Here are some key points about the solfeggio methodology and use of hexachords in 18th century Neapolitan conservatories:

- Solfeggio was a core part of the training in Neapolitan conservatories like the Conservatorio di Santa Maria di Loreto. Students would practice singing various intervals, scales, and vocal exercises using solfeggio syllables (do, re, mi, etc.).

- The solfeggio system was based on hexachords, six-note segments of the musical scale. The central hexachord was ut-re-mi-fa-sol-la. Different hexachords could start on different pitches, using mutations like soft (b) and hard (♯) to transition between hexachords.

- Singing exercises would utilize different hexachord mutations to ascend and descend through the full range of the voice. This allowed training in sight-singing and ear training through various keys and intervals.

- The solfeggio method trained musicianship and voice technique but also reinforced understanding of music theory and the hexachord system which was foundational to 18th century harmony and composition. 

- Important Neapolitan teachers who helped develop and spread the solfeggio method were Francesco Durante, Nicola Porpora, Leonardo Leo, and Giovanni Battista Pergolesi at the Conservatorio di Santa Maria di Loreto.

So in summary, solfeggio using mutable hexachords was integral to the conservatory training system in 18th century Naples, helping build musicianship skills and reinforce musical knowledge. The Neapolitan schools and teachers were very influential in advancing this methodology.

Here is an outline for a series of introductory solfeggio lessons using the 18th century Neapolitan conservatory hexachord methodology:

Course 1:  

- Explain the hexachord system (ut-re-mi-fa-sol-la) using C major scale as example 

- Have student sing major scale ascending and descending using solfeggio syllables

- Introduce the mutations - hard (♯), soft (♭) - to transition between hexachords

Course  2: 

- Practice solfeggio major scale exercises focused on working across mutations between hexachords

- Introduce minor scale degrees using solfeggio syllables 

- Exercise on solfeggio minor scales and mutations between hexachords

Course  3:

- Work on interval exercises using solfeggio syllables, focusing on transitioning between hexachords

- Practice identifying intervals by ear using solfeggio syllables 

Course  4: 

- Introduce Solfeggio exercises with arpeggios and basic melodic patterns within a hexachord

- Add rhythmic notation to solfeggio singing exercises 

Course  5:

- Begin simple 2 part exercises singing each line separately using solfeggio syllables 

- Then combine 2 parts singing together using solfeggio syllables

This foundational series focused on scales, intervals, arpeggios, and basic two-part singing provides an introduction to solfeggio technique using the hexachord system. Additional lessons can further expand into more advanced melodies, counterpoint, and composition exercises building on this base.

LESSON ONE  The Hexachord on C (The natural hexachord)

- Solfege uses six syllables: do, re, mi, fa, sol, la.

- The interval between each syllable is a whole step (whole tone) except between mi-fa and fa-mi which is a half step (semitone). 

- On the musical staff with no sharps or flats added, the half steps occur between E-F  (mi fa) and B-C (mi fa) ascending and descending FE (fa mi) and CB (fa mi). (i.e. in C major scale)

- So the solfege syllables mi-fa or fa-mi must be used on those half step intervals to sing them accurately.

-Sing a scale using the solfege syllables, starting without any time signature or rhythm.

- Sing a scale up and down from C to C with the syllables: 

do re mi fa sol la where Do is C, and the scale ascends

la sol fa mi re do where la is A and the scale descends

- This scale allows the student to practice hitting the half step intervals correctly using mi-fa and fa-mi.

- After singing the scale with syllables, the teacher can add in a time signature and rhythm, keeping the body still and just using fingers to indicate the beat. 

- The goal is for the student to differentiate whole steps and half steps and sing intervals accurately using solfege syllables before moving on to singing letter names or lyrics.

So in summary, this chapter introduces solfege as a key technique for interval training and sight singing. The syllables provide markers to identify and sing the half step intervals correctly.