Origins
Pythagoras taught in southern Italy during the sixth century BCE, and his followers formed communities with a shared way of life. Later philosophers developed this mathematical and religious inheritance. 1
← The traditionsThe Pythagorean
Ancient Greece · Number & harmony
The Pythagorean
Way
Number, harmony, and a life brought into measure
For the Pythagoreans, the study of number and music belongs to the purification of the soul. The pupil learns to hear the proportions that give music its harmony, while a disciplined life prepares the mind to understand the cosmos. This education concerns an immortal soul whose present body is one dwelling in a succession of lives.
Beliefs · Diagrams · Practice · Primary texts
The essentials
At a glance
These are the tradition’s central teachings.
Number
Number makes the world intelligible. The proportions between musical notes offer a direct example: an octave sounds harmonious because its pitches stand in a ratio of two to one. 2
The soul
The soul survives death and enters other living bodies. Porphyry links this teaching to the kinship of living beings, extending moral concern beyond one human life. 1
Harmony
Harmony joins different things into a whole. Philolaus describes the cosmos through limiters, which give definite shape, and unlimiteds, which receive it; harmony fits them together. 3
The work
Learning changes the way a person lives. The Golden Verses require measured habits, justice toward others, and an examination of the day before sleep. 4
The aim
Purify the soul and learn its place among divine beings. The Golden Verses end with the promise of immortality after the mortal body has been left behind. 4
A small figure with a large claim
Ten points. Four rows.
The tetractys arranges the first four numbers into a triangle. Open the explanations to see how its numbers also describe musical consonances.
drawn with equal distances between neighboring points.
The monochord · sounding length
Unison1 length · 1:1 pitch
Fourth¾ length · 4:3 pitch
Fifth⅔ length · 3:2 pitch
Octave½ length · 2:1 pitch
The string and its tension stay the same. Shorter sounding length gives higher pitch. The lengths above are drawn to scale.
The historical image
Fludd’s world monochord
Robert Fludd, 1617. A later image of musical cosmology. The complete plate retains the hand, scale, divisions, and labels.
The unitOneA beginning from which counting proceeds.
Beginning with one point, each row adds another until four rows make a triangle of ten points. The figure shows how a complete arrangement can grow from a single unit while retaining a clear place for each of its parts.
The octave · 2:1TwoDoubling establishes the octave.
An octave relates two pitches at a frequency ratio of two to one. On an ideal string under the same tension, halving the sounding length raises the pitch by an octave.
Fifth · 3:2 / Fourth · 4:3Three & fourThe next numbers hold two more consonances.
The ratios three to two and four to three give the fifth and fourth. Together they span an octave: 3/2 × 4/3 = 2. The first four numbers contain these central musical relations.
The completed figureTenThe four rows gather into one whole.
The first four numbers add up to ten, forming the complete triangle. In the Golden Verses, the pupil swears by the one who gave the tetractys to the soul. Its place in the oath shows the religious importance of a figure that brings number, musical harmony, and the origin of nature together.
The points show the traditional tetractys. Pitch ratios and string lengths explain its musical relations in modern terms. Philolaus’s surviving fragment on harmony names the octave, fifth, and fourth. 436
The work of transformation
The discipline of the Golden Verses
The Golden Verses begin with duties to the gods and other people, then turn to food, sleep, anger, and the examination of one’s actions. The poem prepares the pupil for knowledge of divine things through a sustained reform of daily life.
Examine the day before sleep.
Honor gods and family
The poem opens with reverence for the gods, respect for an oath, and duties toward parents and friends.
Practice moderation
The pupil learns to govern appetite and anger, with care for food, sleep, and the health of the body.
Review what happened
Before sleep, recall actions, errors, and neglected duties. Acknowledge good conduct as well as failures.
Learn from the review
The pupil is told to regret wrongful actions and take encouragement from good ones. Repeated examination makes the day’s experience part of the education of the soul.
The Golden Verses transmit the later school’s ethical teaching under Pythagoras’s name. Their surviving form belongs to the tradition that followed him.
The teaching in the texts
Look more closely
Number heard by the earWhat a single string can teachA monochord makes the proportions of music audible. The Pythagoreans understood harmony as a clue to the structure of the cosmos and used music in the education of the soul. The listener who studies a consonance is also someone whose own life can be brought into harmony.
A monochord consists of a single stretched string with a movable bridge. Sound the whole string, then divide its length in half: the shortened string sounds an octave higher. Two-thirds of the original length gives a fifth, and three-quarters gives a fourth. The ear hears consonances that the eye can follow as measured lengths.
This correspondence between number and sound helps explain the Pythagorean interest in music. The notes differ, yet their differences allow them to sound together in harmony. Philolaus gives harmony a similar role in his explanation of the cosmos. Things of different natures can form one world because they are fitted together in a definite way.
Iamblichus describes music as part of the school’s daily life. Pythagoras chose melodies to calm disturbances of the soul. Evening music prepared pupils for sleep, while morning music prepared them to resume activity. In this biography, he understands how sound affects a person and selects music for the change needed. Learning harmony involves hearing its proportions and experiencing its effects.
Fludd’s world monochord, published in 1617, extends the musical image across the heavens and the elements. A divine hand holds the tuning peg above the whole arrangement. His engraving belongs to a later Hermetic and Christian cosmology, where the proportions of the world are given by God. The instrument familiar to the student has become a picture of creation, with the sounding string passing through its several regions.
Transmigration and kinshipThe soul through its successive livesPorphyry connects the immortality of the soul with its passage into other living beings and the kinship of all life. These teachings change the significance of a creature’s present body. The Pythagorean discipline prepares the soul for a destiny that extends beyond the life now being lived.
Porphyry attributes four teachings to Pythagoras: the soul is immortal, it enters other living beings, events recur, and living things are kin. These teachings describe a life whose history extends beyond a creature’s present appearance. A human body is one form through which that longer life passes.
The teaching about kinship follows from this understanding of the soul. Human beings and animals belong to a living family whose members can appear in different bodies. The way a person treats another creature therefore has religious significance. Restraint toward living beings follows from recognizing something shared beneath the differences in their visible forms.
The Golden Verses describe the preparation of a soul with such a future before it. The pupil is instructed to examine conduct, govern appetite, and seek an understanding of divine and human things. Near the end, the poem speaks of humanity’s divine ancestry and of sacred Nature revealing what is hidden. The daily discipline prepares the pupil to receive that knowledge.
The poem concludes with the soul leaving its mortal body for divine and immortal life. Its earlier instructions prepare the pupil for this destination. The person reviewing the day’s deeds is also preparing for a change in the soul’s condition after death. Porphyry’s teaching makes this care for conduct intelligible within a school devoted to number and contemplation. The pupil’s habits and treatment of other beings matter to a life that continues beyond the present body.
The formation of attentionHow the pupil is preparedIamblichus describes an education that begins with the character of the person seeking admission. Silence and music prepare the pupil to receive instruction; the Golden Verses add an examination of each day. Sacred knowledge is entrusted to a soul undergoing a deliberate course of purification.
In Iamblichus’s Life of Pythagoras, admission to the school follows a period of testing. Pythagoras studies the character of those who come to him, and pupils undergo a discipline of silence. Their capacity to receive instruction has to be cultivated before they are entrusted with the fuller teaching.
Silence gives listening a sustained place in the pupil’s life. The student hears the teaching without immediately using it to display an opinion. Iamblichus also describes music that settles passions and prepares the soul for sleep or waking. These practices prepare the person who will study divine things. A troubled or undisciplined soul brings its disturbances into what it hears.
The Golden Verses extend the preparation into a nightly examination. Before sleep, the pupil is to recall what happened during the day, consider errors, and remember duties left undone. Good actions are to be acknowledged along with failures. The review makes the student responsible for seeing the day clearly and learning from what has been done.
The poem places these instructions on the way to knowledge of the gods, the human soul, and the nature of the world. Prayer accompanies the undertaking, with the gods asked to bring it to completion. The pupil’s effort prepares for an understanding that sacred Nature can reveal. The promised conclusion is the soul’s immortal life, for which the discipline of the school has been preparing it.
Across the centuries
The historical outline
- 6th century BCE
Pythagoras teaches in Italy
The Samian teacher establishes the community remembered at Croton. 1
- 5th century BCE
Philolaus writes on nature
Limiters, unlimiteds, and harmony become the terms of a developed cosmology. 3
- 4th century BCE
Aristotle describes the school
His account preserves the Pythagorean connection between number, music, and the heavens. 2
- 3rd–4th centuries CE
The philosophical biographies
Porphyry and Iamblichus gather traditions about the teacher and his way of life. 5
- 1617
The world becomes a monochord
Fludd gives the musical cosmos a memorable printed image. 7
Continue into the illustrated journey
One string across a world.
Continue into Robert Fludd’s later vision of cosmic harmony. Four scenes explain the divine hand, the sounding string, and the proportions of his universe.
Explore the Monochord ↗Read at the source
Texts & objects
- Life of Pythagoras, especially §§18–19 ↗
The Italian community, transmigration, recurrence, and kinship of living beings.
- Metaphysics I.5, 985b–986a ↗
The Pythagoreans on number, musical ratios, and the cosmos.
- Philolaus: fragments 1, 4, and 6 ↗
Ancient fragments within a modern study of limit, number, and harmony.
- The Golden Verses ↗
Conduct, evening examination, the oath, and the soul’s destination.
- Life of Pythagoras, chapters XV–XVIII ↗
Music, preparation, and the testing of pupils in the late antique biography.
- Standing Waves and Resonance, §16.6 ↗
The physical relation between an ideal string’s length and frequency.
- The world monochord, 1617 ↗
A later reception of musical cosmology, reproduced in the Monochord guide.