, au dit plus grand nombre, proportion 'sous' double : comme six à trois ont proportion double : ou bien deux à un. Car six font deux fois justement aussi grands, comme trois. Et deux contiennent un deux fois. Et ainsi faut-il entendre de tous les autres nombres, lesquels ont entre eux semblable proportion.

[Proportion sesquialtère et sous sesquialtère](cf Note A)
Quand quelque plus grand nombre contient un plus petit nombre une fois et moitié du dit plus petit nombre justement, - ce plus grand nombre, a proportion sesquialtère au dit plus petit nombre : et ce plus petit nombre a proportion sous sesquialtère au dit plus grand nombre : comme douze à huit, ou bien six à quatre, ont proportion sesquialtère ; et huit à douze, ou bien quatre à six, ont proportion sous sesquialtère : car douze contiennent huit une fois (8x1) et quatre davantage (+4) qui font la moitié de huit [12=8+4]
Pareillment six contiennent quatre et deux davantage qui sont la moitié de quatre [6=4+2]

Note A
A. − MATH. Qui est dans le rapport de 1,5 à 1. Nombres sesquialtères; rapport sesquialtère. Six est à quatre en raison sesquialtère (Ac.1798-1935).Ercole: Vous savez le secret de sa beauté [d'Imperia]. C'est que son corps offre en tout la proportion sesquialtère. Jacinto: Comment diable avez-vous fait pour le mesurer? (Renan, Drames philos., Caliban, 1878, II, 1, p. 393).
B. − MÉTR. ANC. Rythme sesquialtère. Rythme comprenant des pieds où l'un des demi-pieds vaut une fois et demie l'autre. (Dict. xxes.).
C. − MUS. ANC. Mesure sesquialtère. Mesure dans laquelle la note principale vaut une fois et demie sa valeur ordinaire. On peut les écrire [les branles gais] de trois manières différentes: le (...) 2ePar mesure sesquialtère 3/2 (Écorcheville, Suites orch., 1906, p. 53).
Prononc. et Orth.: [sεskɥialtε:ʀ]. Att. ds Ac. dep. 1762. Étymol. et Hist. 1. Ca 1375 math. proporcion sequialtere (N. Oresme, Livre du ciel et du monde, 194a, éd. A. D. Menut et A. J. Denomy, p. 700, 35); 1572 mus. id. (Amyot, Les Œuvres morales, Propos de Table, p. 663 G-H); 2. 1933 « jeu de fourniture dans l'orgue » (Lar. 20e). Empr. au lat.sesquialter, en parlant d'un nombre « qui en contient un autre une fois et demie », comp. de sesqui, adv. empl. surtout en compos. « un demi en plus », dér. de semis, propr. « moitié de l'unité, de l'as » et de alter « l'un des deux, l'un, l'autre ». Bbg. Baldinger (K.). Zum Übergang von der lateinischen zur französischen Fachterminologie im 14. Jahrhundert. Z. rom. Philol. 1975, t. 91, p. 489.

[Proportion sesquitierce et sous sesquitierce] '(Cf Note B)
Quand un plus grand nombre contient une fois un plus petit nombre et la troisième partie du dit plus petit nombre justement, ce plus grand nombre a proportion sesquitierce au dit plus petit nombre. Et ce plus petit nombre a à ce plus grand nombre proportion sous sesquitierce.


Page 19
Comme ont huit à six, ou bien quatre à trois : car huit contiennent six une fois (6x1) et deux davantage (+2) soit [8=6+2] qui sont la troisième partie de six (6:3=2).
Semblablement, quatre contiennent trois, une fois,(3x1) et un davantage (+1) qui est la troisième partie de trois (3 : 3= 1) [4=3+1].

Note B
(Mathématiques) Se dit de deux nombres, ou de deux lignes, dont l’une contient l’autre et un tiers de plus.

[Proportion triple et sous triple]
Quand un plus grand nombre contient un plus petit nombre trois fois justement a proportion triple à ce plus petit nombre. Et le dit plus petit nombre, a proportion sous triple à ce plus grand nombre : comme quinze à cinq ou bien six à deux, (ils) ont proportion triple.
Et cinq à quinze, ou bien deux à six ont proportion sous triple.

[Proportion quadruple et sous quadruple]
Quand un plus grand nombre contient un plus petit nombre quatre fois justement, ce plus grand nombre a proportion quadruple au dit plus petit nombre : et ce plus petit nombre a proportion sous quadruple au dit plus grand

Page 20

nombre : comme douze à trois ou bien huit à deux ont proportion quadruple. et trois à douze et aussi deux à huit ont proportion sous quadruple.

[Proportion octuple et sous octuple]
Quand un plus grand nombre contient un plus petit nombre huit fois justement, le dit dit plus grand nombre a proportion octuple à ce plus petit nombre. Et le dit plus petit nombre a proportion sous octuple au dit plus grand nombre : comme seize à deux, ou huit un, ont proportion octuple. Et deux à seize ou bien un à huit ont proportion sous octuple.
Car seize contiennent deux huit fois (16=2x8), ainsi que huit contient un huit fois (8=8x1).

[Proportion sedicuple et sous sedicuple]
Et quand un plus grand nombre contient un plus petit nombre seize fois justement, ce plus grand nombre a relation sedecuple au dit^plus pe...

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...tit nombre et ce plus petit nombre a proportion sous sedecuple au dit plus grand nombre : comme trente deux à deux ou bien seize à un ont proportion sedecuple : deux à trente deux ou bien un à seize, ont proportion sous decuple. Car trente deux contiennent deux seize fois justement (32=16x2), semblablement seize contiennent un justement seize fois (16 = 16x1)

Bout of insomnia at the moment, so whiled away the early hours having a go at above section. (MikeH can hopefully offer an improvement).

Pages 11
PREFACE: GETTING TO KNOW THE MEANING OF THE GAME OF CARDS
[...]

p.15
(The Game of Tennis is of such a nature as we said because it is based on arithmetic and geometry): Similarly the ancient Game of Cards which ends its number and wins the great victory at thirty-one (31 ) by showing us the perfect chords and harmonies of music composed in four (4) parts: which represent to us the temperaments of the four elements of which all natural things are made and composed by certain proportions and harmonies: and we will declare (by the grace of

p.16
God) in this little treatise, demonstrating via the numbers that we practice in the game of cards; the rules and harmonies the four elements have between them for the generation and composition of each natural thing and for the conservation of its being. But before entering into this discourse, it is necessary to know certain numbers and proportions that come from the perfect rules of music, which musicians were formerly accustomed to use. The numbers of the game of cards as customarily used in the game, which we represent by images and other figures painted on each card.

Declaring the numbers and proportions on which the meaning of the game of cards depends.

[Duple and subduple proportions (or double and half proportions)]
One should know that when something is precisely twice as large as another smaller number, the greater number is said to be double the proportion of the smaller numbers. And the smaller number is said to have a proportion half that of the greater number: as six to three is double in proportion: or two to one. For six is precisely twice as large as three. And both are contained two times. And so must we hear of all the other numbers, which have between them like proportion.

[Sesquialter and sub-sesquialter proportions]
When a larger number is one and a half times a smaller number precisely – the greater numbers proportion is sesquialter to said smallest number: and this smaller numbers proportion to the greater number is called sub-sesquialter : as twelve to eight, or six to four, have sesquialtera proportion; and eight to twelve, or four to six, are in the proportion sub-sesquialter: for twelve contain eight once (8x1) and four more (4) that half of eight [8 + 12 = 4].(sic) Similarly six contain four and two more which are half four [6 = 4 + 2].

[Sesquitertian and sub-sesquitertian proportions]
When a larger number contains both a smaller number and the third part of said smaller number precisely, the proportion of the greater number is sesquitertian to said smaller number. The proportion of the smaller number to the greater number is sub-sesquitertian.

p.19

As have eight to six, or four to three, for eight contain six times one (6x1) and two more (2) or [8 = 6 + 2], which are the third part of six (6:3 = 2). Similarly, four contain three once, (3x1) and a further (1) which is the third part of three (3:3 = 1) [4 = 3 + 1].

[Triple and sub-triple Proportion (or threefold and thirds)]
When a larger number contains a smaller number three times it is in a triple proportion to the smaller numbers. And said smaller number, a proportion sub-triple to the greater number: as fifteen to five or six to two, have triple proportion. And five to fifteen, or two to six are of sub-triple proportion.

[Quadruple and sub-quadruple Proportion (or fourfold and fourths]
When a larger number contains a smaller number four times precisely, the greater number is in quadruple proportion to the smaller number: and this smaller number is sub-quadruple (a fourth) the bigger

p.20

number: as twelve to three or two to eight have quadruple ratio. and three to twelve and two to eight are sub-quadruple.

[Eightfold and eighth proportions]
When a larger number contains a smaller number precisely eight times, the greater has an eightfold proportion to the smaller number. And said smaller number is an eighth in proportion to the greater: as sixteen to two, or eight to one have eightfold proportion. And two to sixteen or one to eight are an eighth in proportion. For sixteen contains two times eight (16 = 2x8), and eight contains one eight times (8 = 8x1).

[Sixteenfold and sixteenth proportions]
And when a greater number contains a smaller number sixteen times precisely, the greater number has a sixteenfold relation to said smaller

p.21

number and the smaller number is a sixteenth in proportion to the greater number: as thirty-two to two or sixteen to one have a sixteenfold proportion: two to thirty-two or one to sixteen are a sixteenth in proportion. For thirty-two contain two exactly sixteen times (16x2 = 32), similarly sixteen contain one exactly sixteen times (16x1 = 16).

At this thread ...

viewtopic.php?f=11&t=1102&hilit=popess&start=60#p17062

... SteveM wrote ...

Croce also wrote a tarocchi appropriati in ottava rima, in which he puts the matto at the bottom, so I don't think we were in any way meant to take from the riddle the placement of the fool, which elsewhere he clearly and without riddles puts at the bottom :

(There are only 20, no pope or papessa, or four papi - just the emperor and empress)

Angelo
Mondo
Sole
Luna
Stella
Saetta
Diavolo
Morte
Traditore
Vechio
Ruota
Carro
Fortezza
Giustitia
Temperanza
Amore
Imperatore
Imperatrice
Bagattino
Matto

I din't find this Tarocchi appropriati (without pope and popess) in the text ... is it somewhere?

Huck wrote:At this thread ...

viewtopic.php?f=11&t=1102&hilit=popess&start=60#p17062

I din't find this Tarocchi appropriati (without pope and popess) in the text ... is it somewhere?

It is in the book I referenced in the post. It is also reported on by Giordano Berti here:

http://www.associazioneletarot.it/page.aspx?id=268

He specifically addresses the question of there being 20 trumps in Bologna if that is the reason for your interest (I presume it is, following your recent thread), and rejects this on the basis that both prior to and after the poem the Bolognese tarot had 22 trumps, and becuase of the two trumps excluded. The exclusion of the Papesse and Pope is probably due to the regional sensitivities about the inclusion of these trumps. It is in Italian and, as far as I know, is not one of the pages that has been translated as yet; but google translate is sufficient to give you the gist of it.

Well, you beat me to it, Steve. I had a translation ready in "draft" form, but wanted to let it sit a bit before posting it.

Well, I'm not going to let mine go to waste. Here it is.

Translation of above (viewtopic.php?f=11&t=1102&start=140#p17209 and the one after), based on Gosselin at https://play.google.com/books/reader?id ... g=GBS.PT21, pp. 15-21. Paragraphing and parts in brackets are added by Alain.

page 15
...The Game of Tennis is of such a nature, as we have said, because it is based on Arithmetic and Geometry: Similarly the ancient Game of Cards, which ends its number and gains the great victory at thirty-one [31) shows us the perfect Chords, and the Harmonies of Music composed in four [4] parts, which represent for us the temperaments of the four Elements of which all natural things are made and composed with certain proportions and harmonies, and we will declare (by the grace of

page 16
God) in this little treatise, showing by the numbers that we apply in the game of cards; the Proprieties [* (*?)] and Harmonies, which the four Elements have among them for the generation and composition of each natural thing and also to keep it in its being.

But before entering into this discourse, one should know certain numbers and proportions that produce the perfect Proprieties [* (*?)] of music: which Musicians were accustomed to use in ancient times.
In the game of Cards the numbers are represented for us by images and other figures that are painted on each Card that is customary to use in the game.
_________________
[* In French, Convenance: quality of that which is appropriate, that which is in keeping. In sacred liturgy, we speak of the criteria of convenance of a musical piece.]

Then on p. 17 a new paragraph begins, the first such break in the original text, with a heading before it. Here again, the parts in brackets are Alain's. His "Note A" , which is not part of Gosselin's text and seems to be from the late 20th century, was too difficult for me to translate accurately, due to idiomatic quotations, abbreviations, and technical terms. It seems to me that Gosselin defines the terms he is using clearly enough without these notes. Note A explains that Gosselin's terms are used in music theory, both about notes and about rhythm. Note B only mentions mathematics.

The Pythagorean basis of what Gosselin is saying is evident from the page from Plato's Timaeus that I cited earlier, https://books.google.com/books?id=mwZOB ... ls&f=false, which describes the Creator ("Demiurge") as using most of these ratios, and others, as basic building blocks of the universe. In the translation, I could not find an English equivalent for "sesquitierce", so I left it in French.

p. 17
[In this First Part, Gosselin enumerates 7 proportions (actually 14 counting the under-proportions in each).]

Declaring the numbers and proportions on which the meaning of the game of cards depends.

[Double and under-double proportion]
One should know that when some larger number is precisely twice as large as another smaller number, said large number has a proportion of double to the smaller number. And said smaller number has, to said larger number, a proportion of 'under' double: as six to three has a double proportion: or two to one. For six is precisely twice as large as three. And both contain one two times. And so must be understood of all other numbers, which have between them similar proportion.

[Sesquialter and under-sesquialter proportion (see Note A [omitted here: mikeh]).]
When a larger number contains a smaller number one and a

Page 18
half times said smaller number precisely – this larger number has a sesquialter proportion to said smaller number: and this smaller number has an under-sesquialter proportion to said larger number : as twelve to eight, or six to four, have a sesquialter proportion; and eight to twelve, or indeed four to six, have an under-sesquialter proportion: for twelve contains eight once [8x1] and four more [4], which is half of eight [12 = 8 + 4]
Likewise six contain four and two more, which is half of four [6 = 4 + 2]

[sesquitierce and under-sesquitierce proportion. (See Note B [Omitted here: mikeh)]]
When a larger number contains both a smaller number and the third part of said smaller number precisely, the larger number is in sesquitierce proportion to said smaller number. And this smaller number is to the larger number in under-sesquitierce proportion. As has

Page 19
eight to six, or four to three: for eight contains six one time (6x1) and two more (+2) is [8 = 6 + 2], which is the third part of six (6: 3 = 2). Similarly, four contain three once, (3x1) and one more (1) which is the third part of three (3: 3 = 1) [4 = 3 + 1].

[Triple and under-triple proportion]
When a larger number contains a smaller number three times precisely, it has a triple proportion to this smaller number. And said smaller number has a proportion under-triple to the larger number: as fifteen to five or six to two, has a triple proportion.
And five to fifteen, or two to six has an under-triple proportion.

[Quadruple and under-quadruple proportion]
When a larger number contains a smaller number four times precisely, the larger number has a quadruple proportion to said smaller number: and this smaller number has a quadruple under-proportion to said larger

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number: as twelve to three, or eight to eight, has a quadruple proportion, and three to twelve and also two to eight have a quadruple under-proportion.

[Octuple and under-octuple proportion]
When a larger number contains a smaller number precisely eight times, said larger number has an octuple proportion to this smaller number. And said smaller number has an under-octuple proportion to said larger number: as sixteen to two, or eight to one, have an octuple proportion. And two to sixteen or one to eight have an under-octuple proportion.
For sixteen contains two eight times (16 = 2x8), and eight contains one eight times (8 = 8x1).

[Sixteen-fold and under-sixteen-fold proportion]
And when a larger number contains a smaller sixteen times precisely, the larger has a sixteen-fold [Fr. sedecuple] relation to said smaller

Page 21
number, and this smaller number has a sixteen-fold proportion to said larger number: as thirty-two to two or sixteen to one have a sixteenfold proportion: two to thirty two or one to sixteen have an under-sixteen-fold proportion. For thirty two contain two just sixteen times [16x2 = 32], similarly sixteen contains one exactly sixteen times [16x1 = 16].

As you see, I was as literal as possible, except for not using the word "duple". There are some small differences besides, notably the translation of "convenances" (your "rules" and my "proprieties") and "anciennes" (your "formerly" and my "in ancient times").

Thanks MikeH - as I said, insomnia and stuck for something to do to while away the hours -- I have managed to have a bit of a sleep now

mikeh wrote:In the translation, I could not find an English equivalent for "sesquitierce", so I left it in French.

I struggled with that too. Turned out the English and French are virtually the same (both being based on Latin terminology)

English (derived from Latin)

Sesquialter (for one and one half)
Sesquitertian (for one and one third)

The English prefix for the French 'sous' in this context (prefix to a mathematical or musical ratio) would normally be sub-

As duple and subduple (double and half) are used in English.

sub- +‎ duple
Adjective
subduple ‎(not comparable)

(mathematics) Indicating one part of two; in the ratio of one to two.

duple
adj.
1. Consisting of two; double.
2. Music Consisting of two or a multiple of two beats to the measure.

1:2 & 3:6 are subduple ratio, while 2:1 & 6:3 are duple ratio.

Gosselin II partie 21-30

Dans cette Deuxième partie, Gosselin lie les proportions des nombres aux intervalles musicaux.

Page 21
Démontrer comment des Consonances et de l' Harmonie naissent des proportions qui appartiennent à la Musique

(Introduction)
Pour connaître par expérience aussi bien que par raison comment des consonances de musique sont produites des proportions ci-desssus déclarées, il convient d'avoir une corde uniforme faite de boyau (comme les

Page 22
cordes des violons) ou de métal (comme les cordes d'un cistre) laquelle est bien tonante et bien tendue dessus quelque coffre bien égal et un par dessus et résonnant comme le monocorde dont les Anciens usaient : ou bien le coffre d'une épinette quand il est fermé. Et il faut que la dite corde soit rendue en telle manière que l'on puisse mettre entre elle et ce coffre en divers endroits plusieurs Magades appelés vulgairement chevalets : comme sont ceux qui portent les cordes des violons : lesquels chevalets on puisse changer d'un endroit de la dite corde en un autre afin de distinguer et partager la longueur de cette corde en autant de parties qu'il sera nécessaire-lesquelles parties nous mesurerons avec une mesure connue est le Pied du Roy ou une Paume qui est la quatrième partie de la

Page 23
longueur d'un pied.A finalité d 'accomoder aux parties de la dite corde les proportions des nombres ci-devant déclarées.

[A. Proportion double = Diapason ou Octave]
Donc pour accomoder sensiblement la proportion double, à deux parties de cette corde il faut les mesurer de telle manière que l'une d'elle soit deux fois aussi longue justement, comme l'autre : c'est à dire que la plus grande des dites parties ait proportion double à la plus petite en mettant trois chevalets dessous la dite corde – c'est à savoir deux chevalets aux extrémités des dites parties et un en la division de ces parties comme si la plus longue des deux avait six paumes de longueur, l'autre partie n'aurait trois paumes de longueur justement.
Et par ainsi il y aurait proportion double de la plus grande des dites parties à la plus petite.Et de cette plus petite partie à la plus grande, il y aurait proportion sous

Page 24
double, et alors, si l'on touche l'une et l autre partie de la corde ainsi divisée pour les faire sonner : on connaîtra que leurs sons seront distants 'de bas en haut) l'un de l autre par l'intervalle du Diapason lequel est une consonance parfaite appellée par les Musiciens modernes une Octave.

Et ici en droit et en ce qui s'en suit ci après, il faut noter un Axiome de musique universel : c'est que la plus longue partie d'une corde uniforme tonante rend un son plus grave c'est à dire plus bas que la plus courte partie de la dite corde. Semblablement, que la plus courte partie de la dite corde rend un son plus aigu c'est à dire plus haut que ne le fait la plus longue partie de la dite corde.

[B Proportion sesquialrtere = Diapente ou Quinte]
Pour appliquer la proportion sesquialtere à deux certaines parties de la dite corde, il convient en mesurant celle-ci et aussi en changeant les Magades

Page 25
d'un lieu à un autre, (de) faire deux parties de cette corde dont la plus longue ait proportion sesquialtere à l'autre partie qui est la plus courte comme si la plus grande partie avait six paumes de longueur, il faudrait que la plus courte n'en mesure que quatre paumes de longueur;
Et alors, si l'on touche ces parties pour les faire sonner, on connaîtra quelles rendront deux sons distants l'un de l 'autre, de bas en haut, par l'intervalle de Diapente : qui est une consonance parfaite nommée un quinte par les Musiciens modernes.


[C. Proportion sesquitierce = Diatessaron ou Quarte]
Pour accomoder la proportion sesquitierce à deux certaines parties de ladite corde : il convient (de) distinguer deux parties en cette corde, en les mesurant et en appliquant les Magades à ce cette dernière (comme cela est dit) desquelles parties la plus grande ait proportion sesquitierce à la plus courte : comme si la plus grande
partie avait huit paumes de longueur et la plus petite, six paumes de longueur. Et alors, si l'on touche ces parties pour les faire sonner, on connaîtra que leurs sons seront éloignés l'un de l'autre, selon le bas et le haut, par l'intervalle de Diatessaron : laquelle est appellée par les musiciens modernes une Quarte.

[D. Proportion triple = Diapason Diapente ou Douzième]
Pour appliquer une proportion triple à deux certaines parties de ladite corde, il convient en mesurant et en changeant les chevalets, (de) faire deux parties en cette corde dont la plus grande ait proportion triple à la plus courte comme si la plus grande de ces parties contenait douze paumes, alors la plus petite n'en contiendrait que quatre : alors si l'on touche les dites parties de cette corde pour les faire sonner, leurs sons seront distants l'un de l autre, de bas en haut, par l'intervalle de Diapa-

Page 27
-son Diapente, [Diapason Diapente] lequel est composé d'un Diapason et d'un Diapente et nommé par les Musiciens modernes une Douzième.

[E. Proportion quadruple = Deux Diapasons ou Quinzième]
Pour accomoder une proportion quadruple à deux certaines parties de cette corde, il faut en mesurant et en changeant les chevalets, distinguer en la dite corde, deux parties : dont la plus grande ait proportion quadruple à la plus petite comme si la plus grande des dites parties contenait douze paumes de longueur , (tandis que) la plus petite n'en contiendrait que trois paumes : et alors si l'on touche l'une et l autre de ces deux parties de corde pour les faire sonner, on connaîtra que leurs sons seront distants l'un de l autre par l'intervalle de deux fois (le) Diapason) lequel est une consonance parfaite et appellée par les Musiciens modernes une Quinzième.

Page 28

[F. Proportion octuple = Trois diapasons ou Vingtdeuxième]
Quand on veut appliquer une proportion octuple à deux certaines parties de cette corde, il faut, en mesurant et en changeant les chevalets, faire deux parties dans cette dernière, dont la plus grande ait proportion octuple à la plus courte : comme si la plus grande contenait seize paumes et la plus petite en contenait deux.
Et alors, si l'on touche l'une et l autre de ces deux parties, leurs sons seront éloignés l'un de l autre de trois fois le Diapason qui est une consonance parfaite nommée par les musiciens modernes une Vingt-deuxième.

[G. Proportion sedecuple = Quatre Diapasons ou Vingtneuvième]
Quand on veut accomoder une proportion sedecuple à certaines parties de la dite corde, il convient, en mesurant et en changeant les chevalets de faire en cette corde deux parties dont la plus grande ait proportion sedecuple à la plus petite partie

Page 29

comme si la plus grande de ces parties contenant seize paumes (et) la plus petite, une. Et alors, si l'on touche ces parties de la dite corde, elles rendront deux sons qui seront distants l'un de l autre, de bas en haut, par un intervalle qui contient quatre fois le Diapason qui est une consonance parfaite nommée par les musiciens modernes, une Vingt-neuvième.


[Conclusion]
On peut donc remarquer, en ce qui concerne ce que nous avons ci-devant déclaré touchant les sons et les voix de Musique qu'il n'en faut que deux seulement qui l'accordent ensemble selon l'une des proportions ci-dessus décrites, pour faire une consonance. Mais davantage, quand il s'en trouve trois ou plus qui s'accordent toutes ensemble :
un tel accord ne s'appelle pas seulement consonance mais pour son

page 30
Excellence, on le nome Harmonie car il est bien plus parfait qu'une consonance simple.

Next chapter of Gosselin

III partie
Déclarer la signification des Images, ...des caractères du Jeu des Chartes : comment le dit jeu nous représente la composition de chaque chose naturelle
pp 30-40

Coming soon!

Here is a quick and very rough go -- Mike's will hopefully clear up any errors or areas of confusion.

To demonstrate how Consonances and Harmonies are born of the proportions belonging to Music

To know by experience as well as by reason how musical consonances are produced as per the proportions declared below, one should have a uniform string made of gut (like the strings of violins) or wire (like the strings of a cittern) which is taut and tonal above a fine resonant box like the monochord which the ancients used: or else the case of a spinet when it is closed. And it must be said that the string is made in such a way that we can put between it and the box in various places several Magades, which are vulgarly called bridges: like those which bear the strings of violins: bridges whose position can be changed beneath said string in order to distinguish and divide the length of the string in as many parts as necessary – parts which we measure with a known measure as the Royal Foot or a palm, which is the fourth part of the length of a foot. The aim is to accommodate the parts of the said string with the proportions and numbers heretofore described.

[A. Proportion double = Diapason or Octave]
Thus to reasonably accommodate the doubles proportion, two parts of the string must be measured so that one of them is exactly twice as long as the other: that is, the larger of said parts is twice the proportion of the smallest, putting three bridges below the said string - that is to say two bridges at the ends of said parts and one in the division of these parts, e.g., if the longer of the two was six palms in length, the other part would be precisely three palms long. And thus there would be double proportion of the largest of said parts to the smaller. And this smallest part to the biggest, would be in half proportion, and then, plucking one and then the other part of the string thus divided so it rings: one may know that their sounds are distant from low to high, one from the other by the interval of a Diapason, a perfect consonance which is called by modern musicians an Octave.

And here in law and in what follows below, note an axiom of universal music: that the longest part of a uniform string’s tone makes a deeper sound that is lower than the shortest portion of said string. Similarly, the shorter portion of said string makes a higher pitch that is higher than does the longer part of said string.

[B = Proportion sesquialter = Diapente or Fifth]
To apply the proportion sesquialter to the two parts of said string, it is accommodated by measuring thereof and also moving the bridges from one place to another, to make both parts of the string that which is longer in proportion sesquialter to the other part which is the shortest, e.g., if the largest had six palms in length, the shortest would have a length of four palms; And then if you pluck these parts for the tones, will be known two sounds distant one to the other, from low and high, by the range Diapente: that is a perfect consonance called a fifth by modern musicians.

[C. Proportion sesquitertian = Diatessaron or Fourth]
To accommodate the proportion sesquitertian to both parts of said string: it is necessary to divide two parts in this string, measuring and applying bridges to this latter (as has been said) with the largest part in proportion sesquitertian to the shorter, e.g., if the greatest part had eight palms in length then the smallest has the length of six palms. And then if you pluck these parts to make them ring, you may know that their sounds are distant from one another, from the low and high, by the range of Diatessaron: which is called by modern musicians a Fourth.

[D. Proportion triple = Diapason Diapente or Twelfth]
To apply a triple proportion to the two parts of said string, measure and move the bridges to make two parts in which the largest string is in triple proportion to the shortest, e.g., if the largest of these parts contains twelve palms, the smallest contains four: so if we cause the parts of the string to ring, their sounds are spaced from each other, from the low to the high, by the interval Diapason Diapente, which is composed of a Diapasson and a Diapente, and named by modern musicians a Twelfth.

[E. Proportion quadruple = Two Diapasson or Fifteenth]
To accommodate a fourfold proportion to both parts of this string, measure and move the bridges to divide the said string into two parts, of which the largest has quadruple proportion to the smallest, e.g., if the largest of the parts contained twelve palms in length, the smallest contains three palms: and then if one touches one and the other of these two string portions to make them sound, know that their sound will be distant from the other by the interval of two times Diapasson, which is a perfect consonance and called by modern musicians a Fifteenth.

[F. Proportion eightfold = Three Diapasson or Twenty-secondth]
When we want to apply a proportion eightfold in two parts of the string, one must, by measuring and changing bridges, make two parts in the latter, the greater in proportion eightfold to the shorter, e.g., if the greatest contains sixteen palms the smaller contains two. And then, if one touches one and the other of these two parts, their sounds are distant from the other three times the Diapason, a perfect consonance modern musicians call a Twenty-secondth.

[G. Proportion sixteen-fold = Four Diapasons or twentieth-ninth]
When we want to accommodate a sixteen-fold proportion to parts of said string, it is by measuring and moving the bridges to make this string two parts, the greater in sixteen-fold proportion to the smaller part, e.g., if the largest of these parts contains sixteen palms the the smallest contains one. And then, if these portions of said string are touched, they make two sounds that are distant from the other, from low to high, by a gap which has four times the Diapasson, which is a perfect consonance called by modern musicians, a twenty-ninth.

[Conclusion]
We can remark, regarding what we have heretofore stated touching sounds and the voices of music that it takes only two which agree together according to the above described proportions, to make a consonance. But greater, when one finds three or more that fit all together: such an agreement is not called just consonance, but because of its Excellency, we call it Harmony, because it is more perfect than a simple sound.

A poetic interlude, or Poets number games

Sonnet VIII by W. Percy 1594

SONNET VIII.

Strike up, my Lute! and ease my heavy cares,
The only solace to my Passions :
Impart unto the airs, thy pleasing airs!
More sweet than heavenly consolations.

Rehearse the songs of forlorn amor'us
Driven to despair by dames tyrannical!
Of Alpheus' loss, of woes of Troilus,
Of Rowland's rage, of Iphis' funeral!

Ay me! what warbles yields mine instrument!
The Basses shriek as though they were amiss!
The Means, no means, too sad the merriment!
No, no! the music good, but thus it is

I loath both Means, merriment, Diapasons;
So She and I may be but Unisons.

Shakespeare too, in the numeration of his sonnets, used pythagorean number/music references, his sonnet 8 (diapason/octave) is also about music:

8
Musick to heare, why hear'st thou musick sadly,
Sweets with sweets warre not, ioy delights in ioy:
Why lov'st thou that which thou receavst not gladly,
Or else receav'st with pleasure thine annoy?
If the true concord of well tuned sounds,
By unions married do offend thine eare,
They do but sweetly chide thee, who confounds
In singleness the parts that thou should'st beare:
Mark how one string sweet husband to an other,
Strikes each in each by mutual ordering;
Resembling sier, and child, and happy mother,
Who all in one, one pleasing note do sing:
Whose speechless song being many, seeming one,
Sings this to thee thou single wilt prove none.

The last couplet too, may reference that in pythagorean theory one is not a number. His sonnet one was originally published without number for, and as he mentions in sonnet 136:

If thy soul check thee that I come so near,
Swear to thy blind soul that I was thy Will,
And will, thy soul knows, is admitted there;
Thus far for love, my love-suit, sweet, fulfil.
Will, will fulfil the treasure of thy love,
Ay, fill it full with wills, and my will one.
In things of great receipt with ease we prove
Among a number one is reckoned none:
Then in the number let me pass untold,
Though in thy store's account I one must be;
For nothing hold me, so it please thee hold
That nothing me, a something sweet to thee:
Make but my name thy love, and love that still,
And then thou lovest me for my name is 'Will.'

Number symbolism in Shakespeare's Sonnets 8 and 128 by Fred Blick.